Quantum tunneling transport of electrons in double- barrier heterostructures : theory and modeling Noteborn, H.J.M.F. DOI: 10.6100/IR396859 Published: 01/01/1993 Document Version Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication Citation for published version (APA): Noteborn, H. J. M. F. (1993). Quantum tunneling transport of electrons in double-barrier heterostructures : theory and modeling Eindhoven: Technische Universiteit Eindhoven DOI: 10.6100/IR396859 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 06. May. 2018
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Quantum tunneling transport of electrons in double-barrier heterostructures : theory and modelingNoteborn, H.J.M.F.
DOI:10.6100/IR396859
Published: 01/01/1993
Document VersionPublisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differencesbetween the submitted version and the official published version of record. People interested in the research are advised to contact theauthor for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
Citation for published version (APA):Noteborn, H. J. M. F. (1993). Quantum tunneling transport of electrons in double-barrier heterostructures : theoryand modeling Eindhoven: Technische Universiteit Eindhoven DOI: 10.6100/IR396859
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ?
Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.
... de tunnel. En dit was de zaak van de twine!. Terwijl ... /de houweel, een man tot z'n na.aste, en terwijl er drie ellen wa.ren om te worden doorb(oord werd gehoor)d de stem van een man roe- / pend tot z'n naaste. Want er was een resonantie in de rots aan de zuidkant ... En op de dag van de / tunnel sloegen de houwers, een man z'n naaste tegemoet, houweel tegen (hou)weel. Toen gin$en / de wateren vanuit het vertrekpunt naar het reservoir in tweehonderd en du1zend el. En hon- / derd el was de hoogte van de rots boven het hoofd van de houwer(s).
lnscriptie uit de Shiloah-tunnel van koning Hizkia (ca. 715-687 v.Chr.) te Jeruzalem, thans in het Museum van het oude oosten te. Istanboel; transliteratie en vertaling van de oudhebreeuwse tekst. Uit: K.A.D. Smelik, Behouden Schrijt, Ten Have, Baarn, 1984.
aan Corine
PREFACE
This thesis consists partly of new material and partly of published papers. This
set-up slightly sacrifies the systematics in favour of the diachrony of the
research. Thus the notation may differ slightly from one chapter to another.
Another consequence concerns the method of reference to the literature. In the
papers, references are indicated by numbers between square brackets [ ], and
listed at the end of the text. In the remaining sections, references and footnotes
are designated by superscripts, and given at the bottom of the page, thus
allowing a quick look-over. A general and complete list of references is added at
the end of the thesis. Gratefully I acknowledge friendship and support from the members of the
Theoretical Physics Group at the Physics Department of the Eindhoven
University of Technology. Three people I would like to thank in particular: my
two graduate students Guido van Tartwijk and Rene Keijsers for their valuable
contributions to the project; and Benny Joosten, who bas been a true colleague
and a good friend near by and far off.
Eindhoven, March, 1999 Harry Noteborn
v
vi
CONTENTS
chapter 1 General Introduction 1
1.1. Evolution of quantum devices 1 1.2. Double-barrier resonant tunneling 3 1.3. Modeling of DBRT structures: short survey 6 1.4. Modeling: present approach 11
1.5. Outline of the thesis 18
chapter 2 From Bloch to BenDaniel-Dnke 21
2.1. Introduction 21 2.2. Bloch waves in bulk materials 22 2.3. Envelope functions 24 2.4. Lowdin renormalization and effective mass 26 2.5. Kane model 27
2.6. Heterojunctions 33
chapter 3 Coherent Tunneling 41
3.1. Introduction 41 3.2. Transfer matrix approach 42 3.3. The resonance energy; dependence on barrier parameters 48 3.4. Current density expression 54 3.5. Chemical potential 59 3.6. Inelastic scattering in the Jonson-Grincwajg model 62
vii
viii
chapter 4 The Self consistent Electron Potential
4.1. Introduction 4.2. Accumulation and depletion region
4.3. Selfconsistent study of double-barrier resonanttunneling (Phys. Scripta T33, 1990)
chapter 5 Current Stability and Impedance of a DBRT-diode
5.1. Introduction 5.2. Stability of the selfconsistently determined current
in a double barrier resonant-tunneling diode (J. Appl. Phys. 10, 1991)
5.3. Alternative for the quantum-inductance model in resonant tunneling (Superlattices and Microstractures, 1993)
chapter 6 Effects of parallel and transverse magnetic fields
6.3. Magneto-tunneling in double-barrier structures: the B.LJ configuration (J. Phys.: Con.dens. Matter4, 1992)
Evalna.tion and outlook
References
Summary
Samenvatting
List of publications
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171
chapter 1
GENERAL INTRODUCTION
1.1. The evolution of quantum devices
The last two decades have witnessed the revolutionary development of a new
class of electronics devices, the operation of which is directly controlled by
quantum phenomena such as tunneling. It has been the strong interplay between
technology and physics, theory and experiment, that has enabled the rapid
growth of this new field of quantum rnicrostructures1. One of these
semiconductor heterostructure devices that has attracted a lot of interest, is the
Double-Barrier Resonant-Tunneling (DBRT) diode, and the understanding of
its physics is the subject of this thesis.
The birth of what is now called "band gap engineering" is usually considered
to be the publication of the Tsu and Esaki papers on semiconductor superlattices
1The history of this development has been discussed by several authors, among which are pioneering workers. See e.g. L. Esaki, IEEE J. Quant. Electron. QE-22 (1986) 1611; F. Capasso, in: Physics of quantum electron devices, ed. F. Capasso, Berlin: Springer, 1990; C. Weisbuch, in: Semiconductors and semimetals 24, ed. R. Dingle, San Diego: Academic Press, 1987; ch. 1.
1
2 ChQ,pter 1
and negative differential conductivity2• The quantum-size effects envisioned in these papers were soon experimentally demonstrated in resonant tunneling,
superlattice transport and optical absorption measurements3•
A real breakthrough of nanostructure devices had to await the progress in
layer growth techniques. Both MBE (molecular beam epitaxy) and MOCVD (metal-organic chemical vapor deposition) matured in the seventies, emerging as
precisely controlled and well monitored growth processes with accuracy up to
one atomic layer. They have allowed the design and fabrication of various
structures, lattice-matched or with strain, type I or type II, from single interface to multiple quantum well and supperlattice, for parallel or vertical transport.
The artificial tayloring or engineering of quantum structures has led Esaki to
speak of "do it yourself quantum mechanics11 4.
An important class of quantum devices that emerged in the early eighties,
many of them denoted by acronyms as e.g. HEMT, MODFET, TEGFET and
SEED, exploits the formation of a 2DEG near a heterointerface. Besides this
technological application, heterostructures have been of important relevance to
the fundamental research on phenomena like the (integer and fractional) quantum Hall effect.
In many of the nanostructure--based devices that were realized in the
eighties, as e.g. the RHET (resonant-tunneling hot electron transistor), the
THETA (tunneling hot electron amplifier) and the RT-diode and RTBT
(resonant tunneling bipolar transistor), the phenomenon of resonant tunneling
plays a central role. Renewed interest in this phenomenon was triggered by the
terahertz experiment of Sellner et al in 19835 and by the discussions about intrinsic bistability in DBRT diodes6• Experiments were performed in magnetic
fields, new materials (GainAs/ AllnAs, Si/GeSi) were studied, different doping
2L. Esaki and R. TsuJ IBM J. Res. Develop. 14 (1970) 61; R. Tsu. and L. Esaki, Appl. Phys. Lett. 19 ll971) 246; -, Appl. Phys. Lett. 22 (1973) 562. 3L.L. Chang, L. Esaki and R. Tsu, Ap_l?l. Phys. Lett. 24 (1974) 593; L. Esaki and L.L. Chang, Phys. Rev. Lett. 33 (1974) 495; R. Dingle, W. Wiegmann and C.H. Henry, Phys. Rev. Lett. 33 (1974) 827. 4L. Esaki, in: Proc. Srd Int. Symp. Foundations of Quantum Mechanics, Tokyo: Phys. Soc. Jap., 1990; p.369. 5T.C.L.G. Sellner, W.D. Goodhue, P.E. Tannenwald, C.D. Parker and D.O. Peck, Appl. Phys. Lett 43 (1983) 588.
sv.J. Goldman, D.C. Tsui and J.E. Cunningham, Phys. Rev. Lett. 58 (1987) 1256.
General Introduction 3
types investiga.ted7.
Finally the development of quantum interference semiconductor devices is
mentioned. Advances in nanolithogra.phy have ma.de it possible to build
semiconductor structures of low dimensionality, the physics of which reveal analogies between optics and micro-electronicss. In all fields of the quantum
device physics there can be observed an ongoing development in technology,
experiment and theory.
1.2. Double-Ba.uier Resonant Tunneling
Quantummechanical tunneling, or barrier penetration, is a. subject encountered
in a.11 textbooks on quantum mechanics. Most often this paradigm is treated in
the chapter on one-dimensional problems, a.t the beginning of the book9• It is a
standard example for showing how classical common sense is overtaken by
quantummechanical reasoning. Nuclear alpha decay is often used to give the
mathematical exercise some physical meaning.
Also double-barrier tunneling is encountered in textbooks a.s early as 1951,
when Bohm's Quantum Theory10 appeared. There the interest is in the resonant,
virtual and metastable states, treated within the WKB- approximation. A
connection between resonant tunneling and solid state physics was ma.de in the
book by Duke in 196911, reviewing the tunneling in solids.
In the same year, Esa.ki and Tsu put forward their proposal of an engineered
7E.g. GalnAs/AllnAs: S.Ben Amor, K.P. Martin, J.J.L. Rasco!, R.J. Higgins, R.C. Potter, A.A. Lakhani and H.Hier, Appl. Phys. Lett. 54 (1989) 1908; S. Ben Amor, J.J.L. Rascal, K.P. Martin, R.J. Higgins, R.C. Potier and :fl. Hier, Phys. Rev. B 41 (1990) 7860; L.A. Cury, A. Celeste, B. Goutiers, E. Ranz, J.C. Portal, D.L. Sivco, A.Y. Cho, Superlattices and Microstructures 1 (1990) 415. For Si/GexSi1.u see: H.C. Liu, D. Landheer, M. Buchanan and D.C. Houghton, Appl. Phys. Lett. 52 (1988) 1809; S.S. Rhee, J.S. Park, R.P.G. Karunasiri, Q. Ye and K.L. Wang, Appl. Phys. Lett. 53 (1988) 204; For p-type DBRT structure, see: R.K. Hayden, D.K. Maude, L. Eaves, E.C. Vala.dares, M. Henini, F.W. Sheard, O.H. Hughes, J.C. Portal and L. Cury, Phys. Rev. Lett. 66 (1991) 1749. 8W. van Haeringen and D. Lenstra eds., Analogies in optics and microelectronics, Kluwer, 1990; -, Proc. Int. Symp. Analogies in optics and micro-electronics, North-Holland, 1991. 9See e.g. S. Gasiorowicz, Quantum physics, New York: Wiley, 1981; ch. 5. 10D. Bohm, Quantum Theory, New Jersey: Prentice Hall, 1951; ch. 9. uc.B. Duke, Tunneling in solids, Solid State Phys. Suppl. ,10, New York: Academic, 1969; ch. x.
4 Chapter 1
semiconductor superlatticel2. This initiated a research on semiconductor
quantum structure design, which resulted in the first experimental observation
of resonant tunneling in a GaAs-AlGaAs double-barrier heterostructure by
Chang et al. in 197413. After a calm ten years, the work of Sollner and coworkers
in 198314 initiated an outburst of publications on design, experiment and
modeling of resonant tunneling heterostructures.
The structure studied by Chang et al. was of the compositional type, making
use of the fact that different semiconductor materials have different band gaps.
At each interface between a layer of material A (e.g. GaAs) and one of material
B (e.g. AlxGa1.xAs), there is a conduction band discontinuity that serves as a
potential step to the conduction electrons15• Thus a simple two-terminal
double-barrier structure (DBS) consists of the following layers (see Fig.1): a
central layer of material A, called the well; two sandwiching layers of material
B, called the barriers; in turn sandwiched between heavily doped contact layers
of material A, termed emitter and collector18. The thicknesses of the central
layers are typically several nanometers.
The function of the doped layers is to provide a Fermi sea of electrons; a
donor density of 1011 .. 1ou/cm3 corresponds .to a Fermi energy of about 10-50
meV. The function of the well is to define a narrow quasi-bound or resonant
state; a 5 nm wide GaAs well supports a resonance energy of 84 meV, while a
second resonance is found at 310 meV. Applying a bias voltage between collector
12In an IBM Research Note RC-2418 (1969) by L. Esaki and R. Tsu, Superlattice and negative conductivity in semiconductors, refered to by L. Esaki, in: Proc. 3rd Int. Symp. Foundations of Quantum Mechanics, Tokyo: Phys. Soc. Jap., 1990; p.369. 13L.L. Chang, L. Esaki and R. Tsu, Appl. Phys. Lett. 24 (1974) 593. 14T.C.L.G. Sollner, W.D. Goodhue, P.E. Tannenwald, C.D. Parker and D.O. Peck, Appl. Phys. Lett 43 (1983) 588. . 15In the same way, the valence band discontinuity is a barrier to the holes. Whether tunneling is by electrons or by holes, depends on the type of doping in the contact layers. Both types of tunneling have now been observed. For holes see e.g. E.E. Mendez, W.I. Wang, B. Ricco and L. Esaki, Appl. Phys. Lett. 47 (1985) 415. We will concentrate on electron tunneling, which implies n-type doping. 18In some structures the contact layers are separated from the barriers by undoped spacer layers; see H.M. Yoo, S.M. Goodnick and J.R. Arthur, Appl. Phys. Lett. 56 (1990) 84. Sometimes medium doped buffer layers are added; see: L. Eaves, E.S. Alves, M. Henini, O.H. Hughes, M.L. Lea~beater, C.A. Paylii:g, F.W. Sheard, G.A. Toombs, A. Celeste, J.C. Portal, G. Hill and M.A. Pate, m: High magnetic fields in semiconductor physics II, ed. G. Landwehr, Berlin: Springer, 1989, p.324.
General Introd'UCtion 5
Fig.1.1
Ohmic contact
u-Al(Ga)As barrier t::===j-- u-GaAs well u-Al(Ga)As barrier
n+ GaAs substrate
Ohmic contact
Cross-sectional view of DBRT diode/not to scale). Typical mesa diameter "' 5 µm. Typical widths: o contact layers "' 1 µm, of undoped (= u-)layers"' 5 nm.
and emitter tilts the potential profile, see Fig. 2, and lowers the resonance energy with respect to the emitter band edge. Thus above a certain voltage, say
V10 , the resonance channel is accessible to electrons from the Fermi sea, that can
now carry a substantial particle current from emitter to collector {and an
electrical current in opposite direction). At higher voltages the resonance level is
pulled below the emitter band edge, and the channel is blocked. Thus above V up
all. current is due to off-resonance tunneling corresponding to much lower
transmission probability. At Yup there is therefore a steep descent in the current
and a negative peak in the differential conductance of the DBRT diode. This
negative differential conductance {NDC) makes the DBRT diode into a very
interesting electronic component, promissing possible application in amplifiers,
transistors, mixers, detectors and oscillators.
Also for the theorist, the DBRT structure represents an interesting challenge, its physics involving coherent wave propagation, space charge in the well and
contact layers, transport in an nonequilibrium open system, hot-electron effects,
and scattering. With regard to all this, the DBRT diode is very properly called17
17D.K. Ferry, in: Physics of quantum electron de'llices, ed. F. Capasso, Berlin: Springer, 1990; p. 77.
6
Fig.1.2
Chapter 1
v Operation principle of DBRT diode. Left: conduction band driagrams at three different bias volta!f_es. Dashed areas indicate Fermi seas, line in the weU indicates first resonance. In upward direction: the resonance is brought in and out of tune with the emitter Fermi sea. Right: the corresponding point ( o) in the I - V characteristic.
the 11fruit fly11 for quantum studies of device dynamics.
1.3. Modeling of DBRT stmctures: short survey
Modeling of DBSs and other quantum structures has concentrated on the
computation of the current-voltage (I-V) characteristics of these devices. A voltage difference V between collector and emitter induces an electric current I
due to electrons tunneling through the ha.triers from emitter to collector. Two
main approaches to calculating the tunneling current have been developed. The
first one (called the coherent-tunneling (CT) picturelB) i& based on the
calculation of the transmission coefficient for. the full structure, regarding the
18Already in: R. Tsu and L. Esaki, Appl. Phys. Lett. 22 (1973) 562. See also: B. Ricco and M.Ya.. Azbel, Phys. Rev. B 29 (1984) 1970; H. Obnishi, T. lnata, S. Muto, N. Yokoyama and A. Shibatomi, Appl. Phys. Lett. 49 (1986) 1248; and E.E. Mendez, in: Physics and applications of quantum wells and superlattices, eds. E.E. Mendez and K. von Klitzing, New York: Plenum, 1987, p. 159. This thesis adheres to this CT picture.
General Introduction 7
tunneling from emitter to collector as one coherent wave propagation. This
transmission coefficient as a function of the energy of the incoming electron
shows a sharp peak at energy Eres of width r < Eres· For a symmetric structure having barriers of equal width the peak. height is unity. The CT description is a
truly wave-mechanical approach to tunneling: the resonance is due to multiple
reflections of electron waves in the well, wherefore the DBRT structure is
sometimes called the electronic analogue of the Fabry-Perot interferometer in opticsts. Though extendible to cover time-dependent tunneling2° or elastic
interface roughness21, the method has its limitations when inelastic scattering or
many-body effects come into play. Its strong points are the computational
feasibility and the relative ease with which new concepts can be incorporated. A more detailed account of the CT approach, which is the basis of the present
study, will be presented in the next section.
The second approach (the sequential-tunneling (ST) picture22) considers the
tunneling as two separate and subsequent processes, from emitter to well and from well to collector. The approach goes back to an argument of Luryi23,
explaining the NDR as solely due to tunneling of electrons from three
dimensional states in the emitter to two-dimensional states in the well. No
coherence of the wave function is required in this reasoning. This qualitative argument can be made quantitative by using the tunneling-Hamiltonian
method, a well known approach in the field of superconductivity24. Let us give a
brief sketch of this approach following Payne25• The first step is to replace the
19See e.g. L. Eaves, in: Analogies in optics and micro-electronics, eds. W. van Haeringen and D. Lenstra, Kluwer, 1990, p.227. 20H.C. Liu, Appl. Phys. Lett. 52 (1988) 453. 21H.C. Liu and D.D. Coon, J. Appl. Phys. 64 (1988) 6785.
22F.W. Sheard and G.A. Toombs, Appl. Phys. Lett. 52 (1988) 1228; L. Eaves, F.W. Sheard and G.A. Toombs, in: Band structure engineering in semiconductor microstructures, eds. R.A. Abram and M. Jaros, NATO-ASI, 1989; S.M. Booker, F.W. Sheard and G.A. Toombs, Superlattices and Microstructures 9 {1991) 111; V.J. Goldman, D.C. Tsui and J.E .. Cunningham, Phys. Rev. B 35 (1987) 9387. 21s. Luryi, Appl. Phys. Lett. 47 (1985) 490;
24L. Solymar, Superconductive tunnelling and applications, London: Chapman and Hall, 1972; ch. 2. 25M.C. Payne, J. Phys. C: Solid State Phys. 19 (1986) 1145. See also: T. Weil and B. Vinter, Appl. Phys. Lett. 50 (1987) 1281; G.A. Toombs and F.W. Sheard in: Electronic properties of multilayers and low-dimensional semiconductor structures (Proc. NATO-AS!), New York: Plenum, 1990.
8 Chapter 1
one Hamiltonian for the full double-barrier structure by three Hamiltonians for
the emitter, the well and the collector separately. In the Hamiltonian for the
well, the barriers are made infinitely wide, so that the well becomes a true well
with a bound (in stead of a resonant) state. The current from emitter to well
(and mutatis mutandis from well to collector) is now considered to result from
electron transitions from emitter states to the bound state in the well, calculated
from the Fermi Golden Rule:
Here, Pe(Eb) and fe(Eb) are the density of states and the Fermi function for the
emitter evaluated at the bound state energy Eb. In the same way, Pw = 1 and fw
are the density of states and the occupancy for the well. The matrix element
Me-w for the transition from 'l/Je(z) to 'l/Jw(z) can be calcutated according to
Bardeen's prescription26:
to be evaluated at any z inside the emitter barrier. The wave functions 'l/Je(z) and
'l/Jw(z) are the eigenfunctions at energy Eb of the Hamiltonians for the
disconnected subsystems. Since both functions decay exponentially inside the
barrier, Me-w will depend exponentially on the barrier width b. A complete
calculation yields for the transition probability per unit time, We-w=
where w is the well width, and k1i i11:2, k3 and i11:4 are the (real and imaginary)
local wavenumber in the emitter, emitter barrier, well and collector barrier,
respectively27. In the last factor of this expression a well-known approximation
for the transmission probability P 1(Eb) of a thick barrier can be recognized28 , so
26J. Bardeen, Phys. Rev. Lett. 6 (1961) 57. 27This expression is more general than Payne's Eq.(18), since his requirement that k1=k3=k5, 11:2=11:4 is unnecessary and, in the case of a biased structure, incorrect. 28Any t.ext book on quantum mechanics, e.g. S. Gasiorowicz, Quantum physics, New York: Wiley, 1981; p. 85.
General Introduction 9
that we find that:
hka/m p (E) 10e-w !i:I 2(w+l/K2+ 1 /x4)' 1 b
In the same way, we find for the current from the well to the collector:
in self-explanatory notation. The steady;tate occupancy of the bound state is
obtained from the .condition that Je-w = Jw-c=
which yields a steady;ta.te current J0 of:
This one-dimensional result ca.n easily be generalized to 3D, in which case the
Fermi-Dirac functions fe and fc are to be replaced by Fermi-Dirac integrals. The
important point however remains that J0 ,. ~e-w ~ tDw-c , and that all probabilities f.l.:.W '°w-C
must be evaluated at the bound-state energy J:!jb·
In the next section, we will see that the CT approach yields a. current density
expression, that is very similar to the ST result given above. The slight
difference is related to the fa.ct that the bound-state energy Eb differs from the
true resonance energy Eres• and to the fact that the proportionality of We-w to
P 1(Eres) is only approximate. However, for the usual structure parameters these theoretical differences have no numerical consequences. Hence, the CT and ST
pictures yield the same I-V chara.cteristic29 for the experimenta.lly relevant cases.
And the shortcoming of coherent tunneling that it predicts too large pea.k-to
va.lley ratios30 is not remedied by a. mere switchover to the sequential approach. The accordance of the CT and ST models is perhaps not as surprising as it
may seem. Of course, the unity transmission probability of CT cannot be
29Payne (1986), and Weil and Vinter (1987); but in fa.ct already in Solymar (1972), p. 24. 3op, Gueret, C. Rossel, W. Schlup and H.P. Meier, J. Appl. Phys. 66 (1987) 4312.
10 Chapter 1
reproduced in the sequential approach, where the coupling between the well and
the electrodes has to be small in order for the Fermi Golden Rule to be
applicable. However, as will be seen in the next section, not the peak height but the area under the peak is the relevant quantity in the current calculations. And
since the distribution of incoming energies is much broader than the peak width
r, the total transmission is small, and the tunneling-Hamiltonian approximation
valid. Furthermore, the (inelastic) scattering of tunneling electrons, that is assumed in the sequential reasoning, is not taken into account explicitly in the
tunneling-Hamiltonian calculations. In fact, the sequential aspect refers to the
coupling of the structure to the electrodes that serve as reservoirs31, rather than
to some scattering mechanism in the well or the barriers. And from this point of view, the CT and the ST pictures are divided on the where and how of the
coupling between structure and reservoirs, rather than on the (in)coherence of
the electron wavefunciions. The tunneling-Hamiltonian method of ST is a
convenient way of describing the coupling, but can only be applied inside the
barriers, where the wavefunction is small. For situations in which the coherence
length exceeds the structure length, a different description has to be considered.
On the other hand, a realistic description of the physics of a DBRT diode will
include explicitly some actual scattering processes (interface roughness, alloy, impurity, phonon, etc.). Then, both the CT or ST models described above can
serve as a starting point for further study. In such descriptions tunneling will
always be partly coherent and partly sequential i.e. scattering-assisted or
-hampered. An advantage of the sequential-tunneling model may be the fact that it is easily extendible to non-stationary state situations32.
In addition to the CT and ST models, a third approach, using the Wigner
distribution function or the density matrix, has to be mentioned 33. The Wigner
function is a Fourier-transformed density matrix, written in a mixed
representation of both position and momentum. Though not positive definite in
31Conducta.nce between reservoirs conceived as a transmission problem was proposed by Landauer; for a review, see: R. Landauer, in: Analogies in optics and micro-electronics, eds. W. van Haeringen and D. Lenstra, Kluwer, 1990; p. 243. The coupling of the transmittive structure to the reservoirs is not a trivial matter, and deserves more attention in the literature than given to it hitherto. 32F.W. Sheard and G.A. Toombs, Solid-State Electron. 32 (1989) 1443.
aaw.R. Frensley, Phys. Rev. ~ 36 (1987) 1570; -, Ap 1. Phys. Lett. 51 (1987) 448~· N.C. Kluksda.hl, A.M. Knman, D.K. Ferry and C. fer, Phys. Rev. :a 39 1989 7720; R.-J.E. Jansen, B. Farid and M.J. Kell hysica B 175 (1991) 49; . ~izuta and C.J. Goodings, J. Phys.: Condens. Matter 3 (1991) 3739; K.L. Jensen and F.A. Buot, Phys. Rev. Lett. 66 (1991) 1078.
General Introduction 11
all phase space, the Wigner function is the closest parallel to the classical
distribution function, that quantum mechanics ha.a to offer. Consequently, many
results of classical transport theory can be transferred to a Wigner-function
based quantum transport theory. This approach is troubled by several problems,
one of them being the proper choice of a basis set of functions for evaluating the
Wigner function. Another problem is the proper boundary conditions that
describe an ohmic contact in a quantum system. Connected with this matter are
the difficulties of introducing dissipation in quantum transport. Because of these
theoretical problems, in conjunction with the computational complexity of the
method, the Wigner-function approach is still far from being completed.
Finally we mention the studies of resonant tunneling using a Green's function
formalism34, combined with transfer or tight-binding Hamiltonian, Feynman
path integral theory or otherwise. They have been able to include into the
description all kinds of scattering mechanisms, from elastic .interface roughness
to inelastic electron-phonon interaction. In all cases, restriction to one- or two
dimensional systems, or to the use of simplified interaction models is necessary
to keep the numerical computation feasible. The use of the transfer Hamiltonian
method places these studies in the sequential camp.
The future of quantum structure theory and modeling is to be sought in a
complementing of, rather than a competition between, the above methods.
1.4. Modeling: present approach
In this section, the spirit of the present study is outlined: starting from a
. coherent-tunneling description of the DBRT structure, a number of simple and
easily interpretable rules-of-thumb are derived. As an example of our method, a
clear expression for the peak current is presented. In the proces, we will calculate
Ww-ci the rate of decay of charge stored in the well into collector states, which in
the ST picture is given byH:
34L. Brey, G. Platero and C. Tejedor, Phis. Rev. B 38 (1988) 10507; G. Platero, L. Brey and C. Tejedor, Phys. Rev. B 40 l1989) 8548; J. Leo and A.H. MacDonald, Phys. Rev. Lett. 64 (1990) 817; H.A. Fertig and S. Das Sarma, Phys. Rev. B 40 (1989) 7410; H.A. Fertig, S. He and S. Das Sarma, Phys. Rev. B 41 (1990) 3596; L.Y. Chen and C.S. Ting, Phys. Rev. B 43 (1991) 2097; X. Wu and S.E. Ulloa, Phys. Rev. B 44 (1991) 13148. 35F.W. Sheard and G.A. Toombs, Appl. Phys. Lett. 52 (1988) 1228.
12 Chapter 1
(1.1)
where w is the well width, vb: v'{2~/m} is the velocity of the boUnd state in
the well, m is the effective mass of the well material, and P 2(~) is the
transmission probability of the collector barrier.
In CT this decay rate is evaluated as follows 3B. Both the (z-component of the
electrical) current density J and the (areal) charge density in the well ct are
written as a sum over incident states, labeled by a wavevector k and Fermi-Dirac distributed over energy:
J = 2e I 2!'u(1Pk*Vth:--'¢kV1/Jit);f(E1t) It
ct = 2e l J dz (th: *th:) · f(E1t) It well
(1.2)
Assuming that the tunneling is quasi-ID: 'lh:(r) = "°k (r11 )·xk (z) where the ' . d" ul h b . . h' II z z-a.xis is perpen ic ar tot e a.rners, we rewrite t is as:
(1.3)
where /J = 1/k8 T is the inverse temperature and Er is the Fermi level of the
reservoir. The zero-order Fermi-Dirac integral -'Q(x) equals ln(l+exp(x)). The
last factor in both equations is the 2D channel density at finite temperature,
obtained from integrating the Fermi-Dirac distribution over the parallel
wavevector kn. A parabolic conduction band is assumed. Since the main
contribution to the sum over kz comes from states with E(kz) ::: Eres• the only
states that can penetrate substantially into the well and collector, this channel
density can be approximated by its value at the resonance energy, and placed in
front of the summation:
36A more detailed account can be found in H.J.M.F. Noteborn, H.P. Joosten and D. Lenstra, Phys. Scripta T33 (1990) 219, which is reproduced in this thesis (Sect. 4.4).
General Introduction 13
(1.4)
In this scheme the quotient J / <1 does not depend on temperature or Fermi level,
or the lateral motion, but is determined solely by the lD resonant state. The
current density can be evaluated at any position, the easiest being the collector
where the wavefunction reads:
where r is the transmission amplitude, kz" is the local wavenumber (this
component of the wavevector is not conserved), and L some normalization
length. Substitution in (1.4) yields:
J = 211"1i1~;i1p .5b(P(Er-Eresn J dE P(E) (1.5)
where P = (kz" /kz)· I rl 2 is the transmission probability, and the integration of
P is over the (first) resonance. In the same way, the charge density can be evaluted: in the well, the wavefunction is:
where r 2 and p2 are the transmission and reflection amplitude of the second
barrier, and kz' is the local wavenumber in the well. Again, substituting this in
(1.4) we find:
where P 2 = (kz" /k7.') • I r 212 and R2 = I p2 I 2 = ( 1 - P 2) are the transmission and reflection probability of the second barrier, evaluated at the resonance energy. In
writing (l+R2)/P2 we have neglected cross terms. Combination of (1.5) and
(1.6) yields (cf. (1.1)):
14 Chapter 1
(1.7)
provided that P 2 < 1. From the derivation of (1. 7) it can be seen that this decay
rate is independent of the exact shape of the transmission peak P(E), as long as
its width r < Er < Eres· The current and charge densities can be evaluated further, by substituting for P(E) the expression37:
that for sma.11 P11 P2 describes a peak at ct(E) = 0. We obtain:
(1.8)
where a' is dct{Eres)/dE. Again, for sma.11 P 11P 2 this reads:
(1.9)
This CT expression for the current density is just as easily interpreted
"sequentially": the current is proportional to the probability of a.n electron
having energy Eres to reach the collector, which is the product of the probability to cross the first barrier, P 1> times the probability to leave the well through the
second barrier, P2/(P1+P2). The only difference here between CT and ST is in
the calculation of Eresi where CT does, and ST does not, take into account the
leakage of the resonant state out of the well. In the case of not too thin barriers, however, this difference is negligible.
The conductance of the DBRT structure is obtained from differentiating the
current expression (1.5) with respect to the bias voltage Vb. In (1.5) both the
resonance energy Eres and the integral J dE P(E) depend on the bias voltage. In good approximation, Eres is constant with respect to the band edge in the well.
H a linear potential profile is assumed between emitter and collector, Eres is
linear in Vb. For a symmetric structure, i.e. a structure with barriers of equal
37Derived in ch. 3.
General Introduction 15
1.0
0.05
i - 'J .J ir' o.s
o.oo
0.0 0.00 0.10 0.20 0.00 0.10 0.20
v. (V) V,,(V) 0.04 6
+ ...
5
0.03
i 4
i .... s :I 0.02 & :i: -~ .... .,
2
0.01
0.00 1...-~~~~...._~~~..____J
0.00
Fig.1.3
0.10 0.20 0.1 0.2 o.s
v. (V) v.cv>
Calculations for a symmetric GaAs-AZxGa1_xAs DBRT diode {x=O.SS, b-w-b=5-5-5nm}, assuming a linear potential drop across the undoped layers. Contact doping is 5· 1011/cm3, {a) Resonance energy {with respect to the emitter conduction band} vs. bias voltage. {b} Ma:r:imum transmission probablity vs. bias voltage. Dashed line: the square-root appro:r:ima:tion of Eq.(1.12). (c} Half width at half ma:r:imum {HWHM} or resonance width r vs. bias. Dashed line: r-value at zero bias. { d} Current density. J vs. bias at two different temperatures. The + indicates the simplified appro:r:imation {1.1S} to the current maximum.
16 Chapter 1
width, this dependence is simply:
(1.10)
where E1 is the (first) resonance energy of the unbiased structure, depending
only on the structure parameters. From Fig. 1.3a where the numerically
determined Eres(Vb) is plotted, it is seen that (1.10) is indeed a good
approximation. At zero temperature, resonant current is possible when 0 <
Eres(Vb) <Er, or, in terms of Vb:
(1.11)
Hence, the width of the current peak in the I-V characteristic is of the order of
2Erfe. The integral J dE P(E) is written as rrPmax• where r and Pmax are the
width and height of the transmission peak. This expression, exact in the case of
a Lorentzian peak, approximates the integral quite well38• r is roughly
independent of the bias voltage, but Pmax has a square-root behaviour near eVb
= 2E1:
(1.12)
where a is a constant of the order of 1. Both r and Pmax are plotted in Fig. 1.3
as a function of Vb·
Substituting (1.10) and (1.12) in (1.5), we obtain an explicit expression for
J(Vb), which function turns out to have a maximum Jp:
(1.13)
and to yield an average negative conductance Gn of39:
38Derived in ch. 3. 39Cf. D.D. Coon and H.C. Liu, Appl. Phys. Lett. 49 (1986) 94.
General Introduction 17
Fig.1.4
7.0 4.2 K
I .., 8.5 -Q
.9
8.0
-2.00 -1.75 ·1.50 ·1.25
Log(EJ
Logarithm of current density ma=mum (in A/m2) vs. logarithm of Fermi energy (in e V}. Dots correspond to different contact doping densities. The slope of the regression line is 1.555.
(1.14)
In particular, (1.13) predicts that Jp N Er3/2. Calculations for the structure of
Fig. 1.3, shown in Fig. 1.4, yield a.n exponent of 1.555, in reasonable agreement
with (1.13).
One of the approximations in the derivation of (1.13-14) ha.s been shown to
be too drastic: the linear potential profile, which leaves out all space charge
effects in the emitter a.nd collector electrode a.nd in the well, cannot reproduce
the correct voltage scale•0• Thus a. prerequisite for a.n adequate DBRT model is a.
description of these space charge effects on the electron potential. Furthermore,
the effect of the nonresona.nt tunneling, small with respect to Jp but significant
where Gn is concerned, is not ta.ken into account. Nevertheless, (1.13-14)
4DSee e.g. M. Ca.hay, M. McLennan, S. Datta. a.nd M.S. Lundstrom, Appl. Phys. Lett. 50 (1987) 612; cf. N. Yokoyama., S. Muto, H. Ohnishi, K. Imamura., T. Mori and T. fna.ta, in: Physics of quantum electron devices, ed. F. Capasso, Berlin: Springer, 1990, p. 253.
18 Chapter 1
provide us with a first understanding of the determinant factors in calculating
the DBRT I-V characteristics.
1.5. Outline of the thesis
From the discussion in the previous section, it is not surprising that part of this
thesis deals with the modelling of space charge effects in a DBRT structure. In Chapter 4, both the charge accumulation and depletion in the electrodes and the
electrostatic feedback due to charge storage in the well are studied. The latter
charge build-up is responsible for the intrinsic bista.bility in the I-V
characteristic of the DBRT diode. The discussion of this phenomenon is presented in the form of the paper published in Physica Scripta T33 (1990) 219.
In Chapter 3 the coherent-tunneling method is described, with special
emphasis on the determination of the position and width of the transmission
peak. As can be seen from (1.13), these quantities E1 and r, together with the
Fermi energy Er, are important to the current scale in the I-V characteristic.
The high estimates in CT for the peak-to-valley ratio are discussed . in
connection with a simplified method to take into account the effect of scattering
within the structure. Chapters 3 and 4 then contain the presentation of our
model for the DBRT structure, and thus form the kernel of this thesis.
The theoretical background for the simple wavemechanical approach to
tunneling in semiconductors is provided in Chapter 2. A rigorous derivation of
the SchrOdinger-like equation for tunneling electrons is out of the question. Only a didactical presentation of the commonly accepted model for heterostructures is
to be expected.
The second part of this. thesis ( Chs. 5 and 6) describes some applications of the DBRT model developed in Chapters 3 and 4. It is based on four previously
published papers. In Chapter 5 we discuss the stability of the current solutions
obtained from the static model, its relation to the DBRT impedance and the
equivalent circuit that describes the diode. Here the charging of the well offers
an alternative explanation for the low cut-off frequency, that is sometimes
ascribed to a quantum-inducta.nce41• Chapter 6 covers the application of
quantizing magnetic fields, both perpendicular and parallel to the barriers. In
the former configuration, magneto-oscillations in the current provide direct
41E.R. Brown, C.D. Parker and T.C.L.G. Sollner, Appl. Phys. Lett. 54 (1989) 934.
General Introduction 19
evidence for the charge build-up in the structure. The latter configuration,
theoretically more complicated, necessitates a distinction between (what Eaves
et al.42 have called) 1traversing1 and 'skipping' resonant states.
An evaluation of the DBRT model presented and an outlook on possible
developments and applications conclude the thesis.
421. Eaves, E.S. Alves, M. Henini, O.H. Hughes, M.L. Leadbeater, C.A. Payling, F.W. Sheard, G.A. Toombs, A. Celeste, J.C. Porta.I, G. Hill and M.A. Pate, in: High magnetic fields in semicond'UCtor physics II, ed. G. Landwehr, Berlin: Springer, 1989, p.324.
20 Chapter 1
chapter2
FROM BLOCH TO BENDANIEL-DUKE
2.1. Introduction
DBRT structures a.re commonly ma.de of III-V semiconductorst, the most
important of which are the compounds of 13Al, 31Ga and 49In (III), and 15P, 33As
and 51Sb (V). Well-known binary materials a.re Ga.As, InAs and AISb, while also
ternary (AlxGa1.xAs) and even quaternary (Gaxin1.xAs1P1•1) solutions are used. The III-V compounds crystallize in the zinc-blende structure, which consists of
two interpenetrating fee lattices, one occupied by the III-a.toms and one by the
V-atoms, and displaced from each other by a quarter diagonal2. The first
Brillouin zone of the reciprocal (bee) lattice is a truncated octahedron. High
symmetry points a.re indicated by r, L (111), X (001) etc. Although global description of the dispersion relations over the whole Brillouin zone a.re available
(e.g. tight binding), for most semiconductor electronic proporties a local
1For an overview of the material properties of the III/V semiconductors, see Landolt-Bornstein New Series 1Il/17a, Berlin: Springer-Verlag, 1982. 2C. Kittel, Quantum theory of solids, New York: Wiley, 1963.
21
22 Chapter 2
description of the band structure suffices.
In many resonant-tunneling structures, only a small interval around the r point comes into play. In some structures a second valley may be of importance3•
If electron tunneling is considered, we can further restrict ourselves to the
conduction band. For the description of such heterostructures, the effective mass
approximation (EMA) is the most common and widely used approach4 5 s.
In this chapter we sketch a route through band structure theory leading us
from Bloch states in bulk material to the BenDaniel-Duke model in EMA for
heterostructures. The emphasis will be on the approximations needed to end up
at the simple heterostructure model. Many side-issues, however interesting,
remain unmentioned, and difficult derivations are perforce treated without rigor
or detail. A didactical treatment is aimed at, giving insight in the merits and
limitations of the methods and models that are discussed.
2.2. Bloch waves in bulk materials
A crystal is a complex system of nuclei and electrons, exerting electro-magnetic
forces on each other. The electrons can be divided into core and valence
electrons: the latter are important in electrical transport, while the former plus
the nuclei are considered as ions. The many-body Hamiltonian describing the
crystal energy consists of kinetic energy terms for the ions and (valence)
electrons, and potential energy terms, taken into account the interionic,
electron~lectron and electron-:-ion interaction.
Since for most semiconductors the ion mass M is about a factor of 104-lQS
greater than the electron mass m0, it is not too drastic an approximation, if
terms of the order m0/M in the Hamiltonian are neglected. In this so-called
adiabatic approximation 1 the total wavefunction can be written as a product of a
wavefunction for all ions, and an electronic wavefunction, and the SchrOdinger
equation for the crystal splits into a purely ionic and a purely electronic
3E.E. Mendez, E. Calleja and W.I. Wang, Phys. Rev. B 34 (1986) 6026. 4S.R. White and L.J. Sham, Phys. Rev. Lett. 47 (1981) 879. 5G. Bastard, Phys. Rev .. B 24 (1981) 5693; Phys. Rev. B 25 (1982) 7584.
6M. Altarelli, Phys. Rev. B 28 (1983) 842.
7B.K. Ridley, Quantum processes in semiconductors, Oxford: Clarendon Press, 1988.
From Bloch to BenDaniel-Duke 23
equation.
If in the Sclu:Odinger equation for the electrons the fluctuating part of the
electron-electron interaction is disregardedt the electronic wavefunction can be
written as a Slater determinant of one-electron functions. Each electron is then
regarded as an independent particle moving in the potential of the ions. All
organized or collective effects are now out of reach. For low temperatures the
vibrations of the ions around their equilibrium positions are small, and the
potential energy in the one-electron Schrodinger equation is essentially periodic.
Thus the electronic band structure of a crystal is obtained from the
single-particle Sclu:odinger equation:
[ 2£: + V(r)] Vi{r) = E Vi(r) (2.1)
where V(r) is the ionic potential, having the same periodicity as the ion lattice.
As usual, p is the momentum operator (h/i)V of the electron. The solutions of
(2.1) can be written in Bloch formB:
1/ln1h) =ii exp(ik·r) Un1h) (2.2)
where Unt(r) is a periodic function with the same periodicity as V(r). A Bloch state 1/lnt(r) can thus be labelled by a discrete band index n and a wave vector k,
restricted to the first Brillouin zone. If the Unt'S a.re normalized over the unit
cell (volume 0 0) to 0 0, the Bloch states are normalized over the whole crystal
(volume 0) to unity. Inserting (2.2) in (2.1) yields an equation for the functions
Unt:
(2.3)
or:
(2.4)
where H0 is the crystal Hamiltanian of (2.1), the eigenfunctions of which are Uno=
Ho Uno = Eno Uno·
8See e.g. W. Jones and N.H. March, Theoretical solid state physics, Vol. I, New York: Dover Publications, 1973.
24 Chapter 2
2.3. Envelope funclions
If a perturbing non-periodic (but local) potential U(r) is added to (2.1) (describing impuritiesi external fields, etc.), the wave function 1/J(r) can be
expanded in terms of the Bloch waves (2.2):
1/J(r) = l A11(k) Wnk(r) nk
since these functions form a complete orthonormal set. A set of slightly different
functions Xnk(r) = ~ exp(ik·r) u110(r) using the periodic Bloch functions at the
r point k = 0 works equally well, and turns out to be advantageous when
focussing on this special point9• In stead of k = 0 any k0 in the first Brillouin zone
can be chosen, which would however only burden the notation. With this set we
write the wavefunction as:
1/J(r) = l A11(k) Xnk(r) = l A11(k) ~ exp(ik·r) u11o(r) (2.5) nk nk
or, defining coefficients f11 (r):
as:
f11(r) = l A11(k) ~ exp(ik·r) k
1/J(r) = l f11(r) u11o(r) n
(2.6)
(2.7)
The coefficients f11(r) of (2.6) are termed "envelope functions": they vary slowly
and smoothly with position, contrary to the strongly :fluctuating Bloch functions
u110(r). From the SchrOdinger equation for 1/J(r):
[ 2£: + V(r) + U(r)] 1/J(r) = f 1/J(r) (2.8)
we can obtain an equation for A11(k) in (2.5)10:
9J.M. Luttinger and W. Kohn, Phys. Rev. 97 (1954) 869.
10J. Cuypers and W. van Haeringen, Physica B 168 (1991) 58.
From Bloch to BenDaniel-Duke
+ l l Bnm(K) l U{k'-k-K)Am(k') = 0 ][ m k'
where Pnm is the momentum matrix element:
Pnm =fro J d3r Uno*(r) ~ Umo(r) ,
no
K is a reciprocal lattice vector, Bnm(K) is the matrix element:
Bnm(K) =fro J d3r Uno*(r) exp(iK·r) Umo(r) ,
no
25
(2.9)
and U( q) =ft J d3r U(r) exp(ik·r) is the Fourier coefficient of U(r).
n Using the definition (2.6), we rewrite (2.9) in terms of the envelope function
fn(r):
+ l l Bnm(K) J d3r' A(r-r' )exp(iK· r' )U(r' )fm(r') = 0 (2.10) ][ m
where A(r-r'): ft l exp(ik·r) is a sharply peaked function11• With (2.10) we hav k
succeeded in getting rid of the fluctuating part Uno(r), at the cost, however, of
having to deal with a non-local equation. Fortunately, (2.10) allows for a local
approximation in the case of gentle, slowly varying potentials U(r). If in (2.9)
U(q) is only appreciable for small lql, and if the corresponding Am(q+k+K) is
negligible for K # 0, then we can in (2.10) approximate A(r-r') by a Dirac-6
and restrict the summation to K = 0, obtaining:
11See also M.G. Burt, Semicond. Sci. Technol. 3 (1988) 739.
26 Chapter B
(Eno+ 2£: + U(r)- f)fn(r) + l Pnm·&; fm(r) = 0 (2.11) m
Thus we have found in (2.11) an envelope-function equation that is very
SchrOdinger-like. It can be cast in matrix form:
where J. is the unity matrix, ! is a vector with components fm(r), and ~ is a
matrix with components Hnm: (Eno+ p2/2mo + U(r)) Dnm + Pnm·p/mo. In
(2.11) the crystal potential is only indirectly present through the band edges Eno
and the interaction Pnm·P·
2.4. LOwdin renormaliza.don and effective mass
Although the infinite linear system (2.11) is valid for all momenta., its usefulness
is apparent ma.inly in combination with perturbation theory. In many cases we
a.re interested in only a few of the infinitely many bands in (2.11). Dividing all
states into two classes, one of states min which we a.re interested (A), and one
of states µ in which we a.re not interested (B) but which have a nonnegligible
effect on the states in A:
1/l(r) = l f1n(r) Umo(r) + l ~(r) u110(r) me A µeB
we can truncate the matrices and vectors in (2.11) to have only components in
class A, provided the interactions a.re Lowdin renorma.lized12;
(2.12)
With this renormalized interaction the secular equation Det II ;fj; - f J II.= 0
yields the eigenvalues fnk for n e A. In the case that class A consists of a single
state n this procedure is simple. Omitting all terms involving two or more
t2P.-O. Lowdin, J. Chem. Phys. 19 (1951) 1396.
From Bloch to BenDaniel-Duke 27
intermediate B states (and putting for tli.e moment U(r) = 0), the energy Enk can
be expressed implicitly as:
which up to second order ink can be written as:
(2.13)
Eq. (2.13) is a parabolic approximation of band n, valid only in the vicinity of
the r point k = 0. Here we can define an effective mass tensor through the
relation:
fnJi: = fno + 2!: l ka. (iii!Jo(3 Jca , a,/3 = x,y,z
a. .i:i
so that:
(2.14)
Hence, the difference between this effective mass and the bare electron mass m0
is due to the coupling of band n to the other bands. For n corresponding to the
conduction band with r 6 symmetry, the effective mass is a scalar. Since the
main contributions to me come from the valence band states, lower in energy,
the effective mass is positive.
2.5. Kane model
In the case that class A consists of the r 6 conduction band and the r 7, r 8
valence bands, a different approach due to Kane is takent3. This procedure is
appropriate when studying optical properties, or hole tunneling in
13E.O. Kane, J. Phys. Chem. Solids, 1 (1956) 83; Ch.4A in Handbook on Semiconductors, Vol. I, Amsterdam: North-Holland, 1982.
28 Chapter!
heterostructures. But also in the case of a single conduction band, the Kane
model is very convenient to go beyond the quadratic dispersion relation of the
previous section t4 15. This application is the main reason for discussing the model
in this section, as an intermediate step towards an energy dependent effective
mass. The consequences of this nonparabolicity for tunneling will be discussed in
the next chapter.
In the Kane model, the coupling within the set of topmost valence bands and
the lowest conduction band is treated exactly, whereas the coupling between
these bands and all other r edges is treated perturbatively. Let us first consider
the case without external fields, i.e. put U(r) = 0 in (2.8) or (2.11). For III-V
compounds it is necessary to add a spin-orbit term to the crystal Hamiltonian in
(2.8), a correction of relativistic origin16, taking into account the coupling betwee
the electron spin and the orbital angular momentum. The spin degener-
Table I. Kane-model and other foara.meters tor Ga.As and AlAs and their d~endence on the mole action :i: o/ Al] in the ternary compound. A er Adachi11, and Eppenga et al.1 , Tcible Ill/IV.
parameter y unit Y[GaAs} Y[AlAs] dY/dx
lattice constant a nm 0.56533 0.56611 0.00078 stat. diel. const. K. 13.18 10.06 -3.12 band gap ~,l!)) eV 1.430 3.002 1.572 spin-orbit en. eV 0.343 0.279 -0.064 Kane energy Ep eV 28.8 28.8 s coupling s -3.849 -2.655 effective masses: conduction band IIlc 0.0667 0.1500 0.0833 light holes }1101) m1h 0.0870 0.2079 s.o. split-o Illso 0.1735 0.3147 heavy holes f 001~ mhh 0.3799 0.4785 heavy holes 111 mhh 1 0.9524 1.1494
14G. Bastard, Wave mechanics applied to semiconductor heterostroctures, Halsted Press, 1988. 15M.F.H. Schuurmans and G.W. 't Hooft, Phys. Rev. B 31 (1985) 8041. 16See e.g. W. Jones and N.H. March, Theoretical solid state physics, Vol. I, New York: Dover Publications, 1973. 17S. Adachi, J. Appl. Phys. 58 (1985) RI. 18R. Eppenga, M.F.H. Schuurmans a.nd S. Colak, Phys. Rev. B 36 (1987) 1554.
From Bloch to BenDaniel-D'Uke 29
acy is removed and class A now contains eight bands. The 8>< 8 Hamiltonian
matrix for the A bands is diagonal in a basis of eigenvectors lj,mj> of the total
angular momentum J and its projection Jz along the z-a.xis. For the s bands, this
has no consequences, but the p bands are split into a j = 3 /2 quadruplet of r 8
symmetry, and a j = 1/2 doublet of r 7 symmetry. The former is shifted upwards
in energy by an a.mount of 6. /3, whereas the doublet is lowered by twice this
amount. This spin-orbit energy 6. between the doublet and quadruplet is one of
the three basic Kane para.meters. The other two are the band gap energy Eg
between the conduction band and the quadruplet, and the interband matrix
energy Ep: 2m0P2/0.2, where Pis the velocity matrix element betweens and p
states. If all coupling to remote edges outside the r 6r 7r 8 subspace is excluded,
three light bands and one heavy band are found, the effective masses of which
can be expressed in terms of E,, Ep and 6.:
heavy holes: (2.15)
light holes: !!!JI_ = - 1 + i~ m1h g
split-off band:
electrons:
Inclusion of remote bands19 means Lowdin-renormalizing the 8><8 Hamiltonian.
This introduces four new para.meters, s, 111 12 and 73, that replace the 11 's in the
effective-mass expressions of (2.15). In addition, the light-hole and heavy-hole bands turn out to be anisotropic.
In stead of calculating the seven Kane parameters from expressions like
(2.14), we use the model as a description of the bands near k = 0, fitting the
parameters to reproduce the correct gaps and masses. In fact, this "empirical"
picture of the Kane method is the usual one. Accuracy of the Kane model can be
ascertained by comparing its results with calculations from global band structure
descriptions20. In Table I a set of parameter values is summarized for both GaAs
and AlAs, together with the effective masses that result from these parameters.
19R. Eppenga, M.F.H. Schuurmans and S. Colak, Phys. Rev. B 36 (1987) 1554. 20M.F.H. Schuurmans and G.W. 't Hooft, Phys. Rev. B 31 (1985) 8041.
30
Fig.2.1
Chapter 2
GaAs Al As
-0.2 o.o 0.2 ·0.2 0.0 0.2
lm(k) Re(k) lm(k) Re(k)
Complex band structure E{O,O,k) for GaAs and AlAs. Positive (negative) k correspond to rea( (imaginary) wave number. The horizontal scale is in terms of 27r/a, a being the lattice constant.
In Fig. 2.1 the various bands are plotted as function of the wave number: the
positive horizontal axis corresponds to real wave numbers and envelope functions
that oscillate in space, the negative horizontal axis to imaginary wave numbers
and envelope functions that are exponentially damped. In this complex band
structure, the conduction and the light-hole band turn out to be one branch.
The zero of energy is chosen at the top of the GaAs valence band, and the offset
for the AlAs bands is in accordance with the 67 /33 value for the band-edge
discontinuity ratio of GaAs/ AlxGa1:xAs21•
If we are only interested in the conduction band (as in the case of electron
tunneling), we can reduce the Hamiltonian matrix from 8x8 to 2x2 by
renormalizing once more. However, the valence bands are treated differently
21For GaAs-AlxGa1_xAs values in the range of 65/35-fi9/31 are reported. See J. Menendez, A. Pinczuk, D.J. Werder, A.C. Gossard and J.H. English, Phys. Rev. B 33 (1986) 8863.
From Bloch to BenDaniel-Dv.ke 31
from the remote bands: for the latter the denominator ( fnk - Hl4J.) is
approximated by ( fno - f"'0), whereas for the valence bands this becomes
( fnk - £"'0). The non-diagonal elements now remain zero, and the diagonal
elements become:
where the zero of energy is ta.ken at the conduction band edge. Thus we find for
both spin-up and spin-down a simple 11Schrodinger" equation:
(2.16)
with an energy-dependent effective mass given by:
m 2Ep Ep ~ = 8 + 3(£+E8) + 3{e+E8+ZS) (2.17)
In Fig. 2.2 the complex band structure according to (2.16-17) is plotted,
together with the full Kane solution for electrons and light holes {Fig. 2.1) and
the dispersion corresponding to the constant Ille of (2.15). In contrast to Fig. 2.1
k is shown as a function of f. Both the m( f) and me dispersion relations are
expansions a.round the conduction band minimum, where all three relations
coincide. Eqs.{2.16-17) however follow the Kane band structure remarkably well
over a large energy interval, and the difference with the constant me case is quite
clear. Especially in the bandgap where the wavenumber is imaginary, there is an
appreciable difference. Although these· energies a.re not very important in bulk
materials, they do play an important role in heterostructures and hence in
resonant tunneling.
In the case of a perturbing nonperiodie potential U(z), the same recipe that
led to (2.16) ean be applied22. It is found that, even in the ease of a spin
independent U, the two spin states a.re now coupled via a nondia.gonal term:
Complex band structure k(f} for GaAs and AlAs. The dots correspond to the m(f) model of {2.16-17}; the upper cu.rue is the Kane solution for electrons/light holes, the lower curve is the EMA using me of {2.15). Positive (negative} k correspond to real (imaginary) wave number. The vertical scale is terms of 27r/a, a being the lattice constant.
where M(f) is defined as:
while the diagonal term changes to:
From Bloch to BenDaniel-'Duke 33
(2.19)
Eq.(2.16) now reads:
(2.20)
From the definition of M(E) it can be seen that the coupling between spin-up
and spin-down states of the conduction band is due to the spin-<lrbit coupling: if
a = 0 the nondiagonal term vanishes. The above Kane theory is applicable to elemental semiconductors like Ge, or
binary compounds like GaAs and AlAs. Also ternary solutions like AlxGa1_xAs,
which, strictly speaking, do not have a periodic crystal potential, can often be
described within energy band theory by use of the socalled "virtual crystal
approximation"2i: the random potential is replaced by a periodic one: x• Vu + (1-x)- V Ga + V As• where x is the mole-fraction of Al. The difference between this potential and the actual one is then responsible for the alloy scattering and
the zero temperature resistance24. Thus the virtual crystal approximation makes i
· possible to introduce composition dependence of the band gap, Eg(x), the
spin-<lrbit energy, .6(x), etc. Often this dependence is linear, so that we can
write Eg(x) = Eg(O) + x·dE8/dx. For the case of AlxGa1.xAs these linear coefficients are given in Table 12s.
2.6. Heterojunctions
Based on the simple conduction band model (2.16-17) that we arrived at in the previous section, we will now consider the case of a heterostructure.
In a heterojunction of two materials A and B the potential V(r) that enters
23According R.H. Parmenter, Phys. Rev. 97 (1955) 587, this approximation goes back to L. Nordheim, Ann. Physik 9 (1931) 607; 641. 24N.F. Mott and H. Jones, The theory of the properties of metals and alloys, Oxford University Press, 1936. 25The parameters in Table I pertain to the r point. The X valley has a much smaller composition dependence. For x ~ 0.43 the band gap at the X-point is smaller than E (r), and here AlxGa1_xAs is an indirect semiconductor. See e.g. Landolt-Borns~ein III/17a. Transport properties of heterostructures depend much on whether the component materials are direct or indirect. We consider here only direct semiconductors, so that xis confined to 0 < x < 0.43.
34 Chapter 2
the one-electron SchrOdinger equation:
[ 2£: + V(r)] ,P(r) = E ,P(r) (2.21)
is no longer periodic in general:
{
(1)
V(r) = V (r) , r in material 1 (2)
V (r) , r in material 2
(2.22)
and the solutions of (2.21) cannot be written in Bloch form. We can however
make an expansion of ,,O(r) in both materials in terms of the periodic functions ( 1l ( 2)
Uno (r) and Uno (r):
,P(r) =
{
~ (ll (ll it fn (r)·uno (r) , r in layer 1
~ (2) (2) ff fn (r)·Uno (r), r in layer 2
(2.23)
( 1,2) where the envelope functions fn (r) are slowly varying functions, and the
summation over n runs over all included band edges. For each layer the envelope
functions fn follow from (2.11):
[
- (1,2) ] (1,2) ~ -EJ, f =.Q (2.24)
Solving (2.21) now amounts to matching correctly the envelope function vectors (1) (2)
f and f at the heterojunction interface.
One connection rule is provided by the continuity of the electron wave
function ,P(r). When the two materials (1 and 2) are not too different (as is the
case with e.g. the lattice-matched GaAs and Al0•3Ga0•7As), the usual
assumption26 that: (1) (2)
Uno (r) =Uno (r) (2.25)
may not be too hazardous. From (2.25) and continuity of ,P(r), it is directly
26S.R. White and L.J. Sham, Phys. Rev. Lett. 47 (1981) 879.
From Bloch to Ben.Daniel-Duke 35
( 1l ( 2) (1) (2) deduced that the envelope functions be continuous: fn (r) = fn (r) or f = f . Let us take z = z0 to be the interface and write r 11 for (x,y). From (2.25) it
follows that the parameter Ep of (2.15) is equal for both materials A and B27• For
all other parameters (Eg, !:>., s, 711 72 and 73), we write:
(1) (2) s = s(z) = s · D(zo-z) + s • D(z-zo) (2.26)
etc., where D(z) is the Heaviside function, D(z) = 1(0) for z H <) O. As a result,
the Hamiltonian matrix i! of (2.24) depends only on z, and the envelope
where Sis the sample area, and k11 is the wave vector parallel to and continuous
across the interface. From (2.25) we obtain a first connection rule for the (1) ( 2)
envelope functions Xn(z): Xn (zo) = Xn {zo) or:
x(z) continuous (2.28)
A second connection rule is obtained as follows: the· matrix ~ can be cast
into the general form of:
(2.29)
where the definition of the~ matrices follows from comparing (2.29) with (2.11). We now replace kz by -io/8z, an operation however without an umambiguous
prescription. Often this replacement is done in such way that the resulting
Hamiltonian is hermitian28; even then there remains some arbitrariness29,
although in the literature a convergence can be noticed on the simple form:
2sa. Eppenga, M.F.H. Schuurmans and S. Colak, Phys. Rev. B 36 (1987) 1554.
29For a general discussion of the form of the Hamilton operator, see R.A. Morrow and K.R. Brownstein, Phys. Rev. B 30 (1984) 678.
36 Chapter 2
(2.30)
We will shortly see how this hermiticity immediately leads to the well-known
1/m connection rule for the derivatives of the envelope functions. Here we stress
that the hermitian form can only be advocated if the Lowdin A-dasses for the
two materials contain all relevant bands, i.e. if there is no significant coupling
between A-states of material (1) to B-states of material (2), and vice versa. Coming back to this question at the end of this section, we now carry on our
discussion of (2.30). Integrating this equation once across the interface yields the
second connection rule:
~(z)x(z) continuous (2.31) where:
(2.32)
We now apply the above recipe for envelope connection rules to the special
case of one conduction band with energy~ependent effective mass, Eq.(2.16) of
the previous section. For a heterostructure (2.16) has to be translated into a
where the conduction band minimum Eco is a step function at the interface z0:
and an analogous expression is valid for the effective mass m(z,e). This steplike
behavior implies that up to the interface the bulk properties are conserved, at
least in terms of the slowly varying envelopes. Put differently, the interface is
assumed to be sharp, and any effect of the interface must decay within one
atomic layera0• The connection rules for x(z) at the interface are readily obtained from (2.28) and (2.31):
3°For a discussion of this socalled flat-band approximation see J. Cuypers, Scattering of electrons at heterostructure interfaces, doctoral thesis (1992).
From Bloch to BenDaniel-Duke 37
x(z) and m(!,e) *must be continuous (2.34)
across the interface.
This socalled BenDaniel-Duke model31 for heterostructures combines
simplicity with versatility. It has proven to be a powerful method for calculating
energy levels in quantum well structures and superlattices32• On the other hand,
it is based on a number of approximations and limitations, some of which are
reasonable, while the status of others is less clarified. As a conclusion of this
chapter, two remarks regarding the validity of the BenDaniel-Duke model are
made. The first one concerns the link between the connection rules (2.34) and
the conservation of probability density and current, expressed in terms of the
envelope functions x(z), and averaged over the unit cell:
p(z) = I x(z)l 2
·c ) _ 1i. [ • d _ d ·] J z - rm(iJI x rzX XrzX
It is easily seen that the connection rules (2.34) ensure the continuity of these
two quantities - a physically appreciable outcome.
A second remark, in the light of a recent study on the connection rules for
envelope functions33, draws in the T-matrix approach, of which (2.34) can be
considered a special case. In this approach, we write at the interface:
[
(2) ] [ (1} ] x x t2l = T Ul
dx /dz dx /dz
where Tis a 2 " 2 matrix containing the connection rules. The rules of (2.34) correspond to a diagonal T-matrix:
(2.35)
31D.J. BenDaniel and C.B. Duke, Phys. Rev 152 (1966) 683. 32G. Bastard and J.A. Brum, IEEE J. Quant. Electron. QE-22 (1986) 1625. 33J. Cuypers, Ph.D. thesis, Eindhoven (1992), ch. 5.
38
Fig.2.3·
Chapter!
1.40 constant-mus ratio: 2.249
.... 1.30
~ !!. 1.20 e -I 1.10
e 1.00
0.90 0.00 0.50 1.00 1.50
ENERQY(eV)
1.15 .---------------, constant-mus ratio: 1.875
1.10
1.05
1.00 .....__._~-~....__~__.._...._~__.....~
0.00 0.10 0.20 0.30 0.40 0.60
ENERQY(eV)
Effective mass 9uotient as a function of energy for (a}. GaAs/AlAs and (IJ) GaAs/AluGauAs. The calcv.lati.ons are based on (2.17). The zero of energy is at the Ga.As conduction band minimum.
Cuypers31 calculated the T-matrix elements from the continuity of the underlying wave function and its derivative, obtained from EPM (empiricalpseudo potential method) calculations. Thus he was able to check, for ea.ch specific heterojunction individually, the validity of the simple diagonal. choice (2.35) .. For the GaAs/AlAs conduction bands (k11 = 0), the situation is quite rosy: the off-diagonal elements are smaller than 10·4, and the first diagonal element is close to unity (0.9 < T11 < 0.92). Also the second diagonal element agrees with the effective-mass quotient of (2.35), provided that the
energy-dependent masses are used. This is illustrated in Fig .. 2.3a, where m(AlAs)/m(GaAs) is plotted as a function of energy, and which agrees quite well with the corresponding plot of T22 (Fig. 5.2 in Cuypers33). Notice that in the case of the familiar energy-independent effective m~s the agreement is
From Bloch to BenDaniel-Duke 39
completely lost. In the case of GaAs/ Al0•3Gao.7As, the agreement between the
EPM calculations of Cuypers and (2.35) is even better: 0.973 < T11 < 0.976 and
1.06 < I T22I < 1.09. Here too nonparabolicity is important, as can be seen from
Fig. 2.3b (cf. Fig. 5.8 in Cuypers33). Hence, in the case of conduction band
matching, the EPM calculations support the simple BenDaniel-Duke model, a
conclusion which remains valid for k 11 # 0. However when the X-valley comes
into play, or when the valence band becomes important (as e.g. in the
InAs/GaSb system), when the two materials are lattice-mismatched or
otherwise dissimilar, the simple envelope-function rules of (2.35) break down,
and a return to the underlying wavefunctions is unavoidable. Since we deal in
the next chapters with the most favourable case of GaAs/ AlxGa1.xAs conduction
bands, we will not go beyond the simple BenDaniel-Duke model. In fact, most
calculations of the next chapters were performed using energy-independent
(equal or different) masses for the heterojunction materials. In Sect. 3.2 we will investigate the numerical consequences of this neglect of nonparabolicity.
40 Chapter 2
chapter3
COHERENT TUNNELING
3.1. Introduction
Detailed studies that compare coherent-tunneling models to experimental
results, like e.g. Gueret et al.1 and Van de Roer et al. 2, invariably end up with the
conclusion that there are large discrepancies between (the bare) theory and
experiment. The general shape of the 1-V characteristic (the resonant peak and
the increase in the Fowler-Nordheim3 regime) is always reproduced by the
calculations. However, the position of the peak, the peak current, and especially
the valley current and the related peak-to-valley ratio (PVR) pose serious
difficulties, and predictions can differ orders of magnitude with measurements. The conclusion that coherent tunneling cannot .be the whole story seems inevitable. Scattering has to be taken into account, either in a sequential
1P. Gueret, C. Rossel, E. Marclay and H. Meier, J. Appl. Phys. 66 (1989) 279; P. Gueret, C. Rossel, W. Schlup and H.P. Meier, J. Appl. Phys. 66 (1989) 4312. 2T.G. van de Roer, J.J.M. Kwaspen, H. Joosten, H. Noteborn, D. Lenstra and M. Henini, Physica B 175 (1991) 301.
ac.B. Duke, Tunneling in solids, New York: Academic, 1969.
41
42 Chapter 3
approach1 or by introducing an "incoherence parameter" 2•
On the other hand, although coherent tunneling may not be the whole story,
it is nevertheless a large part of it. The very existence of a resonant current peak
and of a region of negative differential resistance (NDR) strongly indicates
transport via a resonant state. Therefore, it seems wiser to start with a coherent
tunneling model, in which the scattering has to be incorporated afterwards, than
the other way around. This is at least the approach followed in this thesis.
Coherent tunneling yields sharp peaks in the transmission coefficient, of
which position (Eres) and width (r) depend only on the "intrinsic" properties of
the diode, such as the kinds of material, the widths of the various layers, etc.
The resonance energy Eres is important in determining the position of the
current peak, whereas the width r is of crucial importance to the NDR region.
Scattering effects on the other hand depend also on experimental conditions such
as temperature, on doping concentrations and local variations in the diode's
dimensions. Their effect is mainly to increase the width r, and therefore the
NDR region is the voltage range where to expect substantial deviations from the
coherent picture.
In this chapter we are mainly concerned with the current scale of the I-V
characteristic. First we review the transfer matrix approach that is used to
calculate the electron wavefunctions and hence the transmission and reflection
probability. In Sect. 3.3 the resonance position and width, calculated with TMA,
are closely looked at, especially with respect to the barrier parameters. The
coupling to the reservoirs is treated in the next two sections. Finally the effect of
inelastic scattering is studied within the simple Jonson-Grincwajg model.
3.2. Transfer Matrix Approach
Coherent tunneling in layered heterostructures is easily described using the
Transfer Matrix Approach (TMA), a method already present in the pioneering
work of Tsu and Esaki4• First we will introduce the method on the basis of a
piecewise constant potential. This corresponds to situations without external
fields, or with slowly varying fields that can be considered constant within each
layer. A generalization of this TMA is presented afterwards.
4R. Tsu and L. Esaki, Appl. Phys. Lett. 22 (1973) 562. See also the reviewing paper by G.A. Toombs and F.W. Sheard, in Electronic properties of multilayers and low-dimensional semiconductor structures, eds. J.M. Chamberlain et al., New York: Plenum, 1990, p. 257.
Coherent Tunneling 43
Let us take Eqs.(2.33-a4) of the previous chapter as our starting point for
calculating the envelope functions x(z) in a heterostructure within the flat:--band
approximation. For the moment, we put k11 = 0 for simplicity; the effect of a
nonzero k 11 are treated in the next section. Writing the general solution to
(2.33-34) as:
x(z) = { aL•exp(ikLz) + bL·exp(-ikLz)"' [~~] 1 Z < Zo
a1 ·exp{ik1z) + b8:ex.p(-ik1z)"' [~~ , z ~ z0
(3.1)
where kL = Jf~1·(E-Eko) and ka is defined analogously, we can translate these
connection rules into relations for the a,b-coefficients: aL + bL = aa + ba and
i!~(aL - bi}= i!:<aa - b1), or, in vector notation:
In other words, the connection rules provide a relation between the coefficients
of the envelope function on either side of the interface:
(3.2)
where Mis called the transfer matrix, that in our particular case reads:
[HA 1-A] m k
M = i 1-A l+A • A= mi~
In stead of coupling L(eft) to R{ight) coefficients, we can also connect the
coefficients of the outgoing waves to those of the incoming ones:
(3.3)
where ~ is called the scattering matrix, the elements of which are:
44 Chapters
(3.4)
For an incoming wave from the left, we have aL = 1, ba = 0. From (3.4) the
reflected wave has coefficient bL = p and the transmitted wave has a1 = r. The
amplitude squared of the reflected wave p*p is called the reflection coefficient R,
that of the transmitted wave r*r the transmission coefficient T. For an incoming
wave from the right, we have aL = 0, ba = 1. Now the reflected wave has
coefficient a1 = p and the transmitted wave has bL = 1'. From (3.2) and (3.3) the transfer matrix can be expressed in terms of the scattering matrix elements:
(3.5)
Since both x(z) and x*(z) are solutions of (2.33), because of time-reversal
symmetry, the matrix~ has the property that M22 = M11* and M12 = M21*, so
that p and :; can be expressed in terms of p and 75:
• / • 1-n*n 1-R d M p = -p*1" 1"* , r = ~ = 1"·-y- = 1"• et =
and the reflection coefficient R = p*p and transmission coefficient T = 1"*1' for
right incoming waves in terms of Rand T for left incoming ones:
(3.6)
The two transmission coefficients, from left to right and from right to left, turn
out to be different. Therefore it is convenient, and in fact common practice, to
use the concept of transmission probability& P, defined by:
P:::l-R=l-R
sn. Lenstra and R.T.M. Smokers, Phys. Rev. B 38 (1988) 6452. 6The interpretation of R and P in terms of reflection and transmission probability is limited to situations where both kL and kB. in (3.1) are real. See also any text book on Quantum Mechanics.
Coherent Tunneling 45
and equal to both T·det(M) and T/det(M)· This transmission probability is
indeed symmetric, and the fact that P + R = 1 expresses the conservation of
probability.
This TMA can easily be generalized to be applicable to any kind of band
edge function Ec0(z). Since for every value of£, Eq.(2.33) is an ordinary second
order differential equation· in both the left and the right region, it has on either
side of the interface two linearly independent solutions, {fL(z), gL(z)} and {f1 (z),
g1(z)}, say, the independence of which implies that the wronskian W(f,g):
W(f,g):: f(z)~g(z)-g(z~f(z) = f(z)g'(z)-g(z)f'(z)
is non-zero for all z. It will turn out convenient to define for a given interface at
z = z0 a functional j(f,g) by:
(3.7)
where the limits keep track of the direction from which the interface is reached.
For functions f and g pertaining to the same region, say L, j(fL,gi) is equal to
iti.W(fL,gi)/2mL. For f a.nd g pertaining to different regions (in the case of
j(fL,fa) etc.), however, no such property exists because of the position-dependent
mass. This new functional j(f,g) is a generalization of the probability flux, since
j(f,f") = jc(z) is the flux of proba.lity density in state f. If we now generalize (3;1):
x(z) = { aL.fL(z) + bt•gt(z)"' Ib~] I z < Zo
a11.·f11.(z) + b1 -g1 (z)"' [b!] , z ~ z0
we can express the matrix M of Eq.(3.2) in terms of j:
(3.8)
(3.9)
Again, since the potential energy term of (2.33) is real, the functions f and g can
be choosen real or pairwise complex conjugate. In the latter case, since
46 Chapter 9
{j(f,g)}* = j(g*,f*), we find the same symmetry properties for M as mentioned
below (3.5):
In this TMA, a potential· barrier can now be represented by the matrix
product of two interface matrices, and a double barrier by the product of two
single barrier matrices:
(3.10)
in self-explanatory notation. If the element notation of (3.5) is used for all matrices, the transmission and reflection coefficients of the DBRT-matrix ( r, p) can be expressed in those of the single-barrier B-matrices ( T1i p1 and r 2, p2):
(3.11)
We see that the total transmission r is the product of the transmission
coefficients of both barriers, r 1 and r 2, times a factor which takes into account
the multiple reflections between the barriers, (1-piP2)-1• This interpretation can
be seen to be correct by expanding the factor, yielding:
the n-th term of which corresponds to tunneling through the first barrier, then
being reflected n times in the well, and :finally escaping through the second
barrier. Hence it is this factor (1-piP2)"1 that is responsible for the resonant
Coherent Tunneling 47
character of the tunneling. Indeed, the total transmission coefficient T : T"r is:
(3.12)
where a = arg(pi02), so that, when all quantities vary slowly with incoming
energy, the maxima of T are determined by an = n • 211", with n an integer. These
maxima are sharp resonance peaks if the reflection coefficients are close to unity7,
in which case we can define a peak width (BWHM, half width at half maximum)
r a:
(3.13)
The height of the resonance peak, Tmax• is found to be:
(3.14)
which for the special case that T 1 = T2 = (1-R1) = (1-R2) equals unity. Since:
the area under a resonance peak, Ia, is given by:
(3.15)
Finally, the shape of the resonance peak is approximately Lorentzian, as can be
seen as follows. Substituting in (3.12) for a(e) a Taylor series near the n-th
resonance, a(f) = n·211" + (E-En)a'(En) + ... ,we find for incoming energies near
7For the arcsin in (3.13) to exist, we roust have 17-12./2 S R1R2 S 1; hence, the peak is sharp for R1R2 > 17-12./2 = 0.0294..., or R1 and R2 > 0.172 ...
48 Chapter 9
the resonance En:
T Ill Tmaxr.2
r£2 +(E-En)2 (3.16)
where r£ = 2' !!(~:1/2) 1:1 r a/a'( En) is the resonance width, related to the resonance lifetime To through a Heisenberg-like rela.tion:
(3.17)
Eq.(3.16) expla.ins the success of a Breit-Wigner formulation of resonant
tunneling8•
It is remarked here, that in this section we have made no other assumption
than that there are two non-trivial matrices M of the form (3.5), in other words:
that there are (at least) two discontinuities in the potential. As demonstrated
above, the shape of a resonance is not very sensitive to the exact shape of the potential. When the two reflection coefficients corresponding to the
discontinuities are close to unity, the resonances become sharp and Lorentzian
-like; to them can be attributed a line width r £ and life time To· Thus for small
(1-R1): P 1 and (1-R2): P2, it is found to lowest order in P 1 and P2, that:
3.3. The resonance energy; dependence on barrier parameters
In the case of rectangular barriers of equal width b and height H, an expression
for the resonance energies is easily obtained. Working out arg(pJP2) = O, the
resonance condition, we find:
-::~)tanh(,.b) (3.18)
where k = J 2mwe/h and,.= J 2mb(H-E)/1i, His the barrier height and wand
8D.D. Coon, K.M.S.V. Bandara and H. Zhao, Appl. Phys. Lett. 55 (1989) 2453.
Coherent Tunneling 49
b are the width of the well and barriers9. In the limit of b -1 m, (3.18) reduces to·
the well known relation for the bound states in a finite well:
tan [f!] = :~k or cot (f!] = -:~k (3.19)
for the even or odd states. Just like this relation, (3.18) is a transcendental
equation without analytic solution. In the case of thick barriers, by1(2mbH) > ti., the resonance and the bound state will be very close in energy. It is remarkable
that nowhere in the literature on resonant tunneling Eq.(3.18) is mentioned. In
fact, the only author who offers a resonance condition, Bastard10 , wrongly
presents (3.19) in stead of (3.18), stating that the resonance energies "coincide
with the bound states". For the device of Fig.1.3, varying only b1 = b2, the first
two resonances, Erest and Eres2, are plotted as functions of b in Fig.3.1. Since
Erest decreases with decreasing b1 and Eres2 increases, we have plotted
ln((EbndrEres1)/e) and ln((Eres2-Ebnd2)/e), where Ebnd is the solution of (3.19) and Eres the solution of (3.18). From the straight lines in Fig.3.1. it can be
concluded that the resonance approaches the bound state energy for large barrier
width in an exponential wayll.
Eq.(3.18) is derived for the case of an unbiased structure, and it is not
evident that it would apply to a structure under bias. However, it will be shown
in Sect. 4.3 that the resonance energy with respect to the well potential is fairly
constant, and consequently (3.18) remains important also in the biased situation.
Two clarifications remain to be made, one concerning the masses mw and mb,
and one concerning the barrier height H. From (2.17) it ca.n be seen that the
effective masses in the well and barrier are in fact energy dependent, and they
9 Analogous but more complicated expressions are found for structures with unequal widths. In this section, we confine ourselves to the symmetric case. In numerical calculations (ch. 4-6) also asymetric structures are considered. 10G. Bastardj Wave mechanics applied to semiconductor heterostructures, Halsted ( 1988 . 11From the fact that the resonance energies can increase or decrease with increasing barrier width b, it is induced that for every well width w, there is exactly one barrier height H for which the resonance is independent of b. Indeed, though Bastard's remark is erroneous in general, it is true for the first resanance in the special case that H = ti.2'K2(mw+mb)/8mw2w2. 0£ course, the resonance width does depend on b even in this case. Accordingly, DBRT structures of this kind would offer the possibility to study the specific effect of the resonance width.
50
-Q) -, w
I ... .i w -0 .c
5 c -I
Fig.3.1
0
-5
-10
-15
·20
-25 0 5 10 15 20
BARRIER WIDTH b/a
25
Chapter 9
-o- E rw1
-•- E ml
Dependence of the first two resonance energies on the barrier width b {in units of the lattice constant a = 0.56 nm). Plotted is the logarithm of the energy difference between the resonance Eresfb) and the bound state Eres(m). The lowest resonance decreases with decreasing b, whereas the second one increases.
depend on the total energy, not just the part related to the motion in the
z-direction. This means that the resonance energy, may it be in a. modest way,
depends on k112• Far more important is the fa.ct that H is not just the conduction
band offset ~Eco, but, as can be seen from (2.33), also contains a. part:
which is nonzero wether the masses are energy dependent or not (provided they
a.re different). Thus, the BenDaniel-Duke theory h:ttroduces an effective barrier
potential, the strength of which is k11 2-dependent. As a consequence, the
resonance energies will clearly depend on ku 2, as can be seen from Fig.3.2, where
both Eres(k11 2) and r(k11 2) are plotted. Most authors who use different masses do
not mention this problem and ignore the term. Basta.rd t2 does describe this
t2G. Bastard, op. cit. {1988).
Coherent Tunneling 51
Fig.3.2
0.10
1
> 0.09 0 - 4 I w 2
3 0.08 0.01
0.00 0 2 4 6 8 10
0.10 1
~ E -:I :c 0.05 4 ;:: :c 3
2
0.00 0 2 4 6 8 10
k, • .' (101• m"')
(a) The resonance energp E1es and {b) the HWHM vs. the parallel momentum squared k0
2, for JOur different cases: 1. equal effective masses for both materials; 2. different effective masses; 3. as 2. but now including the in-plane dispersion term; 4. as 9. but now with energy-dependent masses.
52 Chapter 9
"in-plane dispersion effect", as he calls it, and suggests to remove the term to
first order introducing an in-plane mass mn, defined through:
where B(Eres) is the probability of finding the resonant electron in the barriers.
By writing the in-plane contribution to the energy as fi.2k11 2 /2mn in all
layers, the dependence of the resonance energy on k11 2 is indeed removed. This
method is well suited for calculating quantum-well energy levels, for the DBRT
structure however it would destroy an interesting effect. For applied voltages
such that Eres is somewhere in the middle of Fermi window (O,Er) of the emitter
reservoir (Er > r), the exact shape of the resonance peak is unimportant. The
current density is determined by the (total) area under the peak only. However,
when Eres is close to Er or O, only a fraction of the peak area contributes to the
current density, and in these instances the peak shape does matter. At the onset
of resonant current (Eres ~Er), the differential conductance (the slope in the I-V
characteristic) is rather small, and so are the changes when we switch to a k11 -
dependent Eres· At the end of the resonant-current interval (Eres ~ 0), however,
the differential conductance is quite large (and negative). In the case of a k 11 -
independent Eres the peak falls through the conduction band minimum for all k11 simultaneously, and this gives rise to the steep descent in the I-V curve.
Changing to a k11 -dependent Eres in this case means that the peak dissapears at
different voltages for different k11 , which yields a considerable smoothing of the
negative differential conductance. Indeed calculations, the results of which a.re
depicted in Fig. 3.3, show that the maximal (negative) conductance Gmax is
reduced by a factor of 2. And since the cut-off frequency of a DBRT diode is
proportional to ../Gmax 13, the small in-plane dispersion effect is responsible for a
large change in the estimate of this device characteristic.
Fig.3.2 also allows us to study the effect of the conduction band nonpara.boli
city on the resonance position and width. Four different models a.re compared:
(1) The Ga.As effective mass for both Ga.As and AlxGa1_xAs. This yields 0.09302
eV for the first resonance. (2) Different masses for the two materials: the
resonance energy decreases to 0.08445 eV. (3) Ta.king into account the in-plane
dispersion term: as a function of k112 the resonance energy has a small negative
13D.D. Coon and H.C. Liu, Appl. Phys. Lett. 49 (1986) 94; T.C.L.G. Sollner, E.R. Brown, W.D. Goodhue and H.Q. Le, Appl. Phys. Lett. 50 (1987) 332.
Coherent Tunneling 53
tr' E < 0 0 ...... -> I-Cl.) z w c I-z w a: a: :.> 0
Fig.3.3
0 2 ····· tr'
E Cl.)
• 0 --7 ...... -i= -1 1 >
I-0 :.> c z 0 0
0 -2 0.160 0.165 0.170
BIAS VOLTAGE M
. Current density (left am) and differential conductitrity (right axis) vs. bias voltage in the NDR voltage range, with ( dotteil. line) and without (solid line) the in-plane-dispersion term.
slope. (4) Using the energy-dependent effective masses according to Eq. (2.17):
the decrease from (1) to (2) is partly undone. Due to the nonpa.rabolicity of the
conduction bands, we end up somewhere between the case of equal and different
effective masses. This can also be seen from the plot of the resonance width
(Fig.3.2b): the HWHM in ease (4) is roughly twice as small as in case (1) a.nd
twice as large as in ease (2). Since the current density is proportional to the
HWHM, we must expect the same factor of 2 in the I-V calculations. It is
concluded that going from equal (ease 1) to different (case 2) masses does not
necessarily increase the numerical reliability of the model.
54 Chapter 3
3.4. Current density expression
Since the work of Tsu and Esaki1\ the expression for the static current density
reads:
an expression which they adopted from Duke's monograph15• In fa.ct, their
adoption was not completely correct, since they replaced Duke's tunneling
probability P(E) by the tunneling coefficient 1-r{E}l 2, which minor mistake was
already mentioned by Coon and Liu16. It is interesting to look at the derivation
of (3.20} in Duke: the current is obtained by summing the expectation values of
the current operator in the eigenstates of the tunneling Hamiltonian over the
occupied electronic states. Hence the factor f1(E): {1 + exp,8(E-µ}}·1, whereµ is
the chemical potential of the left reservoir. Moreover, Duke adds a factor
(1-fr{E}), where fr(E) = {1 + exp,B(e-µ+eVb)}-1, to "guarantee" the unoccupied
character of the final state of the electron. It turns out, however, that this
addition is of no influence on the final result. From the thus found current
density from left to right, a similar expression for the current density from right
to left is subtracted.
Let us work out Duke's recipe, and make a comparison with the work of
Coon and Liu11, the only authors who have questioned the correctness of (3.20).
In this section we will take the electron mass to be position but not energy
dependent. The solutions to the SchrOdinger or envelope equation:
fi. 2 k 2 fi.2 d( 1 d ) ~x{z)-20z m(z) QiX{z) + Eca{z)x{z) = ex{z) (3.21}
where Eca(z) -i 0 for z -i -11.J and Eco(z)-i --eVb for z -i +Ill, can be written as a
14R. Tsu and L. Esaki, Appl. Phys. Lett. 22 (1973) 562. See also the reviewing paper by G.A. Toombs and F.W. Sheard, in Electronic properties of multilayers and low-dimensional semiconductor structures, eds. J.M. Chamberlain et al., New York: Plenum, 1990, p. 257.
1sc.B. Duke, Tunneling in solids, New York: Academic, 1969.
16D.D. Coon and H.C. Liu, Appl. Phys. Lett. 47 (1985) 172. 11See also: D.D. Coon and H.C. Liu, Solid State Commun. 55 (1985) 339.
Coherent Tunneling
Fig.3.4
1 p
i ~-_......,_......,_.,.__-+----1
p tat well 2nd
bll'l'ler barrier .
Sketch of the left and right incoming envelope f'unctiona. HorizontaUy the z-direction is plotted. ·
55
linear combination of a "left incoming" wave xl k (z) and a "right incoming" t:, n
wave x~ k (z), sketched in Fig.3.4: "' II
(3.22a.)
(3.22b)
where k1 = J 2mt:/'tl,2 - k11 2 and 1tr = J 2m( t:-eVb)/'tl,2 - k11 2. We assume one
effective mass m for the left and right reservoir. The transmission and reflection
amplitudes T, p, 1', p and coefficients T, R, T, R are functions of e and k11 2, a.s
are the normalization constants N and N. The complete envelope functions
ff! k (r) are products of x<; k (z) and the parallell part kexp(ik11 r 11 ), a= l,r. If "' IL, "• U we choose:
56
we have:
for a is either 1 or r. In Dirac notation {3.24) reads:
where we have introduced the notation:
laEku>=Jd3rfa.Ek (r)lr> ' II
Chapters
(3.23a)
(3.23b)
(3.25)
To find out wether this basis {fa. k (r)} is orthogonal, we calculate the overlap
be 1 f d . h . . f, II
tween e t an ng t incoming waves:
(3.26a)
and find the thus introduced 1'J to equal:
Both the normalization constants N and N (3.23) and the overlap matrix
element 'fl (3.26) differ from the ones introduced by Coon and Liuu. Their
approach amounts to putting the structure in a box of length L/2 + D + L/2 l:j
L, where D is the structure width. This yields for the normalization factor N = J{L(l+R+T)/2} and a corresponding expression for N. However, to find the
lBD.D. Coon and H.C. Liu, Appl. Phys. Lett. 47 (1985) 172.
Coherent Tunneling 57
al.lowed energies for this system is now a difficult task depending on the
(periodic or other) boundary conditions imposed. Taking k1 to be n(2r/L), n an
integer, - the usual bulk choice adopted by Coon and Liu -, is now incorrect,
and so is the transition :En .... Jdk1(L/27r), which would finally yield an N of
Coon & Liu: N = Jait~/de (1 + R + T]~
This result differs from our (3.23a) by the ratio ~1}~e in front of T. In the same r €
fashion, they find for 11:
For the important case of time-reversal symmetry, the difference has far
reaching consequences. With our choice of normalization, 1/ turns out to be zero
and, therefore, the basis {I a e k11 >} orthogonal:
1 + R + ~1~ge · T = 1 + R + !t. T = 2 r € k1
whereas Coon and Liu are left with a nonzero 1/ and the complications of a
nonorthogonal basis.
From a physical viewpoint, the 1/ = 0 answer is the most satisfactory. It is
connected with the idea of conductance between two independent reservoirs.
Since the left- and right-incoming states are associated with the left and right
reservoir, respectively, no coupling between these states is expected. Let us work
out this idea of independent reservoirs by introducing a statistical or density
operator p that is the product of the density operators PL and Pa of the left and
right reservoir. Since we assume that both reservoirs are close to equilibrium at
the same temperature 1//3 but with different chemical potential P.L and µ.1:
(Vb being the voltage difference between the reservoirs), we can approximate
58 Chapters
their density opera.tors by that for a grand canonical ensemble with appropriate
µ. Although PL and Pa represent a thermal equilibrium, their product p = PLPR. does certainly not. Using this p we can define the current density as the
statistical average of the current density operator J:
<J> = Tr{pJ} (3.27a.)
and since J is a single-particle opera.tor this expression is equal to19:
<J> = }] <nlflm> <mlJln> (3.27b) nm
where {In>} is any orthonormal set of one-particle states. Ta.king the states of
(3.25) as this set, we translate p = PLPB. into the matrix <nl fl m>:
(3.28)
With respect to the same basis, the matrix <m I JI n> is found to be:
where the current density opera.tor J(z,z') reads:
J(z,z') = ~[8~,fi(z'.z)m(!') + m(!')fi(z'~)8~,] In principle, the Ja13 ca.n depend on z, a.s can be seen from (3.29). In the static
situation considered here, however, all components a.re z-independent. In fa.ct,
we use this property to establish some relations between the transmission a.nd
reflection coefficients. Working out (3.29) we find for Ja13:
19W. Jones and N.H. March, Theoretical solid state physics, Vol. 1, London: Wiley-lnterscience, 1973; Appendix A3.6.
Coherent Tunneling
so that:
N·tN'·te!kl(-p*'f)] •
:&-~!k1(-T)
N· 1 :N-1 e !kt( r*p) l · :N-2e!kr(:R-l)
which relations express the conservation of probability.
59
Z-1-m
(3.30)
Z -1 +m
Combining (3.27-29) we obtain for the current density, since f is diagonal, a
simple sum of only two terms:
(3.31)
which is in fact the Duke expression with which we started this section. Hence
we have found that there is no need for non-orthogonality corrections as
proposed by Coon and Liu, nor for (1-f) factors a.s introduced by Duke.
3.5. Chemical potential
Our next subject, a.gain. receiving little attention in the existing resona.nt
tunneling literature, is the choice of the Fermi level in the reservoirs that
sandwich the DBRT structure. Nevertheless, this para.meter is of utmost importance, both to the width and height of the current peak.
At first instance, the Fermi level may seem an issue of little controversy. It
is simply determined by the condition of charge neutrality20. Denoting the
density of electrons, donors and acceptors a.s n, Nd and Na, respectively, we can
express this condition a.s:
20See e.g. K. Seeger, Semiconductor Physics, Berlin: Springer, 1989; ch. 3.
60 Chapter 3
(3.32)
where Ndn is the number of occupied donor states:
(3.33)
Here, Eco is the conduction band minimum, Ed is the energy of the donor level,
and g is the impurity level spin degeneracy21• /3 is the inverse temperature.
Inserting for the electron density:
(3.34a.)
where Ne is the effective density of states in the conduction band:
[ m ] 2/3
Nc=2~ (3.34b)
and "112(17) is a Fermi-Dirac integral of order':
(3.34c)
we have in (3.32) an equation for the Fermi energy Ef· This solution however is
never used in resonant-tunneling.
The reason for this absence must be sought in the fact that (3.32) is
appropriate only for situations with localized donors, i.e. small Nd· At higher
donor densities, interaction between individual donors becomes significant, the
discrete levels shift towards the base of the conduction band (Eco-Ed ... 0), and
broaden into an impurity band. "Impurity meta.111 is the term used for these
doped semiconductors22. The critical density at which transition to metallic
behaviour takes place is given by the Mott condition 23:
21See e.g. J.S. Blakemore, Semiconductor Statistics, New York: Pergamon, 1962. We will only consider the case of a single simple monovalent donor species and take g to be 2. 22Blakemore, op. cit. (1962). 23B.K. Ridley, Quantum processes in semiconductors, Oxford: Clarendon, 1988.
Coherent Tunneling 61
> CD ->-CJ a: w z w
== a: w LL
Fig.3.5
0.02 0.6 0 -- ........... a:
' 0
" z
' 0 c c
" 0.4 w
0.00 !:::! \ z \ 0 -
\ LL
0.2 0 z 0
-0.02 -I-~ a:
0.0 LL
0 100 200 300
TEMPERATURE (K)
Fermi enerw vs. temperature for the cases of localized donors (Eq. {9.92), solid line}, weak impurity metal (Eq. (9.95}, coinciding with solid line}, and complete ioization of the donors (Eq. {9.96), broken line). Also plotted is the fraction of ionized donors (ri~ht a:r:is) according to Eq. {9.99). The donor concentration is 2· 102 m·3, the donor level is 5.899me V.
where ax• is the effective Bohr radius e li.2 'If~ o"· For GaAs, ax*,., lOn.m, so that
the critical concentration is ,., 2· 101 /cm.3• An experimental estimate of 5·1Q16/cm3 is reported24.
In the case of weak impurity metals, the density of states will be but little
affected, the impurity band essentially adding gNd non-localized states at energy
::: Eco· The condition for the Fermi energy now reads:
24B. Zimmerman, E. Marclay, M. Ilegems and P. Gueret, J. Appl. Phys. 64 (1988) 3581.
62 Chapter 9
(3.35)
Also this solution is not met in the resonant-tunneling literature. For strong impurity metals, there is an appreciable overlap of the broad impurity band and
the conduction band, the density of states of which is now seriously perturbed. A mathematical description of this regime is far beyond the scope of this section25.
The approach generally met in the literature on DBRT modelling26 is to
assume that in the impurity-metallic situation, even at low temperatures, all
donors are ionized, and that the Fermi energy is given by:
(3.36)
which, together with the solutions of (3.32,35), is plotted as a function of temperature in Fig.3.5. This approach, however, neglects the considerable effects
on the density of states of the original conduction band. Most pronounced these effects should be on the lower energy states. Apart from scattering effects, this
impurity banding might be a partial explanation for the large valley currents in
DBRT devices, where low energy states and band tails play an important role.
3.6. Inelastic scattering in the Jonson-Grinewajg model
Although energy band considerations are important, they do not lead to
corrections that can account for the fact that the experimentally observed
peak-to-valley ratios (P.V.R.) are not up to the coherent predictions. Some
inelastic scattering mechanism has to be invoked for that purpose. One approach
to incorporate inelastic effects was proposed by Jonson and Grincwajg27. Without specifying what scattering process ca.uses the loss of coherence, their method
attributes to a DBRT structure a number, that serves as a measure for the
2'See for a theoretical treatment of impurity bands: I. M. Lifshits, S.A. Gredeskul and L.A. Pastur, Introduction to the theory of disordered systems, 1988; ch. 6. 260ften without comment; to mention a few: R. Tsu and L. Esaki, Appl. Phys. Lett. 22 (1973) 562; P. Gueret, C. Rossel, E. Ma.relay and H. Meier, J. Appl. Phys. 66 (1989), 278; L. Eaves, F.W. Sheard and G.A. Toombs, ch. 5 in: Physics of quantum electron devices, ed. F. Capasso, Berlin: Springer, 1990.
27M. Jonson and A. Grincwajg, Appl. Phys. Lett. 51 (1987) 1729.
Coherent Tunneling 63
amount of incoherent tunneling. A number of authors28 have applied this method
to their calculations.
Jonson and Grincwajg begin their considerations with the series expansion of
the transmission amplitude 1' in Section 3.2:
1' = r2(l-PtP2)°1T1 = T2• l (PtP2)n· T1 (3.37) n•O
then-th term of which corresponds to an n-fold reflection (pJP2: back and forth)
between the two barriers. Full coherence (as can be seen from (3.37))
corresponds to adding the amplitudes before squaring:
{3.38)
Full incoherence would come down to the _opposite, i.e. the addition of
probabilities, which yields a transmission probability equal to:
(3.39)
This incoherent transmission probability obeys a real sequential expression: it is
equal to the probability· P 1 for an electron to reach the well, times the
probability P 2/(P1+P2) to leave the well at the collector side, given the fact that
the electron is in the well. As is immediately clear from (3.39), this sequential
expression can never explain a negative differential resistance. The approach by
Jonson and Grincwajg can be situated somewhere in between (3.38) and (3.39):
every internal reflection is associated with a loss of coherence expressed in a
number 7 between zero and unity. Now (3.37) is replaced by29:
28See e.g. D. Lippens, J.L. Lorriaux, 0. Vanbesien and L. de Saint Pol, in: Proc. of the 16th Int. Symp. on Gallium arsenide and related compounds 1989, Bristol: IOP, 1990; T.G. van de Roer, J.J.M. Kwaspen, H.P. Joosten, H.J.M.F. Noteborn, D. Lenstra and M. Henini, Physica B 175 (1991) 301. · The latter reference relates the Jonson-Grincwajg approach to the use of a complex potential (damped wave !unction), a method not unknown to nuclear physicists. See also: P.J. van Hall and J.H. Wolter, Superlattices and Microstructures 8 (1990) 305. 291n Lippens et al. {1989) the h in the denominator is erroneously absent.
64
In the same way, we find for the other amplitudes:
'f( 'Y) = h· 'f 1 'f2 l--YP1P2
Chapter S
{3.40a)
(3.40b)
(3.40c)
(3.40d)
As a consequence, R('Y) differs in general from R('Y). Also, the sum of R('Y) and
P( 'Y) : T( ;)kr/k1 no longer equals unity: the difference (1 - P - R) is interpreted
as the fraction of electrons that are stuck inside the well without the possibility
to leave it coherently. The probability that these electrons (after a sequential
proces) end up in the collector is P2/(P 1+P2). Hence the total transmission and
reflection probabilities are:
p Ptot = P(;) + [1-P('Y)-R('Y)Jrjir;, Rtot = 1-Ptot (3.41a)
These transmission probabilities are equal only in the special cases that 'Y = 1 or
'Y = 0, corresponding to full coherence, Eq.(3.38), and full incoherence,
Coherent Tunneling 65
~ VJ z w c 1-z w cc cc ::::> 0
Fig.3.6
Lcoh•20nm
10• ......... ":
1 o•
104
101
101 ......
........ ·· ..
.·
. . . . . .
······· Lcoh•lnf
101 .......,.·-·~~-'-~~~'--~~~~~~~~~~~
0.05 0.10 0.15 0.20 0.25 0.30
BIASM
1-V characteristics for the case of feU coherence (infinite f.c, dotted line) and finite le {!O nm, solid line).
Eq.(3.39), respectively3o. For 'Y close to unity, both transmissions are peaked at
cosa = 1 ( 'Y has no effect on the resonance position), with a half-width r a.
(HWHM):
r a. = 2· arcsin!fo'C'YJftiR2)) -./( 7./R1R2))] {3.43)
where in the first term the coherent width of {3.13) is recognized, and the second
term "'{14) represents the extra broadening of the transmission peak due to the
inelastic scatteringa•.
In order to make the para.meter 'Ya physically appreciable quantity, Jonson
and Grincwajg relate it to the inelastic scattering time t15 through
'Y = exp(-t/tu), where t is the time associated with one reflection inside the
30Therefore, the current expression as met in Jonson and Grincwajg {1987) and Lippens et al. (1989) is wrong.
UThis result is mathematically more precise than the (14) mentioned in Jonson and Grincwajg (1987).
Coherent Tunneling 67
well: t = 2mww/flkw with kw the local wave number. However, it seems more natural to make a connection 'between 'Y and the path length I. traveled by the electron during one reflection. H le denotes the coherence length, we write:
(3.44)
so that now 'Y will depend 32 on the lateral momentum k112• The large lateral
momenta have smaller 'Y values and are therefore more likely to be transmitted incoherently. Due to this dependence, 'Y(k11
2=0) can be close to unity and yet yield a small PVR. In Fig. 3.6 two I-V characteristics are presented, one for 4. = 20 nm, the other for an infinite le· The temperature is in both cases 4.2 K. The effect of a finite le is to decrease the peak current and to increase the valley current, thus yielding a smaller PVR. These three quantities are plotted vs. 'Y in Fig.3.7 for two different temperatures. At high temperatures, we see that the valley current even decreases with increasing le· At all temperatures, however, the largest effect is on the peak current, that collapses for 'Y just below unity. The resulting PVR is reduced by a factor of 100 for 'Y =i 0.9 (T = 4.2K) compared to the full-coherence situation.
a2le could be inversely proportional. to the cube root of the defect density. When interface roughness is considered, the kw2-term in the denominator of I. should be canceled, ·see Van de Roer et al. (1991). Furthermore, we prefer to regard 'Y as a probability amplitude and 12 as a probability.
68 Chapter9
chapter4
THE SELFCONSISTENT ELECTRON-POTENTIAL
4:.1. Introduction
Application of an electric field introduces a difference in chemical potential
between the contact regions of the DBRT structure, that serve as reservoirs, and
hence causes an electric current to flow through the device. Charge displacement
induces an additional field, which poses the problem of selfconsistency. The
selfconsistent field was shown to be responsible for a bistable1 or tristable2
current. Experimental evidence for this predicted phenomenon was not
considered convincing3, until a simultaneous monitoring of the charge build-up in the DBRT structure was performed4.
1V.J. Goldman, D.C. Tsui and J.E. Cunningham, Phys. Rev. Lett. 58 (1987) 1256; V.J. Goldman, D.C. Tsui and J.E. Cunningham, J. Physique C5 (1987) 463. 2D.D. Coon, K.M.S.V. Bandara and H. Zhao, Appl. Phys. Lett. M (1989) 2115.
3T.C.L.G. Sollner, Phys. Rev. Lett. 59 (1987) 1622, comment; V.J. Goldman, D.C. Tsui and J.E. Cunningham, Phys. Rev. Lett. 59 (1987) 1623, reply.
In this chapter, the effect of an external static electric field is described
within the BenDa.niel-Duke model for the conduction band, introduced in the
previous chapter. In Section 4.2 the effect of charge build-up in the doped
contact layers is studied. This space charge is of importance mainly to the
voltage axis of the I-V characteristic, although the current scale is affected also.
The charge build-up in the well is the topic of Section 4.3, in which its relation
to the current bistability is clarified.
4.2. The accumulation and depletion layer
The electric field in the DBRT structure due to the applied bias voltage Vb is
accompanied by space charge in the doped contact regions to the left and the
right of the central intrinsic layers. Generally, a charge sheet of zero width is
assumed, yielding a constant field inside and no field outside the central
structure5 8• Such a potential profile is reasonable for structures having heavily
doped electrodes that extend up to the barriers. In the case of moderately doped
electrodes or undoped spacer layers, it does not apply. Screening lengths on
either side of the central layers are then to be introduced7 s, A more realistic
charge distribution would extend into the doped layers, causing substantial band
bending according to Poisson's equation:
V[1'(z)· VEco(z)] = ep(z)/ £0 (4.1)
where Eco(z) is the conduction band minimum, and " is the static dielectric
constant that may be different in layers of different material (x1 ,B for material
A, B). p(z) is the charge density that can be written as:
p(z) = e[Nd(z) - n(z)] (4.2)
and G.A. Toombs, Electr. Lett. 24 (1988) 1190; C.A. Payling, E. Alves, L. Eaves, T.J. Foster, M. Henini, O.H. Hughes, P.E. Simmonds, J.C. Portal, G. Hill and M.A. Pate, J. Physique C5 (1987) 289; C.A. Payling, E.S. Alves, L. Eaves, T.J. Foster, M. Henini, O.H. Hughes, P.E. Simmonds, F.W. Sheard and G.A. Toombs, Surf. Sci. 196 (1988) 404. 5R. Tsu and L. Esaki, Appl. Phys. Lett. 22 (1973) 562. 6F.W. Sheard and G.A. Toombs, Appl. Phys. Lett. 52 (1988) 1228. 7V.J. Goldman, D.C. Tsui and J.E. Cunningham, Phys. Rev. B 35 (1987) 9387. 8C.A. Payling, E.S. Alves, L. Eaves, T.J. Foster, M. Henini, O.H. Hughes, P.E. Simmonds, F.W. Shea.rd and G.A. Toombs, Surf. Sci. 196 (1988) 404.
The selfconsistent electron-potential 71
where Nd(z) is the doping profile, and n(z) the density of conduction band
electrons. The boundary conditions for (4.1) are Eco(-m) = O, Eco(+m) = -eVb.
Since the total structure must remain charge neutral, an additional restriction is
obtained from:
J+m
dz p(z) = 0 --Ill
(4.3)
which reads in terms of the electric displacement D(z) : e0x(z)F(z), F(z) the
electric field:
D(-m) = D( +(I))= 0.
To solve (4.1) we need an expression for the electron density n(z). An
approximate expression for n(z) ca.n be borrowed from the well-known Thomas
Fermi screening theory9, generalizing the equilibrium expression:
(4.4)
[ m ]3/2
Nc=2· ~
valid for constant Eco to cases where Eco{z) varies slowly with position. Let us
denote the well and barrier widths by w, b1 and b2, and choose z = 0 to be the
middle of the well. Writing:
and zero otherwise, and:
n(z) = Nc.5'112(/1(µ1 - Eco(z))) ,
= Nc.5'112(/3{µ,;- Eco(z))) ,
z < -b1-w/2
z > w/2+b2
(4.5a)
(4.5b)
9N.W. Ashcroft and N.D. Mermin, Solid sto.te physics, 1976; p.340. For the use of TF screening theory in quantum device modeling, see W.R. Frensley,· in: Nanostructure Physics and fabrication, eds. M.A. Reed and W.P. Kirk, San Diego: Academic, 1889.
72 Chapter 4
where the chemical potentials are given by:
we can solve the system (4.1-2) numerically. There are, however, many reasons
(to be mentioned later) to not take this solution very seriously, especially for
large bias Vb· Here, we will present only a drastic simplification.
In this simplification, we do not link the functions n(z) and Eco(z) to ea.ch
other, but their average values na and Ecoa in the accumulation layer, to the left
of the first reservoir, and nd and Ecod in the depletion layer to the right of the
second reservoir. Now (4.5b) becomes:
n(z) = na = Nc~12(.8{µ1 -Ecoa)), -La-bi-w/2 < Z < -b1-W /2 ( 4.6)
and zero otherwise. In ( 4.6), La and Ld are the widths of the accumulation and
depletion . layer, to be determined by a selfconsistent solution of the system.
Subsituting ( 4.6) in Poisson's equation ( 4.1 ), we obtain the potential drops in
both charge layers:
(4.7)
Charge neutrality ( 4.3) yields:
(4.8)
The electric displacement D in the central layers inside the "capa.citor11 is: D =
e(na-Ndo)La = e(Nd0-nd)Ld, so that the voltage drop across the well and
barriers is found to be:
The total voltage drop across the total structure including the electrodes must
The selfconsistent electron-potential 73
equal the applied bias :voltage, which boundary condition reads:
(4.9)
Eqs.( 4.6-9) constitute a system of six equations with six unknown quantities: na,
Ec0a, ta and nd, Ec0d, td, that can hence be sol:ved10•
The lengths ta and Ld ate of the order of the Debye or Thomas-Fermi
screening length, as can be seen as follows. For the accumulation layer,
Eqs.(4.1-4) can be combined into a single second-order, non-linear differential
equation:
where we have made use of the property of the Fermi-Dirac integrals, that
_(j) dJ :JI n (x) : (1il'"t(x) = "t-J(x). We can formally sol:ve this equation by substituting:
m
Eco(z) = p-1 l amexp(mqoz) m•t
which automatically satisfies one boundary condition: Eco(--m) = O. This lea:ves
one parameter, a11 to be determined by the other boundary condition, while all
other coefficients are recursi:vely related to a1:
The in:verse screening length q0 is found to equal:
(4.10)
19H.J.M.F. Noteborn, H.P. Joosten, D. Lenstra and K. Kaski, SPIE VoL 1675 Quantum Well and Supperlattice Physics IV, 1992, p.57.
74 Chapter 4
which for high temeratures approximates the inverse Deb ye length 11:
and for low temperatures reduces to the well-known Thomas-Fermi expression12:
where kr is the Fermi wave number (311'2Nd0) 113, and aH* is the effective Bohr
radius. Hence for la.rge negative z the band minimum Eco decays exponentially
over a length 1/q0• For small Vb, we find for the system (4.6-9) that:
>. = e • ~312(,BEr) 4qo \1"6+~ob1+qow+~ob2 ~112(.8Er)
Thus La is a decreasing, Ld an increasing function of Vb.
In the same way, the average potential in the accumulation layer is found to
be:
(4.11)
We now introduce an effective Fermi energy by writing (4.6) for the
accumulation layer as na = Nc"'i12(/fEreff). Hence: Ereff = Er - Ec0a. Through
the last term, the effective Fermi energy will depend on the applied bias. For
small Vb, we have, using (4.11):
11B.K. Ridley, Quantum processes in semiconductors, Oxford: Clarendon, 1988. 12c. Kittel, Quantum theory of solids, New York: Wiley, 1963 (p.112 without 11'); W. Jones and N.H. March, Theoretical solid state physics I, New York: Wiley, 1973 (p.408 with 11').
The selfconsistent electron-potential 75
1 6
5 ... 0
z • ;--.. Q.
" ·1 z • =--Q. ·2
!J 4
0 CS'
3 ~
0 CS'
2 a 1
·3 0 0.0 0.1 0.2 0.0 0.1 0.2
V0 M V° (V)
0.6
o.s T•77K
~0.4
> :; 0.3
:: 0.2
0.1
0.0 l"'"=::::::..~-_.._~--X-.........J
0.0
Fig.4.1.
0.1 0.2 0.0 0.1 0.2
v• CV>
(a) Charge densities in acetimulation and depletion region, relative to the doping concentration eNd, w. the potential drop across the central layers VC. (b} Lengths of acetimulation and depletion region, relative to the screening length 1/q0, 118. VC. (c} Potential drops across the acetimtdation region va, the central undoped layers VC, and the depletion region Vd 118. VC. ( d} Effective Fermi level in accumtdation and depletion region 118. vc.
76 Chapter 4
(4.12)
In Fig. 4.1, the screening lengths La and Ld, the charge densities pa : e(Nd0-na)
and pd : e(Nd0-nd), the effective Fermi levels for the accumulation and for the
depletion layer, and the potential drops across the various layers, obtained from
numerically solving (4.6-9), are plotted as functions of the applied bias voltage
Vb· It is seen that the linear dependence of ( 4.12) is quite accurate over a long
range of voltages.
The above model for the accumulation and depletion layers is in fact an
improved version of the one presented by Joosten et al. 13 In the original version,
the depletion layer was left out of consideration. The constituting equations then
become:
(4.13)
Here, Ls is to be considered an adjustible parameter. The system (4.13) was
developed with an eye to DBRT structures having undoped spacer layers
adjacent to the barriers. For such structures Ls is thought to be somehow related
to the spacer width, as can be seen from the fact that Ereff has no doping
concentration dependence. Eq.( 4.12), on the other hand, applies to heavily doped
electrodes with no spacers.
The function Ereff(Vb) can be determined experimentally from magneto
tunneling measurements14• When a magnetic field perpendicular to the barriers is
applied, oscillations in the current are observed that are periodic in 1/B, with
periodicity 1/Br: (mElffjtie)-1• Hence, "measuring" the fundamental field Br at
various biases yields a plot of Ereff vs. Vb· The magneto-tunneling results of
13H.P. Joosten, H.J.M.F. Noteborn and D. Lenstra, Thin Solid Films 184 (1990) 199; H.J.M.F. Noteborn, H.P. Joosten and D. Lenstra, Phys. Scripta T 33 (1990) 219. The calculations presented in chs. 5 and 6 are based on this model. 14See ch. 6.
The selfconsistent electron-potential 77
-t:. a:f
Fig.4.2.
15
10
5
0 0.00 0.05 0.10
.· ..
0.15 0.20
Fundamental field Br, proportional to the effective Fermi energy, vs. bias voltage Vb· The squares are the measurements of Payling et al.; the bold curve is the solution of Eqs.(..1.6-9); the dotted curve is the solution of (..1.13) with L5 = 15 nm.
Payling et al.15 are reproduced in Fig. 4.2, together with the theoretical curves of
both (4.6-9) and (4.13)16.
Although the Thomas-Fermi approach (4.4) can be (and has been17) exploited in
numerically far more sophisticated way than is done in our constant-p model, we
nevertheless stick to the latter crude approximation. The fact is, that
Thomas-Fermi is unable to deal with the quantum effects that are dominantly
present near the barriers1B. Since the electrons cannot penetrate very far into the
1sc.A. Payling, E.S. Alves, L. Eaves, T.J. Foster, M. Henini, O.H. Hughes, P .E. Simmonds, F.W. Sheard and G.A. Toombs, Surf. Sci. 196 (1988) 404.
16The structure parameters used are those of Payling et al. except for the contact doping Ndo = 2.1017 cm-3, which is replaced by 1.1017 cm-3 to give the same Br at Vb = 0. The parameter 15 is chosen 15 nm.
17T.G. van de Roer et al., {Proc. 16th Int. Symp. on) GaAs and related compounds 1989, Bristol: IOP, 1990; p.831. 1BG.A. Baraff and J.A. Appelbaum, Phys. Rev. B 5 (1972) 475.
78 Chapter,/.
barrier, the amplitude of their wave functions will be small near the barrier.
Consequently, the electron density is minimal just before the barrier, where Thomas- Fermi predicts a maximum. The Friedel type of oscillations in the
density that result from this repellence are of course absent in the semiclassical
result. Furthermore, the triangular well in front of the emitter barrier that is formed at finite bias, gives rise to bound states19 20. Hence the accumulation is 2D
in character, while the Thomas-Fermi result is a 3D one. Thus one should not
consider these Thomas-Fermi densities too realistic. Surprisingly, the potential
profiles Eco( z) obtained from the semicla.ssical ( 4.4) are quite a.curate, especia.lly for the accumulation layer, compared to self-<:onsistent quantum-mechanical calculations21. Our constant-p model (4.6-9) can therefore be motivated thus: for
the potential, a crude approximation to ( 4.4) already suffices, for the charge
density, the exact solution of (4.4) still fails.
19Since these states extend into the collector, they are, strictly speaking, not bound states, but resonances with a finite width, very similar to the "well-based" resonances that are central in resonant tunneling.
2osee e.g. T.G. van de Roer, O. Abu-Zeid, H.C. Heyker, J.J.M. Kwaspen, H.P. Joosten, H. Noteborn, D. Lenstra and M. Henini, Procs. 15th annual semicond. conj. CAS 1992, Sinaia Romania, 1992i p.557.
21Y. Ra.jakarunanayake and T.C. McGill, J. Va.c. Sci. Technol. B 5 (1987)1288.
The selfconsistent electron-potential 79
4.3. Se1fconsistent study of coherent tunneling through a doub1e barrier structure22
Abstract - We present a model of the double barrier resonant-tunneling diode (DBRTD), in which the tunneling is described in a 1D transfer matrix approach, based on full wave coherence, and in which the electronic potential is determined selfconsistently from the 3D charge distribution in the structure. Within this simple model, we are able to describe the diode's intrinsic bistability. Results are presented in the form of I-V-<llaracteristics for GaAs-AlGaAs structures. Our approach is evaluated with respect to existing models.
1. Introduction
The Double Barrier Resonant-Tunneling Diode (DBRTD for short) is a new
device with interesting electronical and physical features. Its nonlinear behaviour
with negative differential resistance as well as possible bistable behaviour and.
hysteresis gives it very interesting potential applications [1], [2], [3]. The physics
is interesting as its functional properties are directly based on, and in fact
demonstrate fundamental quantum mechanical phenomena, a characteristic it
has in common with other mesoscopic systems [4]. In this paper we set ourselves two aims. Firstly, we will give a description of the
DBRTD's operation in the context of a fully quantum mechanical treatment,
implying coherent wave propagation and the selfconsistent electronic potential.
Secondly, we will compare our model to other approaches, especially to those not
assuming wave coherence, but using the alternative mechanism of sequential
tunneling. In this respect we mention the work by Luryi [5], Goldman et al. [6], and Sheard and Toombs [7]. Important point of comparison will be the intrinsic
bistability in the I-V-<llaracteristic of the DBRTD.
Let us start by giving a short description of the DBRTD: the diode consists of
several layers of different semiconducting materials (often GaAs and AlGaAs),
doped and undoped ones. We will concentrate on the five central undoped layers:
the well sandwiched between two barriers, in turn surrounded by two so-called
spacer-layers, in all about 200 A long. All other layers in front of (behind) this
central part are conceived of as an ideal reservoir (sink) of thermally distributed
22This section was previously published as a paper: H.J.M.F. Noteborn, H.P. Joosten and D. Lenstra, "Selfconsistent stud)". of coherent tunneling through a double-barrier structure11
, Phys. Scripta T33 l1990), 219-226. · References between square brackets [ ] are listed on page 96.
80
Fig.1
(a) c..
servoir .._Cll ............ QI ltl -sr
re
Ga As GaAs (doped)
Lsp
(b)
energy
Hl
first well barrier
Al0•4G<Jo.6As Ga As
L1 L'w
Eo
Chapter 4
.._c.. second sink ..c QI
barrier en~ ·-c.. "-111
A10.4Gao.£fs GaAs Ga As !dope d)
L2 Lsp
(a) Layer structure of the DBRTD. (b) Schematic diagram of the corresponding electron potential energy under the application of an external bias voltage VB (dashed line); modeUing of the conduction hand edge: in each layer the potential energy is replaced by its average value (solid line).
particles (see Fig. la). This conceptual description or electrical conduction
associated with coherent tunneling or particles between reservoirs was developed
by Landauer [8] and Biittiker et aL [9].
Because of the difference in bandga.p energy between the two semiconducting
materials, electrons experience a transition from one layer to another as a
The selfconsistent electron-potential 81
sudden change in their potential energy (see Fig. lb). We assume an applied bias
voltage VB to this structure to have three effects: first, the reservoir is filled up
to a certain Fermi level (or, at non-zero temperature, in accordance with the
Fermi-Dirac distribution). This defines the electronic input into the structure.
Secondly, the build-up of electronic charge in the undoped layers implies
electrostatic potentials across the barriers (see Fig. lb). Finally, a current I will
result, flowing perpendicular to the barrier layers.
The I-VB curve of the DBRTD exhibits its characteristic features: non-linearity,
negative differential resistance, and a certain interval of VB where every VB corresponds to more than one current value. It is this I-VB curve that we want
our model to explain. Assuming wave coherence means considering the DBRTD a Hamiltonian system,
described by an ordinary ScbrOdinger equation. With a perfect layer structure,
the motion perpendicular to the layers can be separated from the motion parallel to the layers. Thus, we can use a 1D tunneling approach (section 2), restoring
the three-dimensionality when calculating charge and current densities (section
3). The interdependence of the electron potential and the electron density
necessitates a selfconsistent solution (section 4), which is worked out numerically
for a symmetric and an asymmetric structure (section 5).
2. Resonant Tunneling
The DBRTD's operation is based on two quantum phenomena: tunneling through a potential barrier, and resonance in a quantum well. Both are, in the
lD case, easily treated in the transfer matrix approach. To use this approach, we
suppose the electrons in the conduction band to be quasi-free particles of energy
E-Ee = 7i2Jt2/2m*, with m* the effective mass and Ee the bottom of the conduction band (envelope function formalism [10]). Thus, a flux of incoming
monoenergetic electrons can be described by a plane wave exp(ikz). In the
following we take Ee at the left spacer to be the zero of the energy scale (see Fig. ·
lb). Furthermore, we will use only one effective mass for the entire structure.
The electronic potential is approximated by a piece-wise constant potential: this
makes the model easier to handle, while it can be shown that no essential
changes are introduced with regard to the resulting current-voltage features.
[11]. Attributing to the first barrier (width Li. height H1i potential drop across the
barrier V1) a reflection coefficient R1 now means that an incoming wave '¢'1(z) =
82 Chapter 4
exp(ikz) of energy E=h2k2/2m* will be partly reflected by the barrier, resulting
in a wave 1/Jr(z) = JR1.exp(-ikz-iip1-i01), and partly transmitted, leading to a
wave 1/Jt(z) = J(},cl-R1)).exp(ik'z-i01), k'=J(k2+2m*V i/h2). The barrier is
completely described by the three functions R11 rp1 and 01 of wave number k or
energy E. Analogously, the second barrier can be characterized by quantities R2,
02 and rp2• To the two barriers together, seen as one complex barrier structure,
can also be attributed a reflection coefficient R which can be expressed in R1 and
~ of the separate barriers. However, we cannot simply multiply (1-R1) and
(1-R2) to find (1-R): the well between the barriers serves as a resonator, in
which transmitted and reflected waves may interfere destructively or
constructively. Taking into account this interference yields a total (1-R) given
where a= k'Lw and 7 = !{01+0rrp1+1P2+r). Since the energy dependence of 1-R is dominated by sin'( a-,,), the positions of
the maxima of 1-R are fairly well determined by the equation:
a(E) - 1(E) = nr, n=O,:t:l,:1:2,. .. (2)
This is the resonance condition and for every n we may fmd a resonance energy
Eres,n as a solution of (2). In the following we will consider one resonance energy
(the lowest; n=O) only. From (1) we see that:
corresponding to a peak-to-valley ratio of ((HJR1R2)/(1-./R1R2))2, which can
be, for R1=R2=0.998, as large as 10s. The first maximum is thus a sharp peak,
to which (if R1R2>0.0294) can be attributed a line width !::.a (full width at half
maximum), which can be derived straightforwardly from (1):
(4)
The set/consistent electron-potential 83
From (3) we see that for R1=R2 the maximum of (1-R) equals unity. However, a good measure of the resonance's weight is the area under the peak rather than its height. Integrating over a from 'Y-1?!' to 'f+!?r, we find this area S to be:
S('y) = Ida (1-R) = ?r (l-R1 )( l-R2). (l-R1~)
(5)
One might conclude from (3) to (5) that only R1 and R2 are important, and that we can ignore the phase shifts 81,2 and cp1,2 all together. However, R1 and R2 must be evaluated at E=Eres• and since Eres is determined by these phase shifts via (2), it follows that the phases play an essential role in this approach. If both R1 and R2 are close to unity, the peak is so sharp that, in integrations, we can approximate 1-R by a 6--function,
dEI 1-R(E) "'Q(i Eres S(a(Eres)) 6(E-Eres). (6)
From (6) we see that the double barrier filters out precisely the energy Eres= an energy channel is defined, through which the electrons with E=Eres can pass to the well and through the whole structure. The resonance is not a truly bound state, since there is always a possibility for the particle to leave the well by tunneling through one of the two barriers (i.e. the wave function lea.ks out of the well). This implies a broadening LlE of the resonance level. Although working with a time-independent SchrOdinger equation, we can use this LlE to make an estimate of the life time At, writing [12]:
(7)
where Vres = ./ 2CEres+Vi)_ This is by a factor 1/Lla larger than the time m•
needed to traverse the well. We will look at At as a measure of the time that the electrons spend in the well (the so-called dwell time, see Section 6). Let us now calculate the densities in the well and the spacer layers on the left and the right side of the structure. Since the wave functions are plane waves, we have to consider only one z-value in each layer. In the left-hand spacer, there are an incoming and a reflected wave:
I IP1I 2 + I v>rl 2 "' l+R(E) ' (Sa)
S4 Chapter 4
while in the right-hand spacer, there is only a transmitted wave:
(Sb)
In the case of the well, a little algebra leads to:
lv>weul 2111 (l-R(E)).~+l:-J<E!v 1) • (Sc)
Because of the energy filtering by the barrier structure, the electronic densities
both in the well and at the end of the structure contain the factor 1-R, thus
exhibiting the same resonant structure. However, this does not imply that the
electrons in these regions a.11 have the same energy, since the E in (S) is only the
part of the energy associated with the tunneling direction. To find the energy
distribution in the well and at the end of the structure, we have to consider the
problem in all three dimensions.
3. Three Dimensionality
Until now, we have only considered a lD tunneling problem. To calculate charge
densities, however, we need to take into account the other two dimensions as
well. Here, we use the fact that the energy in the tunneling direction is
conserved. Taking z to be the tunneling direction, we can separate in the 3D
Density Of States (DOS), g3, the contributions from the parallel and the
perpendicular directions:
(9)
so that we can write for the 3D electronic concentration n3:
(10)
Here, fFD(E) is the Fermi-Dirac distribution, the moments of which a.re known
The selfconsistent electron-potential 85
as the Fermi~Dirac integrals ~(11) (see [13]):
with f3 = 1/k8T the inverse temperature. Assuming plane waves in the x- and
y-direction, we can express the 2D electronic density n2(Ez) in terms of
.510(/3Er):
with Ne the effective number of states per unit volume in the conduction band
(without spin!).
The lD DOS g1(Ez,z) in (10) can be expressed in terms of the wave functions
W(z) determined by the transfer matrix approach (see section 2):
(12)
Eqs.(10)-(12) allow us to calculate the electronic concentrations at the beginning
of the structure (z=O), at the end (z=L1+Lw+L2), and in the well (z=L1+!Lw)· In the z=O case, the norm squared of the wave function is proportional .to (l+R)
(see (8)), which we approximate by 2 since R(E)d except for E=Eresi the remaining integral is the Fermi-Dirac integral of order ~:
(13)
(g5=2 takes into account spin degeneracy).
In the other two cases (z=L1+iLw and z=L1+Lw+L2), the norm squared of the wave function is proportional to (1-R) (see (8)), which we replace by the
6--function (6); thus we obtain for the concentration at the end ofthe structure:
n(L+L +L)-IJ'_Nf213
.%(R(1<'--E )) l( Eres+V, )~(1-R,fr(l-R2) 3 l w 2 - oi; w 0 fl...,,. res V' Eres+V 1 +V2 {1- 1R2) (l
4)
and in the well:
86 Chapter-I
(15)
In (14) and (15), R1 and R2 are to be evaluated at E=Eres· If we had taken into
account more than one resonance, there would have appeared a sum over all
resonances in these equations. Since ~(,B(ErEres)) is a rapidly decreasing
function of Eresi the contribution of higher resonances to the concentrations can
in most cases be neglected.
In the same way as we determined the concentrations, we can also determine the
current density Jz· Since Jz is independent of z, we can evaluate it at any
position. If we concentrate on the end of the structure, we have n3(L1+Lw+L2)
electrons per unit volume, all with the same velocity component in the
z-direction Vres = J(2(Eres+V1+V2)/m*). Using Jz = e Vres na yields:
J =ea_ N f 213 «s:(,6(1"--E )) t(2(Er es+ V1)) i(l-R1k( l-R2) . (l6) z ~ w vg ""!' res V' m* (1- 1R2)
With (16) we have found the current density as a function of Eres• EF and V 1•
The latter quantities should now be expressed in terms of the bias VB· Before
presenting how to do this selfconsistently, we use (15) and (16) to make a second
estimate of the dwell time in the well, dividing the (areal) charge density in the
well by Jz: 6.t = e.L .... n3(L1+!Lw)/Jz = Cf ~~~).Lw/vres [7],[12]. It is easily shown that this expression coincides with (7) if R2<<R1!::!l, as is the case in a
biased DBRTD. A further discussion is postponed to Section 6.
4.Selfconsistency
Given V 1 and V 2 we are able to determine the resonance levels, and R1 and R2 at
these energies. Given, in addition, the Fermi level EF, we a.re able to evaluate
the electronic concentrations in the structure and the current density. In reality,
the device is part of a.n electric circuit: a bias voltage Vb is applied to it, a.nd the
current is measured. So what we need in order to determine the I-V curve, are
the functions that relate EF, V1 and V2 to VB. Because we also want to include
the phenomenon of bistability, we will not look for Er(VB) etc., for these will
not be single-valued functions. In stead, we will . option for a parametric
description, using EF as a parameter, a.nd determine the functions V 1(EF ),
V2(EF) and VB(EF)·
The selfconsistent electron-potential 87
The central relation of our model then reads:
(17)
It states that both the filling of the left-hand reservoir (characterized by E.,) and the bending of the conduction band edge (expressed by V 1 and V 2) result
from the application of VB. The band bending is due to accumulation up charge in the central, undoped
layers of the DBRTD. Therefore, V1 and V2 can be related to the charge
concentrations at the beginning of the structure and in the well:
Eq.(18) is nothing but a simplified Poisson equation: in fa.ct, we have assumed
the charge to be concentrated in two narrow sheets, one positioned at z=O, the
other at z=L1+!Lw, and calculated the potential drops in capacitor analogy.
Substituting in (18) the results (13) and (15), we find:
(19a)
v = t2+ttw v + e2
tL +1L )g N 2/3~({3(1<'--E ))!(l-R1:k'l+R2) 2 1+ w t %' 2 2 w s c o ..,I!, res (1- 1R 2 ) •
(19b)
This would conclude our task, since (19) suggests that V1(Er) and V2(Er) are found. However, V2 depends on the resonance energy, and, in turn, to determine Eres we need to know V 2• This problem is another aspect of seHconsistency that
every model of the DBRTD encounters. In our simple model it only involves the
potential drop V 2 across the second barrier. More generally, seHconsistency is
the demand to solve the SchrOdinger equation and Poisson's equation simultaneously.
Before presenting our solution of the set.of equations (16), (17) and (19) (see
Section 5), we will introduce some simplifications, giving insight into the nature of the bistability and allowing us to make a direct comparison with literature,
Fig. 2 The I- V characteristics resulting from the simple approximation {20} at zero temperature; the parameter values used are those of the symmetric sample of Section 5.
especially with Sheard and Toombs [7]. Following [7], let us assume the
resonance energy to be constant with respect to the bottom 0£ the well (Eres + V1 = constant = E0); this is a very reasonable approximation as can be seen
from Fig.3a, where E0 vs. Er is drawn. (as calculated with the exact equations
{16)-{19)). Let us further assume zero temperature. We define the constants A,B
where O(x)=O (1) for x<O (x~O) is the unit step function. The two step functions
enter the equation because Eres = E0-V 1 has to be positive in order to have a
resonance, and Eres has to be smaller than Er, otherwise there would be no electrons able to reach the channel. In fact, R1 and R2 are still functions of EF,
thus blocking an easy solution of (20). But if we treat them as constants ( d [7]), we can readily draw the Jz-VB curve from {20); see Fig. 2. The unrealistic (nearly) triangular shape of this I-V curve is due to the fact that the factors
(l-R1){1:1:R2)/{1-R1~) are not constant, as can be seen from Fig.3b, where
these factors (as calculated with the original equations (16) to {19)) have been
drawn.
From (20) and Fig. 3, we see that there exists an interval of V8 for which every
V8 corresponds to three different Er- {and thus Jz-) values. This is the
aforementioned bistability, two of the three solutions being stable, the other
being unstable. The width of the bistability interval !::.. Vbist is according to (20) equal to:
(21)
If we substitute in {21) T1=1-R1 and T2=1-~ (neglecting the difference in wave
number) and further assume these transmission coefficients to be small, !::.. Vbist
turns out to be proportional to Ttf(T1+T2), in accordance with Eq.(8) of Ref.[7].
This proportionality makes clear that bistability is more pronounced in asymmetric structures than in symmetric ones [14]: for an unbiased symmetric
structure, T1=T2, while in the biased situation T2 will be larger than T1 (and
close to unity), so that the factor Tif{T1+T2) is of the order 'I\. For the
asymmetric structure, however, T1<T2 in the unbiased situation, while in the
biased situation T~Tb so that now the factor Ti/(T1+T2) is of the order unity.
The above discussion shows that the results of a sequential tunneling picture (in
90
Fig.3
Chapter 4
:; 0
UJ 0.1000
0.0990 0.060 0.120 0
Ef leVJ
0.600 ---- - - -- ...
~~ ....
...... ..... • q;: ...... = cX ' -;:.. I ,xsooo a: .... . - ' c
' .-JN
' 0300 ' '\ \
&" r£' \
):..
' - ct cF I ' ' .:c \ .... ' .-JN
0 0.()60 0.120
EF !eVJ
(a) The resonance ener!J1J 'With respect to the bottom of the well E0 is drawn as a /'Unction of the fermi level Er.; {b) The factors !(1-R 1 )(l +Rz) (solid line) and 171-R 1 }{l -RzJ (dashed line, z5000}
(1-R1R2) (1-R1R 2 ) as a function of the fermi level.&. The results in (a) and {b) are based on the ezact model el]'IJ.ations (16)-(19}.
this case Ref.[7]) can be retrieved in a coherent tunneling approach. In fact,
these results were found by making crude simplifications in our original
equations (16), (17) and (19). Since for us, th~re is no reason to hold on to these
The selfconsistent electron-potential 91
simplifications, we will drop them in the next section by dealing with the original equations directly.
5. Results
The approximation leading to (20) is very crude, especially in the bistable
interval. Therefore, to get a realistic 1-V curve, we have to solve the original
equations (19) iteratively with the help of a computer. This we have done for a
GaAs-AlxGa1.xAs structure. With x=0.4 a barrier height corresponds of 0.44e V
[lS]. For the lengths of spacers, barriers and well, we take: (L8p)-LrLw-L2 = (28.3)-56.6-50.9-56.6 A. The effective mass and the relative dielectric constant
are ta.ken to be the ones of GaAs at zero temperature: m*=0.067m0 and er=13.18 · [lS]. The results are shown in Fig.4, for T=4.2 K (Fig.4a), T=77 K (Fig.4b),
and T=300 K (Fig.4c). Temperature has the effect of smoothing out the curves,
while the current's maximum remains nearly unchanged. The bistability interval
is hardly noticeable in this symmetric case. The same calculations have been done for a configuration that differs from the
one described above only in one respect: the width of the second barrier L2 is
enlarged to 84.8 A. The results are shown in Fig.4 d, e and f. In this asymmetric
case a bistability interval of about 0.1 V is found, much larger tha.n in the
symmetric case.
Two features of the 1-V curves, the maximum current Jz,max and the width of
the bistability interval !J. Vb!sti are investigated with respect to their dependence on the various structure lengths. We have varied only one structure parameter
at a time, giving all other parameters the same values as in the symmetric
sample described above. The results are presented in Fig.5 a, b a.nd c. From
Fig.Sa. we see that Jz,max depends exponentially on L1• Fig.Sc is another
demonstration of the aforementioned relation between asymmetry and
bistability.
For the symmetric structure, a. direct comparison with experiment (Ref.[3]) is
possible: at 77 K the I-V curve was measured to have a. maximum corresponding to a current density of 1.0x107 A/m2 at a bias of 0.69 V. Our numerical results show a. maximum of 2. 7x107 A/m2 at a bias of 0.27 V. This discrepancy ca.n be
partly explained by the fact that our model does not include a subtle description
of the leads and contacts. A first attempt to take these into account as well led
to encouraging results. A further explanatory factor could be the fact that our
The I - V characteristics resulting from numerically solving the exact equations (16)-(19), for the spmmetric sample {L2=LJ, at T=4.2 K {a), at 77 K (b), and at SOO K (c); and for the asymmetric sample {L2>LJ, at T=,/..2 K {d), at 77 K {e}, and at SOOK(/). For the parameter values, see Section 5.
calculations were done with one effective mass only. Besides, a factor 2 or 3
mismatch is not really alarming, considering the exponential relationships in Fig.5. Therefore, the model is thought to give a good description of resonant tunneling.
The selfconsistent electron-potential 93
6. Discussion
We will now discuss our DBRTD-model in comparison with other approaches.
Pioneering work was done by Tsu and Esaki [1], whose theory was further
developed by Ricco and Azbel [14] and Mendez [12]. In their model, the
transmission is calculated assuming coherent wave propagation, while the Fermi
level is thought to be a constant, in analogy with metals. This would be correct
in moderately to highly doped semiconductors, but is certainly incorrect when
dealing mainly with undoped layers (like we do). The potential drops are not
selfconsistent, bistability is not reported.
A second model is proposed by Luryi [5]: here, the negative differential
resistance is explained by sequential tunneling: during the particles' stay in the
well, wave coherence is supposed to be lost. This picture is used by Goldman et al. [6], who combined it with selfconsistent potential drops and were thus able to
predict intrinsic bistability. Also, Sheard and Toombs [7] worked within the
sequential tunneling approach. Using a constant EF, they too were able to
explain bistability (see also the discussion around (20)).
For completeness, we also mention the work by Ohnishi et al. [16], who worked
out a sophisticated selfconsistent version of resonant-tunneling (no bistability
found); and the work by Kluksdahl et al. [17], who used Wigner distribution
functions (and found bistability).
On four points we will compare our model to the models of Refs.[1], [5], (6], (7],
(12], (14], (16] and (17]. First, we have presupposed wave coherence, and thus
worked within the resonant-tunneling approach. To defend this choice, it may
not be enough to point at the fact that the structure length (typically 200 A) is
much shorter than the electronic mean free path (of the order of lµm., TN77 K),
since, classically spoken, the particles are reflected many times between the
barriers. Therefore, the dwell time of the particles in the well (Eq.7) may be
compared to the scattering time. For the symmetric structure we looked at (see
section 5) this dwell time is circa 2 ps, while the scattering time is about 8 ps.
Contrary to Luryi and Mendez, we therefore conclude that for this kind of
structures, wave coherence is preserved anyway.
A second point of comparison is the choice of the Fermi level in the left
reservoir. Refs.(1] and (7] take this to be independent of the applied bias voltage,
and equal to the Fermi level in the right reservoir, both assumptions being quite
good in the case of metals or moderately to highly doped semiconductors.
Contrary to this (and in agreement with Goldman (6]), our Fermi level is the
94
N" E -~ x IU E N ......
~ .... "' :c
> <I
5 10 30
90
60
30
030 +-
Chapter 4
Lsplll
10 9 10 30 50
b
6 50 70 90 10 30 50 70 90
Ll'L2 ca1 Lwlll
'/ I I +·
I +·
50 70 90 L1.L2 Ill
Fig. 5 {a,b): The maximum of the current density ~.max as a fe,nction of one of the stru.cture parameters L1 {a, 0-0-0), £2 ca, +-+-+), Lw {b, 0-0-0, lower scale} and Lsp {b, +-+-+, upper scale}, keeping aU other parameter values the same as in the symmetric sample of Section 5. (c): The width of the bistability intenJal 6. VbiJt as a function of one of the stru.cture parameters L1 {o-o-o), and L2 c+-+-+), keeping all other parameter values the same as in the symmetric sample of Section 5.
The selfconsistent electron-potential 95
result of (and therefore appears in the formula for) VB (Eq.(17)). When VB=O,
we know the Fermi level to be somewhere below the conduction band edge,_ and
this speaks against a constant~· Thirdly, let us look at how the potential drops in the structure are determined.
In [1] the bias is simply divided by the number of barriers. All others think of
the band bending as caused by charge accumulation in the structure and thus as
determined by Poisson's equation. Only [16] solves this equation by integration,
others simplify it by two (our model, [7]) or five [6] potential steps.
Final point of comparison is the intrinsic bistability: since this is thought to be
the result of the charge build up in the well, only the selfconsistent models are
able to describe this phenomenon. That Ref.[16] did not find bistability must be
due to the fact that in the case of symmetric barriers - the case they examined -the bistable interval is very small, as was discussed below (21). This discussion
showed that the results of the sequential tunneling approach are retrieved in our
resonant tunneling model. This is not so surprising as it may seem. In fact, the name "sequential tunneling11 is not very appropriate: it does not mean that
probabilities in stead of amplitudes should be multiplied. For then there would be no resonant or 2D state in the well. In the resonant tunneling picture, this
state is not a bound state, but a true resonant state, i.e. broadened by the fact
that it can explore its surroundings beyond the barriers. In the sequential
tunneling, the presence of a resonant channel is simply assumed, not explained.
Its broadening, said to be due to collisions or inelastic scattering, is made
proportional to T2, the transmission of the second barrier. This combination,
then, approximately leads to the resonant-tunneling answers. In short, it seems
to us that many of the sequential tunneling approaches are in fact intuitive
versions of coherent resonant tunneling.
In conclusion, we have presented a simple model of the DBRTD, in which lD resonant tunneling based on phase coherence is combined with selfconsistent
potentials and a selfconsistent Fermi level. We have shown that the wave
coherence picture is consistent for the discussed parameter sets. The intrinsic
bistability is an easily understandable consequence of the model. In· spite of its
simplicity, our model yields realistic I-V-characteristics.
Acknowledgments - We thank W. van Haeringen and T.G'. Van de Roer for
stimulating discussions. Part of this work was supported by the Stichting voor
Fundamenteel Onderzoek der Materie (FOM).
96 Chapter 4
References
[1]. Tsu, R. a.nd Esa.ki, L., Appl. Phys. Lett. 22, 562 (1973). Chang, L.L.,
Esa.ki, L. and Tsu, R., Appl. Phys. Lett. 24, 593 (1974).
The revival of resonant-tunneling research in the eighties started with the
experiments of Sellner et al. 1 who used a GaAs/Al(Ga)As DBRT diode as a
detector and mixer of far infrared radiation at f = 2.5 THz. One year later, the
same group reported the observation of active oscillations from a DBRT diode2•
Output power of 5 µ.Wand frequencies up to 18 GHz were achieved with a de to
rf efficiency of 2.4 %. The expected increase in obtainable frequency and power3
1T.C.L.G. Sollner, W.D. Goodhue, P.E. Tannenwald, C.D. Parker and D.D. Peck, Appl. Phys. Lett. 43 (1983) 588. .
2T.C.L.G. Sellner, P.E. Tannenwald, D.D. Peck and W.D. Goodhue, Appl. Phys. Lett. 45 (1984) 1319. 3420 GHz reported in: E.R. Brown, T.C.L.G. Sollner, C.D. Parker, W.D. Goodhue and C.L. Chen, Appl. Phys. Lett. 55 (1989) 1777. 60 µ.W at 56 GHz in: T.C.L.G. Sellner, E.R. Brown, W.D. Goodhue and H.Q. Le, in: Physics of quantum electron devices, ed. F. Capasso, Berlin: Springer, 1990.
97
98 Chapter 5
was soon observed, although the predicted limits off N 1 THz and P N 225 µ.W
a.re not yet attained.
Estimates of the maximum oscillation frequency a.re obtained from the life
time of the resonant state (i.e. the inverse width of the transmission peak), in
combination with equivalent circuit a.rguments4• A Wigner-function study5 on
the temporal behaviour of the DBRT diode yields a. frequency limit of the same
order of magnitude. On the basis of a. time-dependent approach proposed by
Coon and Liu&, the diode impedance can be obta.ined7, yielding a. cut-off
frequency of, a.gain, "' r /21t0.. The method that is described in the next section,
takes a. different pa.th: the small-Bigna.l analysis presented in Sect. 5.2 stays close
to the static model of the previous chapters. The resulting diode impedance
corresponds to a.n equivalent circuit that is larger than the usual dynamic
conductance plus parallel capacitor. An additional RC circuit describing the
space charge effects in the well is found. Sect. 5.3 discusses the relation of this
equivalent circuit model to the quantum-inductance model of Brown et al. 8
The primary objective of the sma.ll-Bigna.l model was to investigate the de
stability of the DBRT diode. The intrinsic bista.bility9, experimentally observed
a.s a. hysteresis in the 1-V curveto, is theoretically described by a. Z-sha.ped I{V)ll.
This yields a. small range of voltages corresponding ea.ch to three static current
solutions, two of which a.re expected to be stable. The question, which of these
three configurations a.re de stable, is studied using a. sma.ll-Bignal analysis. In
addition, the effect of the external circuit on the diode stability is investiga.ted12.
4D.D. Coon and H.C. Liu, Appl. Phys. Lett. 49 (1986) 94.
sw.R. Frensley, Appl. Phys. Lett. 51 (1987) 448.
&D.D. Coon and H.C. Liu, J. Appl. Phys. 58 (1985) 2230; H.C. Liu, Appl. Phys. Lett. 52 (1988) 453. 7R.J.P. Keijsers, H.J.M.F. Noteborn, D. Lenstra., unpublished (1992).
BE.R. Brown, C.D. Parker and T.C.L.G. Sollner, Appl. Phys. Lett. 54 (1989) 934. .
9V.J. Goldman, D.C. Tsui and J.E. Cunningham, Phys. Rev. Lett. 58 (1987) 1256. 10E.S. Alves, L. Eaves, M. Henini, 0.H. Hughes, M.L. Leadbeater, F.W. Shea.rd and G.A. Toombs, Electr. Lett. 24 (1988) 1190.
11F.W. Shea.rd and G.A. Toombs, Appl. Phys. Lett. 52 (1988) 1228· D.D. Coon, K.M.S.V. Banda.ra. and H. Zhao, Appl. Phys. Lett. 54 (1989) 2115; H.J.M.F. Noteborn, H.P. Joosten and D. Lenstra., Phys. Scripta. T 33 (1990) 219. 12See also: H.P. Joosten, H.J.M.F. Noteborn, K. Ka.ski and D. Lenstra., Physica. B 175 (1991) 297.
Current Stability 99
5.2. The stability of the sel:fconsistent]y determined cunent of a
double-barrier resonant-tunneling diode13
Abstract - Double-Barrier Resonant...,Tunneling Devices belong to a novel class of nanoelectronic devices with great potential applications. In these devices, the selfconsistent build-up of charge due to resonant carriers in the well may lead to bistability and hysteresis. To investigate aspects of dynamical (in)stability, a simple set of equations is derived from an extension of the static theory. These dynamic equations adequately describe small and slow (<100 GHz) deviations from the stationary state. This approach is viewed more satisfactory than an equivalent-circuit analysis, but its limitations are also discussed.
1. Introduction
Double-Barrier Resonant-Tunneling (DBRT) structures have attracted much
attention, because they exhibit, among many other interesting features, intrinsic
bistability in the I-V characteristic. Experimentally, this bista.bility is
encountered as a hysteresis 11,2], whereas theoretically a. voltage interval with
triple-valued current (a. ~ha.ped I-V curve) is predicted [3,4]. One way to
harmonise these different data, is to declare one of the three calculated branches
in the I-V characteristic unstable, whereas the other curves are assumed to be
completed in a hysteresis-like way [4]. Another approach is to speak of intrinsic
trista.bility, assuming the third solution to be hard to access experimentally [5].
As the discrepancy between theory and experiments is still too large to
justify a complete identification of the hysteresis interval with the triple-current
interval, a stability criterion based on a theoretical analysis of the DBRT is
desirable. In this pa.per, we present such a stability analysis, resulting from
minor extensions to our static model [4,6]. In the first place, we release the
coupling between the charge densities in the emitter and the well, thus attaining
a. dynamical system with two independent variables. Secondly, we include a
model of the leads connecting the double barrier structure to the battery,
yielding a fair description of the way in which fluctuations are compensated in
this pa.rt of the circuit. In the present study, only the simplest case, that of a
purely ohmic contact, is considered.
13This section was published as a pa.per: H.P. Joosten, H.J.M.F. Noteborn, K. Kaski and D. Lenstra, "The stability of the self-consistently determined current of a double-barrier resonant-tunneling diode", J. Appl. Phys. 70 (1991), 3141-3147. References between square brackets [] are listed on page 114.
100 Chapter 5
2. Static mod.el
In some models of the DBRT [3,4], the bistability in the I-V curve corresponds
to a voltage interval where the voltage Vs across the structure is related to three
values of the current density J. Hence, there exists no single-valued function
J(V5) in the bistable regime, though we can give a parametric description of the
I-V curve. This is done by taking the Fermi energy EF in the emitter as
parameter: (V5(Ep), J(Ep )), where both V5(EF) and J(Ep) are single-valued
(but non-monotonic, hence non-invertible) functions. Expressions for V5(Ep) and J(EF) result from a. combined solution of the Poisson equation and the
Schrodinger equation: the charge densities calculated via. the wave functions
from the Schrodinger equation enter the ·Poisson equation for the electron
potential, while this potential is used in the Schrodinger equation for the wave
functions. Thus, V5(Ep) and J(EF) are determined in a.n iterative scheme, so
that on convergence the ultimate. electron potential and charge density a.re
qualified as "self-consistent".
For V5(EF) the a.hove method can easily be worked out. In the double barrier structure, charge build-up takes place in the emitter, well and collector. Since
the three charge densities are interrelated due to the demand of overall charge
neutrality, we need only consider two of them. Thus, the Poisson equation yields
a. function V5(Ee,Ew), where Ee and Ew are the areal charge densities in the emitter and well, respectively. Considering Ee to result from impinging electrons
in the energy range 0 to EF, and Ew from electrons in the same energy range
tunneling into the well, we can find expressions for Ee(EF) a.nd Ew(EF ). Now we
can write for the voltage over the structure V5(Er )= V5(Ee(EF ),Ew(Er )), where it is the Poisson equation that relates V5 to Ee a.nd Ew, and the time-independent
Schrodinger equation tha.t makes these charge densities in tum related to Er. H
we abandon the latter equation (as we should in a. dynamical analysis), V 5(EF) can be generalised straightforwardly to a. function V5(Ee,Ew) of two, now independent, variables.
The current density J(Er) needs a. different approach. Since J is defined
within the stationary wave function formalism, i.e. for selfconsistent solutions
only, giving up the time-independent SchrOdinger equation means giving up our
expression for the current density. In this case the generalisation, necessary for
the dynamic mod.el, is not straightforward, nor even unique. Let us start from the expression for the selfconsistent current density [4]:
Current Stability 101
(1)
Here, Nc=(2 ~'f) 312
is the effective number of states in the conduction band per
unit volume, ~(e) is the zeroth order Fermi-Dirac integral, Er and Eres are the Fermi energy and the resonance energy with respect to the conduction band of
the emitter; kT is the Boltzmann constant times the temperature. In the second
factor, &=2 takes into account the spin degeneracy, e is the electron charge, and
R1 and R2 are the reflection coefficients on the first and second barrier at the
resonance energy. Finally, m is the effective mass of electrons in the conduction
band, w is the width of the well, and E0 is the resonance energy with respect to
the conduction band in the well. Note that Eres and E0 are different quantities
that refer to the same energy level. The first factor in (1)
(2)
has an easy interpretation: it is the number of points in k-space per square
meter, for which I k I 2~2mEr /h.2 and kz=kres· In other words, N 0 is the areal density of resonant electrons in the emitter. In order to interprete the remaining
factors, a little rearrangement is helpful. The second factor multiplied by u+~2i - 2
yields
(3)
which is the charge per resonant electron located in the well. If the second
barrier were impenetrable i.e. fu=l, then q0 would be &e, as expected. Thus,
the reflection coefficients in (3) take into account the leaky nature of the well.
The product of the first two factors, N0 q0 , is the areal charge density in the well,
Ew· The last factor in (1) becomes now:
1 _ (l=fu) .J(2E 0 /m) ld-"{I+R2J w (4)
and its reciprocal td is called the dwell time of electrons in the well. This dwell
time is equal to the time needed for an electron to traverse the well, \/(2E:/m)'
times a+~2~. that takes into account the multiple reflections within the well. - 2
102 Chapter 5
The quotient q0/td represents the current I0 that each resonant electron
contributes to the total current density. Summarizing, it is found that
(5)
which offers us two starting points for generalising the current density in the
non-selfconsistent situations. Firstly, we can write J(Ee,Ew) =
N0 (Ee,Ew)·I0 (Ee,Ew), in which N0 is the areal density of resonant electrons and Io the current that each electron carries. Secondly, we can write J(Ee,Ew) =
Ew/td(Ee,Ew), as the quotient of the charge density in the well and the time
electrons spend there. For selfconsistent solutions, the functions Ee(Er) and
Ew(Er) are such that both generalisations yield the same relation
J(Er)=J(Ee(Er),l!w(Er)). However, when Ee and Ew are not coupled selfconsistently, the two generalisations differ. In the next section, where the
dynamic model is presented, it will be seen that both formulae are needed.
3. Extensions to static model
The static model of the DBRT outlined in the previous section, amounts to
handling the Poisson equation and the SchrOdinger equation simultaneously.
Since both equations are time-independent, no dynamics is included. In order to
investigate the stability, however, we have to deal with time-dependent charge
and current densities, so we cannot adhere to this set of equations. Nevertheless,
it suffices to give up the time-independent SchrOdinger equation. This is because
the electron potential adjusts itself to changes in the charge distribution effectively instantaneously i.e. with the speed of light.
Instead of replacing the time-independent SchrOdinger equation with its
time-dependent version (which is the formally correct method), we opt for a
different, approximative and intuitive, approach: the time-independent equation
effectuated a coupling between Ee and Ew, so that abandoning this equation
amounts to considering Ee and Ew as henceforth independent variables. Since in the dynamic case the current density has non~ero divergence, we need to
specify it for various positions in the structure: introducing the current densities
lt in the lead, and J 1 (J2) through the first (second) barrier (see Fig.I), we write
for the rates of areal charge density in the emitter and well:
Current Stability 103
Fig.l
emitter well collector
Conduction band minimum in the DBRT-structtr.re as a function of position z. Below: schematic picture of charge b'IJ.ild-up and current densities.
(6a)
(6b}
The third equation, for the charge density in the collector, is equivalent to the
condition of overall charge neutrality, and contains no new information.
If the simplest ex.ample of a lead, a purel! ·ohmic contact, is considered, we
can specify the current density Ji by the quotient:
(7a}
where p is the specific resistance of the lead and I. is its length. Vb(t} is the
extem~y applied battery voltage, and VsfEe(t}.Ew(t}} is the potential drop across the double barrier structure as· obtained from the Poisson equation.
In order to specify the current densities J 1 and J2, we turn to the two
generalisations of the previous section. A natural and appealing choice is:
(7b}
104 Chapter 5
and
(7c)
(7b) represents the current density as seen from the emitter, and (7c) as seen
from the well. In ( 7b) the instream of electrons from the emitter to the well is
proportional to the areal density of resonant electrons in the emitter, but the current they carry depends on the actual amount of space charge in the well. In
(7c) the outstream of electrons from the well to the collector is proportional to
Ew, but the time constant td depends on the actual amount of space charge in the emitter. This intuitive reasoning substitutes a formal derivation.
Eqs.(6) and (7) are an approximative model for the dynamics of the DBRT
structure. It may seem that the dynamics is described by a set of truly
non-linear equations. However, the way in which the expressions for J 1 and J2
are obtained, prevents us from taking (7bc) too seriously for ·large deviations
from the sel:Cconsistent solution. Therefore, a linearised version of (7) will be
investigated in the next section.
4. Linearisation of dynamic equations
Let us suppose that at t=O the DBRT structure has a sel:Cconsistent electron
potential and corresponding charge distribution. A small fluctuation in the
battery voltage Vb(t), e.g. a pulse function of small amplitude and finite
duration, will initiate a dynamical evolution of charge and current densities, that
may or may not be bounded at infinite time. If all quantities remain bounded,
the system originated from a stable point in the I-V characteristic. If, on the other hand, there is an unbounded quantity, the starting point is called unstable.
The boundedness is most easily checked by performing Laplace transforms of
the linearised dynamic equations.
Let for all quantities the difference between the value at time t and the value
at t=O be indicated with the corresponding lower case notation, e.g. v5(t) = V5(t) - V5(0), O'w(t) = Ew(t) - Ew(O) etc. In terms of these differences, and
leaving out all higher order terms, Eqs.(7) read as follows:
Ou.mint Stability 105
(8a)
(8b)
(8c)
where the coefficients Ae to Cw are partial derivatives in the original
selfconsistent solution:
- {JV Ae-~'
C _ -Ew~O),£!d e - td iJf.e '
In these expressions, td and ~ are to be understood as td(Ee(O),Ew(O)) and
~P!e(O),l::w(O)), etc. Substituting (8) in (6), we obtain the linearised
rat~uations:
and
(9b)
which are most easily solved using the Laplace transformation. Denoting the
Laplace transform of O'e(t) by ue(s) etc., we can cast (9) in matrix. form:
A ]["' l [Yll l Pf + Bw O'e = pl
s-Bw +Cw aw 0
(10)
which is solved by inverting the matrix. Thus we find:
106
"' _ s-B +Cw .!b. O'e - (s-s 1) ( s-s2) pl
where s1 a.nd s2 obey:
A St + 82 = - :.= - B + B - C pl e w w
S • 8 _ Aw(Be-Ce)-Ae(Bw-Cw) + {B C _ B C } I 2- pl e w we
Chapter 5
(Ua)
(Uh)
(12a)
{12b)
Eq.(12) is of utmost importance for the stability, as ca.n be seen from the inverse transforms of (11 ):
If s1 or s2 have a p0sitive real part1 the exponential functions in the integrands in (13) increase unboundedly as t goes to infinity. Hence, the criterion for stable charge densities ue(t) and uw{t) must be that both Re(s1) < 0 and Re(s2) < O, or1 since sum and product of s1 and s2 are real, that s1+s2 < 0 whereas s1·s2 > O. Combining this with (12) yields the stability criterion:
(14a)
(14b)
In the next section, the consequences of this stability result for the I-V characteristic will be investigated.
Current Stability 107
5. Stability and imped.a.nee
In order to appreciate the internal stability conditions (14), we will relate them
to the impedance of the circuit. For this purpose, we derive some useful relations
between the coefficients from Ae to Cw in (8). Since we assumed that at t=O the system was in a. selfconsistent point of the
I-V curve, the t=O charge densities Ee(O) and Ew(O) are interrelated in such way
that J 1=J2 (see (7bc)). We use this equation to derive a relation for the Band C
coefficients in (8). At t=O, we have N010 =Ew/td or Ew=N010td· Then for the
total derivative ~ we find ~ = ~(N0I0td) + ~e·4-(N0I0td) or e e e w VlJe
~ = ~(N0Iotd)/( 1 - ~N0I0td)]. The partial derivatives can be worked out ana replaced by the Band C coefficients, yielding:
(15)
The total derivative in (15) describes the change of Ew with Ee along the selfconsistent curve. With the help of this result, two more identifications can be
made. Introducing the abbreviations:
we can express the changes in Vs and J 1 along the selfconsistent curve as:
and
(16b)
As a matter of course,~ is identical with~· The quotient of (16a) and (16b),
rd= &j~~~ = ~, is the differential resistance of the DBRT structure (times
unit area). ~e see that the top of the I-V curve (P 1 in Fig.2) corresponds to
gi 1 = 0, hence A1=0. The point on the I-V curve where the tangent is vertical b .
(P2 in Fig.2) corresponds to gi~ = -m, hence A2=0. Between P 1 and P 2 we have
108
Fig.2
chapter 5
x107
2
1.6
N
1.2 E ...... <(
..... 0.8
0.4
0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 05
Vbias (Volt}
1-V characteristic of the asymmetric DBRT structure {for structure parameters see Section 6), and the loadline (pi= 1osnm2): Between P1 and P2 the differential resistance is negative. P,, is the point of contact between the I- V curve and the loadline, and the turning point from stability to instability.
Ll 1 > 0 and Ll 2 < 0, so that rd is negative here (NDR region). The negative
differential resistance rd is present in the rewritten stability conditions (14):
(17a)
(17b)
which can now be easily related to the impedance of the circuit.
We define the small signal impedance Z(s) as the quotient of vb and jL· This is obtained by combining (Sa) and (12) yielding:
(18)
The first term of this equation is the contribution of the leads. Since we have
assumed purely ohmic contacts, it is simply pi. For more realistic leads, it is to
Current Stabilit, 109
be replaced by some Z1ead(s). The second term of (18) is the impedance of the 'bare' DBRT structure, Zdbrt(s ). It has a simple zero s=!.l. 2/ Ae, so that stability is ensured if !.l. 2/Ae < O. Since !.l.2 changes sign at P2 (see Fig.2), a. stable solution for the bare DBRT can be found throughout the NDR region. This agrees fully with the conditions (17) for pl= 0, thus reading !.l. 2/Ae < 0 ((17a) being trivial for pl = 0 ).
For pl# O, i.e. for the DBRT in circuit, the total impedance must be
considered. An extra zero is introd~ced, and the two stability criteria of (17) result again. It is clear, that for pl not too large, the second criterion (17b) is decisive. For points on the I-V curve which are before P1 (see Fig.2), {17b) is satisfied since here !.l. 1 < 0 and pl+rd > O. Just after P1 (17b) is still satisfied because both factors change sign at the top of the I-V curve. However, at P2 we have pl+rd =pl> 0 whereas still !.l. 1 > 0, so that the stability criterion is no longer met. Therefore, the turning point from stability to instability must be between P1 and P2, in the NDR region. A more precise position can be specified. Since the factor responsible for the change of sign is (pl + rd), the end of the stability interval can be characterised by rd = -pl. The slope of the load line lt(V5) being -1/pl (see (7a)), the turning-point is found to be the point of contact between the I-V curve and this loadline.
From (18) a resistance and reactance for the bare structure are defined as the real and imaginary part of zdbrt(iw):
1:i. ( ) - fk{Z (. )} !.l.~~ !.l. 1 +w2i~ Ae!.l. a .. "<lbrt w = dbrt IW = I +w2) +W2ZS:a 2 (19a)
(19b)
From the condition ~brt(wco) = O, we define for the bare structure a cutoff frequency Wc0
2 = -!.l. 11.l. 2/(!.l.rAe!.l. 3). Analogously, a self-resonance frequency W5r2 = -!.l.1-1.l.2!.l.3/Ae is found from the demand that the reactance Xdbrt(Wsr) = O. The cutoff frequency is positive in the NDR region, as expected. Here, the
DBRT structure can be used as an oscillator for frequencies up to Wco· The series resistance pl will lower the cutoff frequency. Since our model does not contain any inductive elements, only the addition of a.n external inductance can yield a positive Wsr2· At low frequencies the impedance of the structure becomes purely resistive and approaches the differential resistance rd= !.l.2/ !.l.1•
110 Chapter 5
6. Numerical results for asymmetric GaAs-AlGaAs structure
The above theory has been worked out numerically for a GaAs-AlxGa1.xAs
DBRT device with different widths for the two barriers. For the purpose of
calculations we choose ~=0.4, barrier height 0.44eV, well width 5nm, and barrier
widths 5.6 and 8.4nm, respectively. The effective mass of GaAs has been used for
all the layers. Assuming contact layers of 50nm (doping 2.10 16) plus lµm (doping
2.1018) on both sides of the DBRT structure, we can estimate pt to be of the order 10-1onm2 [7].
In Fig.3, the real and imaginary part of the impedance are shown as a
function of the Fermi level Er in the left reservoir, for w = 109rad/s. We know
that for w = 0 we have ~brt=rd and Xdbrt=O, so that the impedance at zero frequency is merely the derivative of V5(J1) shown in the I-V curve of Fig.2.
Hence, Rdbrt(Er) has a vertical asymptot when 6. 1(Er )=0, and a zero when
t:.. 2(Er)=O. As we see from Fig.3a this picture is not changed very much for
frequencies up to a few GHz. The vertical asymptot of Rdbrt(Er) now corresponds to (t:.. 1+w2)2+w2t:,. 32 = 0 and is accompanied by a sharp negative
peak in the reactance Xdbrt(Er ). The zero of Rdbrt(Er) is shifted too (see (19a)),
but both shifts are small, so that Rdbrt(Er) still resembles the differential
resistance rd very well.
Fig.4 shows the impedance as a function of frequency at fixed value of Er
(Er = O.leV). As expected from (19), Xdbrt(w) is zero for w=O and c.Hm, whereas
Rdbrt(w) is rd for w=O and zero for w-iai. Around w/21r ~ 0.3GHz, the imaginary
part shows a peak where the real part increases to zero. This is the cutoff
frequency Wco that separates the amplified oscillations from those that are
damped by the DBRT structure. The cutoff frequency is real for Er values in the
NDR region only. For the bare structure, Wco is zero at P 1 and P2 (see Fig.2),
and increases to infinity when 6.rAe6.a=O. Its typical value is found to be
Wc0 /21r"' 5GHz. Decreasing the barrier width by 10 percent leads to an increase
of this value to "' 15GHz. Both tendency and magnitude agree quite well with
other models [8].
7. Conclusion
Comparison of our results with experimental findings is complicated by the fact
that most impedance measurements were done on symmetric structures showing
no bistability. Besides, our simple model yields a negative differential
The real (a) and imaginary (b) part of the impedance of the DBRT structure as a function of Fermi le11el Er at fixed frequency of 109rad/s. Structure parameters as in Section 6.
112
Fig.4
chapter 5
x10·7
0.5
0
N -1 E c ~-2 ~
QJ
0:::
-3
-4 0 2 4 6 8 10 12 14 16 18 20
Log(w)
x10·7
0
-0.4 N
E c -0.8
"N' ~-12 E .
.......
-1.6
-2 0 2 4 6 8 10 12 14 16 18 20
Log (w)
The real (a) and imaginary (b) part of the impedance of the DBRT structure as a function of frequency at fixed Fermi level of 0.1e V. Structure parameters as in Section 6. .
Current Stability 113
conductance that decreases from zero to ""1D 1 whereas in real devices (due to
non-resonant tunneling) the conductance remains finite. Nevertheless, we find a
qualitative agreement between our calculated impedance as a function of bias
and frequency (Figs. 3 and 4) and the measured curves presented in Refs.[9-11].
Zarea et a.l. [9] studied a GaAs-AlGaAs structure with a 5 nm well and 5.6
nm barriers ( d. Sect. 6), for which they present the real and imaginary part of Z
as function of bias. These curves show the same trends as our Fig. 3. To be able
to compare the horizontal scales we use that Vb ,., 4EF/e (d. Fig. 2), giving a
nice agreement. For the comparison of the vertical scales the impedances of
Ref.[9] are multiplied by the mesa area. It is then found that the scales differ by
a factor of 10. This is however due to our static model (cf. Fig. 2 and Fig. 1 of
Ref.[9]) which overestimates the current, and not to the dynamical extensions.
The cut-off frequency for the device of Ref.[9] was measured to be 1.10 GHz. Gering et a.l. [10] found a value of 88.2 GHz for a similar structure with slightly
smaller barriers (5 nm). The difference is mainly due to a difference in series
resistance (4 and 0.17 0 respectively). Our calculated typical value of 5 GHz has
the correct order of magnitude. Moreover, we find an exponential decrease of Wco
with barrier thickness, in agreement with the results of Ref.[10].
Lippens and Mounaix [11] measured the impedance of a structure similar to
the one of Ref.[10] as a function of frequency, corresponding to our Fig. 4 (but
with linear horizontal scale). Again we see the same trends in the curves. The·
real part starts at the negative differential resistance to become quite small at
higher frequencies; the imaginary part shows a negative peak. The inductive
behaviour at higher frequencies is of course not reproduced by our calculations.
Comparing the horizontal scale we find that our frequencies are smaller by a factor of 3.
We conclude that our dynamical equations are a fair description of the ac
behaviour of DBRT structures in the GHz-regime. Partly this is due to the fact
that the impedance is strongly related to the differential resistance. Hence a
reasonable I-V characteristic will yield a reasonable impedance. Thus the
agreement with experiments is partly owing to our static model. The essentially
new element is the generalized expression for the current density (Eq.(7bc)), making it a time- and position-dependent quantity. This distinguishes our
approach from experimental equivalent-circuit fits, and enables a purely
theoretical analysis of the DBRT dynamics.
Yet one should bear in mind that this double-current definition poses some limitations to the applicability of the model. Its close relation with the static
114 Chapter 5
description leads to a restriction in the frequency domain to values of w for
which the adiabatic approach is valid. A simple estimation shows that the THz
regime is beyond the scope of our approach. Also, only small amplitude
oscillations in the applied voltage Vb are allowed, thus excluding interesting
nonlinear behaviour. Finally, Eqs.(7) cannot describe any inductive effects of the
DBRT structure, that may nevertheless be present [10].
In fact, Eqs.(7) amount to a dynamic description of the DBRT structure in
terms of a couple of shunted capacitors. If there is some charge build-up in the
emitter and well, we expect a connection between the time constants lf:sil and
lf:s2: of (12), and the typical times for charging these two regions, >..e/vF and td
respectively. Here, >..e is the emitter screening length, vF is the Fermi velocity
and td the dwell time already discussed in (4). Indeed, where the coupling is
small, s1 and s2 have the correct order of magnitude. In the general case, the
complete matrix equation (10) must be solved, thus making the task nontrivial.
In summary, we have been able to describe the low-frequency small-signal
part of the DBRT dynamics by minimal extensions of the statit model. From a
theoretical point of view, this approach is an improvement with regard to the
equivalent-circuit method. A more complete account of the dynamics requires
the consideration of the time-dependent SchrOdinger equation and the nonlinear
aspects [12].
Acknowledgments - We would like to acknowledge partial support by the
Academy of Finland, and the "Stichting voor Fundamenteel Onderzoek der
Materie" (FOM), which is financially supported by the "Nederlandse
Organisatie voor Wetenschappelijk Onderzoek" (NWO).
References
[1]. V.J. Goldman, D.C. Tsui and J.E. Cunningham, Phys. Rev. Lett. 58
(1987) p.1256; Phys. Rev. B 35 (1987) p.9387
[2]. E.S. Alves, L. Eaves, M. Henini, O.H. Hughes, M.L. Leadbeater, F.W.
Sheard, G.A. Toombs, G. Hill and M.A. Pate, Electronics Letters 24
(1988) p.1190; L. Eaves, F.W. Sheard and G.A. Toombs, in "Band
Structure Engineering in Semiconductor Microstructures", eds. R.A.
Abram and M. Jaros, 1989.
[3]. F.W. Sheard and G.A. Toombs, Appl. Phys. Lett. 52 (1988) p.1228
Current Stability 115
[4]. B.J.M.F. Noteborn, H.P. Joosten and D. Lenstra, Phys. Scripta T33
(1990) p.219.
[5]. D.D. Coon, K.M.S.V. Bandara and H. Zhao, Appl.Phys.Lett.54 (1989)
p.2115.
[6]. H.P. Joosten, B.J.M.F. Noteborn and D. Lenstra, Thin Solid Films 184
5.3. Alternative for the quantum inductance model in double-barrier
resonant4unneling14
Abstract -A theoretical justification for the phenomenolo~ical equivalent circuit approach in Double-Barrier Resonant Tunneling (DBRT) is given. Starting from the combined SchrOdinger and Poisson equations for the static behaviour of the DBRT diode, we arrive through a small· signal analysis at a. model for the dynamics. This model corresponds to an equivalent circuit consisting of two linked RC-sections. The simple equivalent circuits in which the DBRT structure is represented by a capacitor with parallel resistor, as well as more complicated circuits including a. quantum inductance can all be considered special cases of our model. Furthermore, by studying the regions of stability in the phase diagrams res from our model, the effect of a parallel capacitor on the sta.oility is investi ated.
1. Introduction
Recent literature shows a.. shift in the interest of both experimentalists and
theorists from the static description of Double-Barrier Resonant-Tunneling
(DBRT) structures to the dynamical behaviour of these diodes [1--<>]. This has
yielded a. number of equivalent circuit models [2-4], most of which use two
elements, the differential conductance and the diode's ca.pa.city, to describe the
experimental data. Recently, Brown et al. [7] have suggested that the addition of
a third element, a "quantum well inductance", to such models improves the
agreement with experimental results considerably. However, a fundamental theoretical derivation of the quantum inductance is lacking. An elaboration of
this model has been given by Boudreau and Liu [8].
In a previous pa.per [9], we presented a. description of the DBRT dynamics for the small signal-low frequency regime. This description is based on a small
signal analysis of our static model of coherent tunneling (described by the
Schrodinger equation) in a self- consistent electron potential (described by the
Poisson equation) [10]. It can be translated into an equivalent circuit of four elements: a double RC model [11].
In this paper we show that the quantum inductance model of Refs.[7,8] can
be understood as a special case of our dynamical model. Numerical calculations
14This section will be published as a paper: H.J.M.F. Notebom, H.P. Joosten, K. Kaski and D. Lenstra, 11 Alternative for the quantum-inductance model in double-barrier resonant-tunneling", Superlattices and Mierostructures (1993) (accepted for publication). References between square brackets []are listed on page 127.
Ou:rrent Stability 117
suggest that experimentally this special case is a relevant one. In this way, our
dynamical model provides a theoretical basis for the two- and three-element
equivalent circuits. At the same time, it suggests an understanding of the DBRT
dynamics in terms of coupled RC circuits, rather than a quantum inductance.
2. Dynamical description ofDBRT
For small and slow ( <lOOGHz) oscillations, a dynamical description can be
obtained through minor extensions of the static model. In the static situation,
the static charge densities in the emitter and the well, u6 and uw, are
interrelated through the selfconsistent electron potential [HI]. In the dynamical
case [9], these charge densities become the independent dynamical variables of
the system. Because of the nonzero divergence of the current density J, we have
to distinguish the current in the leads (Ji), and the current through the first and
second barrier (J1 and J2):
(1)
The current density h can be easily expressed in terms of the applied voltage
and the external-circuit impedance. For J 1 and J2, Ref.[9] provides simple and
intuitive expressions based on the quantum mechanical current density
expression for the time-independent situation. The current through the first
barrier, J 11 is proportional to the density of resonant electrons in the emitter,
while the current through the second barrier, J 2, is inversely proportional to the
time spent in the well:
(2)
where N0 is the areal density of resonant electrons, I0 is the current carried by a
resonant electron, Uw is the areal charge density in the well, and td is the dwell
time of resonant electrons in the well. Of course, in the static situation we have:
Uw = N0I0td , so that J 1 = J 2• In the dynamical situation, all quantities are
functions of both u6 and uw· The linearized version of the model made up by (1)
and (2) can then be summarized into a small-signal impedance [9]:
(3)
118 Chapter 5
where s is the complex frequency, and the .four quantities rd, Cd· a and b are
combinations of the partial derivatives of fJh/ fJue, {JJ ii fJuw, etc. (see Appendix),
and therefore functions of the applied voltage V. In particular, rd can be
identified with the differential resistance of the DBRT-structure,
rd = dV dbrtf dJ" and is directly obtainable from the I-V curve. The quantity Cd
is the differential capacity of the emitter, 1/Cd = 8Vdbrt/8ue· For low frequencies
the impedance becomes resistive, ZdbrtUw) :::: rd , for w < < 1/ I a-b-rdcd I, whereas for high frequencies it behaves capacitor-like: zdbrtUw):::: 1/jwcd , for
w >> I b/ ardcd I· A special case occurs when a= b, in which situation
Zdbrt(s) = rd/(l+rd~s) and hence independent of a. In fact, the DBRT
structure is then represented as a capacitor Ca and a parallel resistor rd, as in the
standard equivalent circuits [2,3,4].
3. Biasing Circuit
To describe the effect of the circuit, we introduce a series and a parallel
impedance,
(4)
that take into account the leads and contacts [2-4], and an optional parallel
capacitor, like the integrated Schottky barrier capacitor of Ref.[12]. The circuit
considered is drawn in Fig.1. The zeros of the total. impedance are:
(5)
and are obtained from (3) and ( 4), (or the poles of the corresponding
admittance). They determine the stability of the circuited DBRT: if the complex
frequencies (Laplace variable) s for which Ztot(s)=O have negative real parts,
then stability is ensured. In fact, the negativity depends on the value of Rs, L5
C8 and Cp· In a parameter space set up by these quantities, we can therefore find
regions of stability having a boundary characterized by Re(s) = 0. A graphical
representation of this parameter space could be termed a 'phase diagram' of the
DBRT-device [8].
Apart from their stable or unstable nature, solutions can be classified by
Current Stability 119
Fig.1
Rs Ls DBRT
Cs Cp
DBRT
Circuited DBRT-structure with series impedance (resistance Rs, inductance L5 and capacitor CJ, and parallel capacitor Cp·
their oscillatory or exponential character. In the former case, the zeros of the
total impedance have a nonzero imaginary part. Hence, regions in parameter
space with oscillatory solutions have a boundary characterized by Im(s) = 0. In
the following, we will analyse both classifications, in stable/unstable and in
oscillating/exponential solutions, and calculate the different regions in the phase
diagram.
It is possible to draw a phase diagram for every value of the applied voltage,
i.e. for every set of rd, cd, a and b. However, experimentally only the applied
voltage of minimum negative differential resistance (NDR) is important. Since in
a coherence-based model of the DBRT like ours this minimtim NDR is zero, we
will consider a different working point in the NDR region where rd is negative
but nonzero.
The characteristic equation for the zeros of the total impedance Ztot(s) is a. cubic equation ins (see Eq.(5)):
(6)
120 Chapter 5
From (6) we see that the parallel and series capacitors have the same effect on
the stability. To simplify (6), we introduce dimensionless quantities s = -rdcd · s
and
where we have assumed that we are in the NDR-region. The ,coefficients now
read:
R(l+C-b) 1 1 ai = La( 1 +C) + L(l+C) - a(l+C} (7)
1-R ao = La(I+C)
where we have omitted all tildes. All the dynamics of the structure is now
contracted into the two quantities a and b, while the circuit is described by the
dimensionless parameters R, C a.nd L.
Let us first consider a. special case where analytical answers are possible. In
the case that a = b, the roots of the cubic equation a.re:
S1 = -1/a, (8)
The second and third root are truly complex if 0 5 R < J1~C' ( 2 -J1~c),
Stability chart for the special case that a= b = 1. The bold line separates the stable from the unstable regions {indicated by Sand U}, the thin line separates the region of oscillations from the region of exponential solutions {indicated by o and n, respectively).
which is the case only in the interval 0 < L < 4(1+C). The oscillatory solutions
are stable if R > l~C' whereas the exponential solutions are stable if R < 1. All
boundaries coincide at R = 1, L = 1 +C. In Fig.2 the phase diagram is plotted in
the L-R plane. The effect of the capacitor C on both the stability and
oscillations is clear from all these expressions: (l+C) is a mere scaling factor for
L. A larger C8 or Cp will lead to a larger region of stability in the L-R plane. We
remark here that the region of stability and the region of oscillations are both
independent of a, as it should be, since for b =a the impedance of the DBRT,
Eq.(3), is independent of a, as we have seen in Sect.2. For points on the line
R = ~ + a+i+c (see Fig.I), the root s1 coincides with s2 (if L > a2
( l+C) ) or (a+l+C)2
s3 (if L < a2
( l+C) ). Although this line is not a boundary in the special case (a+l+C)2
a = b, it will become important when a # b. In the case that a # b the expressions for the roots become bulky. However,
we can still find the stability region by putting Re(s) = 0 in (6). This yields two
equations, a0 = 0 and a0 = a1a2, that read in worked-out form:
Stability chart of the asymmetric DBRT structure for the case that C = 0 (a) at a Fermi level of 0.1028 eV (corresponding to a= 1, b = -0.5); and {b) at a Fermi level of 0.1054 eV (a= 5, b = 0). Lines and characters according to the convention of Fig.2.
Stability chart of the asymmetric DBRT structure for the case that C = 10 {a) at a Fermi level of 0.1028 eV (corresponding to a= 1, b = -0.5}; and {b) at a Fermi level of 0.1054 eV (a= 5, b = 0). Lines and characters according to the convention of Fig.2.
In the limit b -+ a the second expression reduces to the previously encountered
R = L/(l+C). The stability region in the L-R plane that corresponds to (9) is
shown in Fig.3, where b S 0 is assumed: it is the closed region { (0,0), <t+c.!6,o), (l+C+a-b,1), (0,1) }. The effect of Con the stability is not much different from
that in the special case b = a. The region of oscillatory solutions cannot be presented in explicit form. In
Fig.3 where this region is plotted, we ca.n however easily recognize the (b = a)
solution: the above-mentioned line R = ~ + a! 1 breaks up into two lines for
L > °'2 ( l+C) , opening up a strip of oscillatory solutions. As a result, a sharp
(a+l+C)2 triangular region of damped stability comes into being. An increase of C
effectuates an enlarging of this region, as illustrated in Fig.4.
4. Numerical results
For the asymmetric Ga.As/ AlxGa1_xAs structure of Ref.[10] (x = 0.3, well width
5 run, barrier height 0.44 eV, barrier widths 5.6 and 8.4 nm, and effective mass
0.067·9.licl0-31 kg) we have calculated numerically the stability chart: for two
values of applied voltage (or Fermi energy in the emitter) the stability region in
the L-R plane is plotted, see Figs.3 and 4. At EF = 0.102eV we have a= 1 and
b = -0.5, whereas at EF = 0.1045eV, a= 5 a.nd b = O. The dimensionless
capacity C is 0 and 100, successively. Also the nature of the current fluctuations
(oscillating/ non-oscillating) is indicated.
5. Discussion
Comparison with the work of Boudreau and Liu [8] is best started at the
characteristic equation (6). Since these authors have not considered a series or
parallel capacitor, we have to put C = 0 in (6). Our equation is then identical
Current Stability 125
Fig.5
-11 -.... -.................. ____ _
-12
-13
a-b---td···
\ _____ ................. .. \ /'\I :1
ii !! i i
-14'--~~'--~~'--~~'--~--''--~--''--~--'
0.06 0.08 0. 1 0. 12
Logarithm of the absolute value of a, b and ~ versus Fermi energy (e V). The sign of b can be seen in the inset where a, b and~ {s} are plotted.
with theirs if we make our a equal their resonant lifetime r, and put our b = 0.
The second identification, b = O; implies a loss of generality. It enables
Ref.(8] to use a three-element equivalent circuit, whereas we need a
four-element one [11]. Following [7], these authors add to the NDR (rd) and the
diode capacity ( cd) a 11qantum (well) inductance11 of magnitude I rd Ir. This
addition is motivated in Ref.[7] by its success to explain experimental data.
The first identification, that a equals r, is not mathematically exact.
However, the two times are of the same order of magnitude, as can be seen from
Fig.5. Substitution of r by a would lead to an equally satisfying explanation of
the data of Ref.[7]. Fig. 4 also has b in the NDR region, showing that b < a and
that for some value of the Fermi level (or applied voltage) b = 0 indeed.
From this we conclude that the quantum inductance models of Refs.[7] and
[8] can be understood as fair approximations of our more general model. The
latter has the· advantage of being derived within a theoretical description of the
DBRT dynamics, rather than being an experimental suggestion.
The proposed scheme of describing stable biasing of DBRT structures is
useful as long as the charge distribution can follow adiabatically the fluctuations
of the applied voltage, i.e. up to a certain frequency. At high frequencies, the
126 Chapter 5
charge in the well will simply be fixed, and Eq.( 4) can be replaced by a
quadratic equation. In that case, the stability analysis of the DBRT-structure is
completely analogous to the stability analysis of the Esaki tunnel diode [13]. The
interesting point concerning the DBRT is the possibility of charge build-up in
the well, resulting in a more complicated equivalent circuit of four components.
Acknowledgments - We acknowledge partial support by the Foundation for
Fundamental Research on Matter (FOM), which is financially supported by the
'Nederlandse Organisa.tie voor Wetenscha.ppelijk Onderzoek' (NWO), and by the
Academy of Finland.
Appendix
The four quantities that determine the DBRT impedance, rd, Cd, a and b, a.re
defined as:
(Al)
(A2)
(A3)
(A4)
where Ae to Cw are partial derivatives of the potential drop acro8s the structure,
V5, and the current though the first and second barrier, J 1 and J 2, with respect
to the areal charge densities Ue and Uw in the emitter and well:
to be evaluated in the self-consistently determined static solution ue, Uw· For a. derivation of these formulae, see Ref.[9].
Ourrent Stability 127
References
[1]. T.C.L.G. Sollner, P.E. Tannenwald, D.D. Peck and W.D. Goodhue,
Applied Physics Letters 4S (1984) 1319; S.K. Diamond, E. Ozbay,
M.J.W. Rodwell, D.M. Bloom, Y.C. Pao and J.S. Harris, Appl.Phys.Lett.
54 (1989) 153.
[2]. J.M. Gering, D.A. Crim, D.G. Morgan, P.D. Coleman, W. Kopp and H.
Morko~, J.Appl.Phys. 61 (1987) 271.
[3]. D. Lippens and P. Mounaix, Electronics Letters 24 (1988) 1180.
[4]. A. Zarea, A. Sella.i, M.S. Raven, D.P. Steenson, J.M. Chamberlain, M.
Henini and O.H. Hughes, Electronics Letters 26 (1990) 1522.
[5]. W.R. Frensley, Appl. Phys. Lett. 51 (1987) 448.
[6]. D.D. Coon and H.C. Liu, J. Appl. Phys. 58 (1985) 2230; H.C. Liu, Appl.
Phys. Lett. 52 (1988) 453.
(7]. E.R. Brown, C.D. Parker and T.C.L.G. Sollner, Appl. Phys. Lett. 54
[8]. M.G. Boudreau and H.C, Liu, Superlattices and Microstructures 8 (1990)
429.
[9]. H.P. Joosten, H.J.M.F. Notebom, K. Kaski and D. Lenstra, J. Appl. Phys. 70 (1991) 3141.
[10]. H.P. Joosten, H.J.M.F. Noteborn and D. Lenstra, Thin Solid Films 184
(1990) 199; H.J.M.F. Notebom, H.P. Joosten and D. Lenstra, Physica
Scripta T 33 (1990) 219. [11]. H.P. Joosten, H.J.M.F. Notebom, K. Kaski and D. Lenstra, Physica B
175 (1991) 297.
[12]. V.J. Goldman, D.C. Tsui and J.E. Cunningham, Phys. Rev. Lett. 58
(1987) 1256; 59 (1987) 1623.
[13]. J.O. Scanlan, "Analysis and Synthesis of Tunnel Diode Circuits", (John
Wiley & Sons, New York, 1966).
128 Chapter 5
chapter6
MAGNETO-TUNNELING
6.1. Introduction
Magneto-tunneling experiments have played an important role in several
resonant-tunneling discussions. Charge build-up in the well, an essential factor
in the explanation of the intrinsic bistability, can be monitored in a B II J configurationt, in which the one-dimensional tunneling remains virtually
unaffected. Thermalization of the space charge in the well of a suitably
asymmetric DBRT structure was observed by Leadbeater et al.2 who studied the
magneto-oscillations in the differential capacitance. The same B II J
configuration is appropriate to investigating LO-phonon emission and
(quasi)elastic scattering3, since the magnetic field increases the amplitude of
1C.A. Pa.yling, E.S. Alves, L. Eaves, T.J. Foster, M. Henini, O.H. Hughes, P.E. Simmonds, F.W. Shea.rd, G.A. Toombs, Surf. Sci. 196 (1988) 404. 2M.L. Leadbeater, E.S. Alves, F.W. Sheard, L. Eaves, M. Henini, O.H. Hughes and G.A. Toombs, J. Phys.: Condens. Matter 1(1989)10605. 3M.L. Leadbeater, E.S. Alves, L. Eaves, M. Henini, O.H. Hughes, A. Celeste, J.C. Portal, G. Hill a.nd M.A. Pate, Phys. Rev. B 39 (1989) 3438.
129
130 Chapter 6
both the resonant and the phonon-assisted current peak. In some structures,
both the GaAs-like and AlAs-like LO phonon modes are observable due to the
resolving effect of the magnetic field. All these effects are based on the
quantization of the in-plane motion of the electrons, which converts the energy
bands into Landau levels. DBRT structures with wide wells (60-120 nm) have
been studied in the B .LJ configuration by Alves et al.4 The resonances in the 1-V
curves evolve into hybrid magneto-electric states of traversing or skipping
character. At high B the tunnel current was seen to be quenched. An initial
quenching of the current is also observed in B .L J experiments with narrow-well
(4.3 nm) structures5.
The study of both the B II J (Sect. 6.2) and the B .L J (Sect. 6.3) case starts
from the effective-mass Hamiltonian for the spherical conduction band6:
where Inc is the effective mass, A the (slowly varying) vector potential for the
magnetic field B :: VxA, a the electron spin (:1:!) and µ,8 = eh/2m0 the Bohr
magneton. The g-factor differs from 2 due to the coupling of the conduction
band to mainly the spin-orbit split valence band. g• can be postive and negative,
and quite large for some materials (InSb: g• = -50). Weisbuch and HermannT
have experimentally determined the g-factor for the GaAs-AlxGa1.xAs system,
and found g•(GaAs) = --0.44, g•(A10•33Gau7As) = +0.48. With such small g*
. values, the spin splitting of the energy levels is unimportant under the usual
experimental conditions. Thus we are allowed to drop the second term of the
Hamiltonian and neglect spin effects8.
4E.S. Alves, M.L. Leadbeater, L. Eaves, M. Henini, O.H. Hughes, A. Celeste, J.C. Portal, G. Hill and M.A. Pate, Superlattices and Microstructures 5 (1989} 527. 5S. Ben Amor, J.J.L. Rascol, K.P. Martin, R.J. Higgins, R.C. Potter and H. Hier, Phys. Rev. B 41 (1990) 7860.
6L.M. Roth, B.Lax and S. Zwerdling, Phys. Rev. 114 (1959) 90. 7C. Weisbuch and C. Hermann, Phys. Rev. B 15 (1977) 816. 8Moreover, the extension of spin-fiPlitting from bulk to heterostructure is nontrivial. In GaAs-AlGaAs heterostructures a reduction of the splitting, dependent on magnetic field and Landau-level, is found. See: M. Dobers, F. Malcher, G. Lommer, K. v. Klitzing, U Rossler, K. Ploog and G. Weimann, in: High magnetic fields in semiconductor physics II, ed. G. Landwehr, Berlin: Springer, 1989, p. 386.
Magneto-tunneling 131
6.2. Two-period magneto-oscillations in coherent double barrier resonant tunnelings
Abstract - Applying a magnetic field B to a Double Barrier Resonant-Tunneling Diode, perpendicular to the layer structure, introduces oscillations in current density and capacitance that are periodic in l/B. A derivation of this periodicity is given, based on coherent wave propagation. Two magneto-periods are found, corresponding to the electron concentration in emitter and well, respectively. Numerical calculations are presented for a semiconductor model with selfconsistently determined electron potential.
The application of a magnetic field in the study of resonant tunneling may reveal
most relevant information, a fact long acknowledged by both experimentalists
and theorists [1-4]. In the case of the double barrier resonant-tunneling (DBRT)
structure, the B II J geometry enables a direct probing of the charge build-up in
the well [2,3]. Since this phenomenon plays a key role in explaining the intrinsic
bistability in the I-V curve of a DBRT structure, magneto-tunneling
experiments have been of importance in the discussion about the nature of the
observed bistability [2-4]. Information about charge density and Fermi level is
contained in the magneto-oscillations in charge and current that result from the
passing of the Landau levels through the Fermi level when varying the magnetic
field at fixed applied bias voltage. These Shubnikov-de Haas-like oscillations are
periodic in l/B with a period I/Brr which is inversely proportional to the space
charge in the well; this periodicity has been reported by a number of authors
[1-3]. The careful analysis of the magneto-oscillation spectrum reveals a second
peak, corresponding to a period that is related to the space charge in the
accumulation layer in front of the structure. This second peak has been reported
by Payling et al.[3,4]. In their description of the magnetospectrum, they start
from a sequential tunneling picture (4,5].
In this paper, we present a derivation of the magneto-spectrum based on the
description of electron transport as coherent wave propagation. To a great
extent, this derivation is independent of the specific model of the ID-tunneling
9This section was published as a paper: H.J.M.F. Noteborn, G.H.M. van Tartwijk, H.P. Joosten and D. Lenstra, 11Two-period magneto-oscillations in coherent double-barrier resonant tunneling", J. Phys.: Condens. Matter 3 (1991), 4249-4256. References between square brackets []are listed on page 141.
132
F'ig.1
Chapter 6
w
Conduction band minimum in the DBRT-strv.cture as a function of position z.
or oi the contact layers: for, both classically and quantum mechanically, the
influence of a magnetic field perpendicular to the layers is on the lateral motion
only. Thus, questions like wether the tunneling is or is not sequential, or wether
the Fermi level is or is not constant, do not affect the following presentation.
In a DBRT-structure, there will be a build-up of charge in three layers (See Fig.
1). In the emitter layer, an accumulation of electrons will give rise to a negative
charge density. Quantum mechanical tunneling enables the formation of an
electron gas, 2D in nature, within the well. Furthermore, the ionised doping in
the depleted collector layer provides a positive space charge. Since the laiter
density is determinable via overall charge neutrality considerations, we
concentrate on the electron densities in emitter and well. Our starting point is a well-known formula for the 3D electron concentration
n(r) at position r:
n(r) = Ik f[~] • 1wk(r)i2 (1)
Here, the 'lfk(r) are the envelope functions describing the electron states labeled
by k=(kuk1,kz). The function f(e,=(l+exp(~))-1 is the Fermi-Dirac distribution.
We take r=O to be the middle of the well. Two remarks should be made about
(1). First, the normalisation of the functions 'lfk(r) is with respect to the
reservoir, formed by the doped layers that sandwich barriers and well, Le. for
Magneto-tunneling 133
large r, the electron concentration should equal the impurity density ND to
ensure charge neutral contacts. Secondly, the label k refers to the allowed states
in the reservoir. If we let the volume of the reservoir tend to infinity, the
Hamilton equation for the envelope functions [6]:
(2)
where Ec0 (r) is the conduction band minimum and A(r) is the vector potential,
related to the magnetic field B via B=VxA, has a continuous spectrum of
allowed electron energies Ek above the conduction band minimum Ec0(r) in the
reservoir.
Our first step is to cast (1) into a quasi 1D form. Let z be the direction
perpendicular to the barriers. A valid choice for the vector potential A is then
(-By,0,0), corresponding to V·A=O and VxA=(O,O,B). Because of the layered
structure, the band edge Eco depends on z only. Separation of variables is
possible:
(3)
replacing (2) by an equation for the lateral state Gk k (x,y) containing all field x y
dependence but no band edge:
~[<~Zx-eBy)2+(~)2]Gk k (x,y) = ~ k Gk k (x,y) (4a) xy xy xy
and an equation for the tunneling state Fk (z) containing the band edge but no z
field dependence:
(4b)
Mter substituting (3) in (1), we do the summation over kx and ky to introduce a
new weighing function g in which the density of the lateral states is incorporated:
(5)
134 Chapter 6
so that (1) now reads:
(6)
.In the perfectly layered structures that we consider, g(x,y;E) will turn out to be
independent of the coordinates x and y, i.e. the electron concentration depends
on z only. Anticipating this, we will write g(E) in stead of g(x,y;E). The function
g( E) has the same weighing role as the Fermi-Dirac distribution function f( E),
but differs from the latter in having a dimension, the dimension of an areal
density. In (6), the ID tunneling described by the functions F1cz(z) is separated
from the effect of the lateral states, incorporated in g. A magnetic field in the
z-direction will introduce no direct changes to F k ( z), but affect only the z
weighing function g.
Our next step is to ensure that in the well only one kz (corresponding to the
resonance energy Er) is present. We indicate this wave number by kzr, leave out
for z=O all terms with kz#kzr• and write nwell = n(z=O) = g(E:fiEr 1·IFkzr{O)l2. This result can be improved by averaging n{z) over the z-interval (-w/2,+w/2),
where w is the well width. Also, the fmite width of the resonance level can be
taken into account, changing the factor I Fk {O) 12• However, the essence of our zr
result: the proportionality of nwell and g(EfiEF]:
1 [Eti:TEr] nwell"' vtg (7a)
will survive these modifications.
Contrary to the well, the emitter may contain electrons with any positive
energy Ek (upto Er at zero temperature). H we neglect the dependence of z
I Fk (z)l 2 on kz, i.e. if we take the reflection coefficient of the structure equal to z
unity, we find:
(7b)
With (7ab), the basis for our discussion of magneto-oscillations is la.id. The
electron concentration in the well depends on 11r=~if: via. the function g( e), which is the Fermi-Dirac distribution dressed with the density of lateral states. The
Magneto-tunneling 135
latter density depends on the magnetic field strength. The electron concentration
in the emitter can be expressed as the integral of the same function, and depends
on the reduced Fermi energy 'f/=EF/kT.
Let us have a closer look at the function g(E), and work out (5) in the case of
zero magnetic field. The function Gk k (x,y) is then a plane wave (see ( 4a)), 2 x y
and the energy Ek k = k<kx2+ky2). Substitution thereof in (5) yields: x y
(8)
where Ne= [mkT ] 312
is the effective number of states in the conduction band per 2 ?rft2
unit volume (no spin degeneracy), and .5j( E) is the Fermi~Dirac integral of order
j [7]. Substitution of (8) in (7ab) yields:
(9a}
(9b)
The difference between (9a) and (9b) is the difference between a 2DEG and a
3DEG. In the case of B#O, we proceed in the same way. Now, the envelope function
in (4a) is essentially a Hermite polynomial H~u), and the energy is quantised
into equidistant levels:
where the quasi-continuous label ky is replaced by the non-negative integer l, expressing the quantising effect of the magnetic field. In fact, l labels the
so-ealled Landau levels of the energy associated with the lateral motion.
Substitution in (5) yields:
136
Fig.2
0.2
0.1
";
iii le; 0 -.., G, c c
· 0·20~~-o~.1--~o.2--~0.30----,0~.4---='o.·s
1.0
0.5
0
I I
/'
I I I I I
118 (1/TI
I ·too~--0-.1~-~nz---o.~3 --o~.4-~o.s·
111! 11/Tl
Chapter 6
(a) The electron density nemitter as function of 1/B, relative to its value at zero magnetic fieltf. (see eq;{11) and {9}), at thr~e temperatures T=0.01K, ,j.2K and 77K respectively. {b) The same for 71.well· Er is taken to be 10meV and Eres=tE'r· mis 0.061 times the electron mass.
(10)
where O=TieB/mkT is the "reduced" magnetic field. Since now g depends also on
B or 0, an extra slot to g is added. Using (10) in (7ab), the electron
concentrations in emitter and well are found to be:
(lla).
Magneto-tunneling 137
(llb)
In Fig. 2 the expressions of (11) are drawn for two values of the temperature.
Two limits of (11) can easily be evaluated. If 0<<1, Le. if the spacing between
the Landau levels is much smaller than kT, then the summation in (11) can be
replaced by an integration, so that the results of (9) are retrieved, as expected;
the effect of the magnetic field is effaced by the temperature. If, on the other
hand, 0>>1, we find:
Dwell "' ~· Nc2/3 0 Int[ rr;a. + t] (12a)
(12b)
The upper limit in the summation of (12b), fmax, is equal to Int[ fJ/0 - i ]. Int[x] denotes the integral part of x. The expressions in (12ab) are independent of
temperature. It is in this limit 0>>1 that the magneto-oscillations are easily
recognised: nwell is a decreasing function of j on every interval ~r < j < ~r' ~O; at j = ~r' however, nwen increases abruptly by Nc21s.m· Hence, the
resulting oscillations in nwell will have a. period ~r [8] as a function of j, corresponding to a period Jie/m(Er-Er) as a function of reciprocal field ft. Because the reciprocal period has the dimension of a magnetic field, we call it a
"fundamental field" [9], and denote it by Brr:
B _ m(EheEr) fr - {13a)
Eq. {12b) can be analysed in the same way: lmax is constant for¥< j < ~· ~O. Now, nemitter is continuous and not monotonous on this interval. However,
its first derivative is discontinuous a.t j = ¥1 ~O, and this results in
oscillations with period l/ 11, corresponding to a. period m~ in ft· The fundamental field, Br, is in this case:
Br=Wfr {13b)
138 Chapter 6
Thus, the two quantities Br and Brr of (13ab) are a brief characterisation of the
magneto-oscillations in nemitter and nwelli respectively. Because the electron concentration in the collector is related to those in emitter and well via the
demand of charge neutrality, it will contain both periods. In the same way, the
capacitance of the DBRT-structure, measured as a function of inverse magnetic
field, will peak in its magneto spectrum at both Br and Brr-
Expressions for the fundamental fields were arrived at, considering only the
lateral motion, and assuming that Er and EF do not change with magnetic field.
We could abstain from considering the motion in the z-direction, i.e. from
specifying the factors of proportionality in (7ab). However, in order to couple the
fundamental fields to the external handle, the bias voltage V applied to the
DBRT-structure, we need to specify the model for tunneling and reservoirs. Let
us first look at what we might term a "metal picture" of the structure: the
Fermi level is determined by the impurity density in the doped contact layers,
and independent of V. The resonance level with respect to the band edge in the
well is a constant, E0 , determined by the structure parameters; with respect to
the band edge in the emitter, however, this level Er decreases with increasing V,
and if the effect of the charge in the well on the band bending is neglected, this
dependence is linear. Thus, in a metal picture, the function Br(V) is a constant
function, whereas Brr(V) is a linear function of V. Since, at zero temperature,
there is resonant charge build-up in the well only if O<Er<EF, the function
Brr(V) is only defined for the corresponding voltage interval, in which the field
increases from 0 to the constant Br. A remark about the condition that EF be independent of B, is in order: in a
metal picture of the contact layers, the electron concentration should equal the
ionised-impurity density, and if the latter does not depend on the magnetic
field, then the former will neither. Hence, magneto-oscillations will now not be
found in Ilemitter but in EF instead [10], and in such a way that (12b) is still valid. Since the oscillations in EF correspond to the. same fundamental field Br
[11], and since quantities like the capacitance or the current depend on EF, we
will still find the two periods derived above.
Characterising the contrasting "semiconductor picture11 by a voltage
dependent Fermi level EF(V), we now find an increasing function Br(V). Also in
this picture, Brr is defined on a small voltage interval only, the beginning of
which corresponds to Er=EF (i.e. Brr=O) and the end of which corresponds to
Er=O (i.e. Brr=Br). It is within this semi-conductor picture that our numerical
{a} Sheet density u~ of the charge in the left spacer as a function of 1/B at fixed bias voitage Vb= 0.156V, determined numerically with selfconsistent electrostatic feedback. Structure parameters: s1-b1-w-b2 = 2.5-5.6-5.0-.5.6nm; barrier height is ta.ken to be o,,14eV; m is 0.067 times the electron mass; Er=19.4 (GaAs - Alo.4Gao.6As values). {b) Its Fourier spectrum.
calculations for a symmetric GaAs/A10•4Gao.aAs structure are done [12]. By
assuming Er(V=O)=O, we neglect all doping effect on the Fermi level. Values of EF(V) for V#O are determined via the selfconsistency demand that V be equal to
the charge induced potential drop plus Er, neglecting however the potential
drops· in emitter and collector. These simplifications yield in general too large
values of EF and henc.e of Bf and Bfr (See Fig. 3). The lD-tunneling through the barriers is calculated in the Transfer Matrix Approach; the sharp peak in the
transmission probability is approximated by a Dirac-delta function with the
correct weight. In this way, the factor of proportionality in (7a}, (9a), (lla) or
(12a) is found to be 1<1-¥::M-i+Rc) , where Re and Re are the reflection coefficients of the emitter and collector barrier for a wave of energy Er. This
"storage factor" expresses the ability of the well to hold the charge: if ~l (at
low bias}, it is unity, but if Rc-+O (at higher biases), it approaches !(1-Re)<<l. In particular, when Re-il (i.e. when Er-10 and Bfr reaches its maximum), the
storage factor, and hence Ilwelh approaches zero. This is in contrast with a true 2DEG characterised by a storage factor of unity. There, the zero field-i:ero
temperature concentration is proportional to ~r (cf. (9a) and (13a)). Here, it is the decreasing storage factor, i.e. the leaky nature of the well, that frustrates such a linear relation between Bfr and the amount of space charge in the well
[13].
140
Fig.4
Chapter 6
90 Q
Q
Q
Q
= 60 Q
of QQ
a; Q Q QQ
Q 30 Q
Q Q
Cl Cl Cl Q
Cl
0 0.1 0.2 0.3 M o.s V IVI
Inverse periods Br and Brr as functions of V. Structure parameters as in Fig.9.
A similar remark as for the metal picture must be made for the semi
conductor picture: since both Er and nemitter appear in the selfconsistency
demand, we are not free to choose one of the two independent of B.
Consequently, both EF and nemitter exhibit magneto-oscillations, that are
restricted by (12b) only. Since also nwell appears in the selfconsistency demand,
the resulting oscillations in all three quantities will contain both periods 1/Br
and 1/Brr (See Fig. 4). In other words, it is the selfconsistency demand in the
semiconductor model, that effectuates the concurrence of the two fundamental
fields in the spectra of all quantities.
If we apply the above discussion to the experimental results of Refs. [3,4], we
find that, for the examined structures, the metal picture is most appropriate: the
slope of Br(V) is found to be smaller than 10-2 T/mV, corresponding to a change
in EF with V of 10-2 eV /V, too small for a pure semi-conductor model. For a
different type of structures, including spacers or moderately doped contact
layers, a semiconductor picture may be more favourable.
Since we started our discussion from coherent wave propagation, we cannot
compare our theory to the experiments of Ref.(14], where thermalisation plays
an essential role. There, the tunneling is not coherent, via a resonant state (i.e. a
state with energy that is positive with respect to the band edge in both emitter
and collector contact), but non-coherent or "sequential", via a "quasi-bound"
state (i.e. a state with energy that is negative with respect to the band edge in
one of the contacts). In the lightly doped emitter, a 2DEG builds up by
M agneto-t'Unneling 141
thermionic processes, whereas in our coherent picture a 3DEG results (cf. (9)).
This indicates once more that our analysis is restricted to the low bias region.
Summarizing, we have demonstrated that coherent tunneling in a DBRT
structure will lead to biperiodicity in the magneto-spectrum of charge and
current densities. In the low bias region, the 3D contact and the 2DEG in the
well each provides its own period. The selfconsistent electrostatic feedback
effectuates the appearance of both periods in all relevant quantities.
Acknowledgments - We would like to thank Professor L. Eaves for elucidating
discussions. This work is part of the research program of the Foundation for
Fundamental Research on Matter (FOM), which is financially supported by the
"Nederlandse Organisatie voor Wetenschappelijk Onderzoek11 (NWO).
References
[1]. E.E. Mendez, L. Esaki and W.I. Wang, Phys. Rev. B 33 (1986) p.2893;
E.E. Mendez, in "Physics and Applications of Quantum Wells and
Superlattices111 eds. E.E. Mendez and K. von Klitzing, 1987, p.159.
[2]. V.J. Goldman, D.C. Tsui and J.E. Cunningham, Phys. Rev. Lett. 58
(1987) p.1256; Phys. Rev. B 35 (1987) p.9387.
[3]. C.A. Payling, E. Alves, L. Eaves, T.J. Foster, M. Henini, O.H. Hughes,
P.E. Simmonds, J.C. Portal, G. Hill and M.A. Pate, J. Physique C5
(1987) p.289.
[4]. C.A. Payling, E. Alves, L. Eaves, T.J. Foster, M. Henini, O.H. Hughes,
P.E. Simmonds, F.W. Shea.rd, G.A. Toombs and J.C. Portal, Surf. Sci. 196 {1988) pA04.
[5]. F.W. Shea.rd and G.A. Toombs, Appl. Phys. Lett. 52 (1988) p.1228.
[6]. In writing down this effective mass equation, we neglect the spatial
variance of the effective mass m entirely. For an overview of the
difficulties that result when this simplification is avoided, see: R.A.
Morrow, Phys. Rev. B 36 (1987) p.4836.
[7]. In this paper, 8](77) is encountered for j=-!,O,+!. For definitions and
approximations of the integrals, see: J .S. Blakemore, Solid. State
Electronics 25 (1982) p.1067.
[8]. Here, the use of the word "period" does not imply invariance under translation, but the presence of a peak in the Fourier spectrum.
142 Chapter 6
[9]. Notation and terminology are adopted from Ref.[4].
[10]. Even when nemitter is constant, the areal charge density in the emitter
will still fluctuate with changing B, due to a fluctuating screening length
for that layer.
[11]. With a field-dependent Fermi level Er(B), Eq.(13) is to be read as:
Br = mEr(O)/ti.e. [12]. For a detailed description of this model, see: H.P. Joosten, H.J.M.F.
Noteborn and D. Lenstra, Thin Solid Films 184 (1990) p.199; and
H.J.M.F. Noteborn, H.P. Joosten and D. Lenstra, Phys. Scripta T33
{1990) p.219.
[13]. Surprisingly, this "coherent" storage factor coincides with the
"sequential" one as given by Sheard and Toombs (see eqs.(5) and (8) in
Ref.[5]), at least for Re and Re close to unity. However, since they regard this factor as independent of the bias voltage, in their theory a
proportionality between Bfr and nwell still holds. [14]. M.L. Leadbeater, E.S. Alves, F.W. Sheard, L.Eaves, M. Henini, O.H.
Hughes and G.A. Toombs, J. Phys.: Condens. Matter 1 (1989) p.10605.
Magneto-tunneling 143
6.3. Magneto-tunneling in double-barrier structures: the B .1. J configurationto
Abstract - The peak in the current-voltage (I-V) characteristic of a Double-Barrier Resonant- Tunneling structure is broadened and lowered by the application of a magnetic field parallel to the layers. The broadening of the peak is r linear in the field. The lowering is completed at a field strength term the quenching field. Both effects are described within a model of coherent tunneling in a selfconsistent potential. The calculated I-V curves agree nicely with experimental data.
I. INTRODUCTION
A Double-Barrier Resonant-Tunneling (DBRT) strueture is a well-known
example of the novel devices based on the vertical transport mechanism. The
possibility of carrying current is based on the existence of a resonant state due to
the quantum well between the two barriers. This state can be accessible to
electrons in the Fermi sea of the reservoirs, formed by the doped regions that
sandwich the DBRT structure. If we assume the tunneling to be coherent,
accessibility amounts to the demand that the energy of the resonant state is in the Fermi window of the reservoir. Since the resonance energy with respect to
the reservoir is tunable by applying a voltage difference across the structure,
there is always an interval of applied voltages where current is possible .. This
vertical transport mechanism is essentially one-dimensional. The lateral
dimensions only come into play at determining the density of states.
When we now apply a magnetic field to a DBRT structure, perpendicular to
the growth axis and parallel to the barrier layers, some basic aspects of the
above picture are changed. The tunneling is no longer dependent on the
transverse motion alone. Accessibility of the resonant state is determined by the
exchange of momentum between transverse and lateral directions effectuated by
the magnetic :field. This leads to a smaller current density at voltages where in
the zero-field case current was possible, since some of the formerly resonant
electrons are now :filtered out on the basis of their lateral motion. At voltages
where in the zero-field case no current was possible, however, there will now be
found some current density, since the magnetic field opens up the resonant
IOThis section was published as a paper: H.J.M.F. Noteborn, G.H.M. van Tartwijk and D. Lenstra, 11Magneto-tunneling in double-barrier structures: the B.LJ con:figuration11
, J. Phys.: Condens. Matter 4 (1992), 4125-4134. References between square brackets [ ] are listed on page 155.
144 Chapter 6
channel for certain lateral momenta. In fact, the application of a perpendicular
field has made the tunneling into an essentially two-dimensional problem. In
comparison with the zero-field case, the resulting current peak is broadened and
lowered. The implications of the magnetic field can also be described in another way.
Let us introduce the difference between resonant states that are extended in both
reservoirs (and which we will call 'extended' states), and resonant states that are
evanescent in one of the two reservoirs (and which we will call 'semi-&tended'
states). The extended resonant states contribute to both the charge density in the well and the current density through the structure, whereas the semi
extended resonant states contribute only to the charge density in the well. In the
zero-field case this distinction is not needed since all resonant states are
extended. The perpendicular field, however, introduces the transformation of
extended states into semi-extended ones, and the larger the applied field
strength, the more complete this transformation is. At a certain field strength,
all electrons are forced into semi-extended states, and hence the current will be
zero, irrespective of applied voltage. This effect of the magnetic field can be
called a quenching of the current, and the field strength above which this takes
place is named the quenching field.
Experimental evidence for these effects of the transverse magnetic field was
presented by Gueret et aL [1,2] and Ben Amor et al. [3,4], both of who reported a
broadening and lowering of the current peak. Some theoretical work along
semi-classical lines was done by Eaves et al. [5,6,7], whose distinction between
'traversing' and 'skipping' orbits parallels our extended/semi-extended states. A
quantummechanical approach was undertaken by Ancilotto [8], considering a
somewhat different structure (showing less interesting properties). We only
mention here the work of Platero et al. [9], that constitutes a totally different
approach.
In this paper, a quantum mechanical description of coherent resonant
tunneling in the presence of a perpendicular magnetic field is presented. Starting
from the Schrodinger equation (Section II), we derive expressions for the voltage
interval where resonant charge build-up takes place, and for the voltage interval
where resonant current is found (Section III). In Section IV, numerical results
are presented for GaAs/ AlGaAs based structures. Finally we will compare our
results with both experimental and theoretical studies (Section V).
Magneto-tunneling 145
II. SCHRODINGER EQUATION WITH MAGNETIC FIELD
The usual way to introduce a magnetic field !} into the Schrodinger equation is
via the substitution ~ -1 ~1 + eA, where A is the vector potential, related to B }N TX. N N ' N
via B = VxA. Let us choose A to be defined by: N N N N
{
(0, 0 ,0) , z<O
~ = (0,-Bz,0) , O<z<L (0,-BL,O) , L<z
(1)
yielding a magnetic field in the x-direction of strength B if O<z<L, and of zero
strength outside this interval. Eq.(1) implies a Coulomb gauge ! · ~ = 0 and a coupling of the magnetic field to the y-<:omponent of the momentum only. This
choice for a magnetic field confined to the interval O<z<L agrees with the usual
and accepted approach in device modelling [1,5,6]. It is also supported by the
physical processes in the reservoirs, where the bulk scattering of carriers will
cause effective broadening of the Landau levels and, eventually a density of
states approaching the zero-field behaviour [5]. The value of L is thus expected
to be related to the mean free path of electrons in the reservoir. In this paper,
however, we will treat Las an extra parameter. Possibilities of determining L by
experiment will be discussed in Sect.V. The choice (1) has the advantage of
enabling a transfer matrix approach, with plane wave solutions in the reservoirs,
which are easy to interpret in terms of current density.
We insert the vector potential of (1) into the Schrodinger equation:
{2)
where '11(~) describes the electrons in the conduction band, m is the effective
mass of this l;>and and Ec0(z) is the band minimum. The z-axis is taken along
the growth direction. In general, the materials of the barriers and the well will
differ in both effective mass and band gap. We will however take into account
only the latter difference, and write Ec0(z), assuming m independent of z. In the
reservoir situated at z<O we have both Ec0(z)=O and ~(z)=g, so that the solutions of (2) are plane waves
exp(ikxx)·exp(ikyy)• [A·exp(ikzz) + B·exp(-ikzz)] at energy
E=~kx2+ky2+kz2). Since the Hamiltonian in {2) is invariant under translation
146
Fig.1
Chapter 6
0.3
> 0.2 -:::> + 8 0.1 w
o.o
0 10 20 30 40
z (nm)
Potential energy of electrons in the DBRT-structure as a function of position z, drawn for three different values of ky {kp, o.4kp, -0.2kp). The applied voltage Va = 0 V, the magnetic field strength B = 5 T.
in the lateral directions, the wave number components kx and k1 (contrary to
kz) are constants of motion. Substituting for w(r) the factorization N
exp(ikxx)·exp(ik1y)-F(z) we have (2) to read:
-fni·~F(z) + Ec0(z)F(z) + U(z;k1,B)F(z) = (3)
where U(z;k1 ,B) is a potential energy term introduced by the magnetic field,
that is quadratic in z and Band linear in k1 (see Fig.l):
z<O
O<z<L
L<z
In absence of Ec0 (z) in (3), we would find for O<z<L that
(4)
F(z) = A'D11(() + B1D_11-1(i(), where D
11(() is the parabolic cylinder function,
11 = Mk2:~kz2) -1, and ( = J 2[B (z -!~Y) [10]. A restriction to non-negative
M agneto-t'Unneling 147
integer v would render the well-known Landau levels. Usually, it is the
requirement for normalization on the z-interval (-ro,+m) that leads to this
quantization of energy [10]. However, since the quadratic potential in (4) applies
only to O<z<L, we have no Landau quantization in this tunneling problem, and
hence no Landau ·levels. Instead, we calculate the transmission and reflection
coefficients for incoming plane waves labeled with kz and scattered by a
ky~ependent potential. The eigen functions F(z;ky,kz) now depend on the
lateral momentum, in contrast to the zero-field case, although the eigenvalues
still depend on kz only. This means that (3) constitutes a tunneling problem
where we have to treat every combination (ky,kz) separately. The resulting
transmission and reflection coefficients will be functions of both transverse and
lateral momentum.
ill. VOLTAGE INTERVAL OF RESONANT CURRENT
To find the voltage interval where resonant charge build-up or current takes
place we make use of the fact that the resonant energy with respect to the band
minimum in the well is almost independent of the exact potential structure.
Hence we determine this energy E0 in the unbiased zero-field situation and treat
it as a constant. In this section, we also assume a constant electric field in the
structure, neglecting the effect of the charge build-up in the well on the band
bending. The potential energy in the well then equals -a·eVa, where Va is the
applied voltage and O<a<l depends on the structure parameters. For identical barriers, a = ,. Let us first consider the zero magnetic field case. For the
resonance energy to be in the Fermi window of the reservoir means:
0 < E0-a·eVa < EF , hence the voltage interval for resonant current and charge
build-up is: (E0-Er )/ea< Va< E0 /ea. In the case of a magnetic field, the potential term U(z;ky,B) of ( 4) should be
included, making these relations dependent on k1. Also, at fixed ky1 the possible
energies related to the transverse momentum are limited to
0 < li.2kz2/2m < EF-li.2ky2/2m, i.e. the window for the resonance energy is
reduced. The condition for resonant charge build-up therefore becomes:
(5)
where Zw is the position of the well. This can again be translated into a
Vadkv), Va2(kv) and Va3f'kv) for three different values of the magnetic field 'B {OT, ff.65T, 13.STJ. The states corresponding to points in the enclosed area contribute to the charge density in the well. The hatched area represents states that in addition contribute to the current through the structure.
(6)
that is sketched in Fig.2. For resonant current to flow, an additional condition
with no zero-field analogue is to be introduced. The transverse momentum in
the collector reservoir should be positive in order to enable an electron to
contribute to the current. This yields:
(7)
which is trivial for B=O. Translated in terms of Va, Eq.(7) reads:
(see Fig.2). If an electron state with momentum ky and resonant kz satisfies
Magneto-tunneling 149
condition (6) but not (8), it contributes to the charge density in the well only.
This is a socalled 'semi-extended' state. If the electron state meets both
conditions (6) and (8), it contributes to both the charge and the current density,
and is called 'extended'. The voltage interval that results from (6) depends on ky. Thus, at a given
applied voltage V a.i (6) will be met by only a fraction of all ky-values. This
implies a decrease in charge density, compared with the zero-field case. Using
the fact that fi.2ky2/2m < EF, we can define voltage intervals where we have
taken into account the contributions of all ky. For the charge build-up this
means that we have to find the minimum of Va.1(ky), denoted by Va. 11 and the
maximum of V a.2(ky ), denoted by V a.2• The latter is equal to V a.2(-kF) for all
field strengths, kF being the Fermi wave number. For B < B0 = tik.F/ezw 1 Va.1 = Va.1(eBzw/fi). For larger field strengths, it is Va.1(+kF)· Hence, we find for the
sum of all ky-eontributions that, in order for the charge in the well to be nonzero
at Va• Ya should satisfy:
Va.1(B) <Va< Va.2(B)
(9)
These interval bounds as functions of B are shown in Fig.3a. From (9) we see
that dVa1/dB ~ 0 and dV a.2/dB > 0, hence both bounds are non-decreasing
functions of B. Since the upper bound increases faster, the total voltage interval
for charge build-up is broadened, see Fig.3b. For B ~ B0 this broadening is
linear in B: l::i. Va = Va.2 - Val = ezw24BB0/2ma = (2n.k:FZw/ma)- B. To find out whether also the total voltage interval for resonant current is
broadened, we have to take into account the effect of condition (8) on the
bounds. This is an easy but complicated matter, depending on a, zw/L and
Er /E0 • Therefore, we will only give the results for the special case that
(Zw/L)2 < a< zw/L and Er < E0/(1-2aL/zw+aL2/zw2). Since experimentally
a ::: i 1 Zw/L ::: i and E0 > > EF, this is the most relevant case. For small field
150
Fig.3
Chapter 6
~ 0.3
0.8 >. . 0.2
~ >1 ii
I 0.1
>. 0.4 ~
>' 6 10
>':;, 0.2 arn
o.o 0 5 10 15
B (T)
cu cu (a) Vai{B} and Va2fB}, as well as VadBJ and Va2{B}. For Va between VatfB) and Va2(B) electrons can enter the well resonantly. For Va
cu cu between Va dB) and Va 2(B} they can leave the well at the collector side. {b) The width of the current peak t:,. Va::: Va2 - Vat as a junction of the magnetic field strength B.
strengths, we find the same bounds as in the charge build-up situation, as
expected. IC, however, B exceeds a value B_:
the lower bound is changed to:
cu 1 [ . ] Va 1(B) = "i;rr;=a - E0/e + eB2zw(L-zw)/2m ,
cu • whereas the upperbound remains unchanged, Va2(B) = Va2(B). This new lower
cu bound Va 1(B) increases more rapidly than the upper bound, so that a.t B = B+
the two bounds coincide and the voltage interval disappears completely.
Therefore, we say that the current is 'quenched' at B=B+ and we call this field
Magneto-tunneling 151
Fig.4
40
30 -E c 20 -N 1st barrier
10
0
-400 -300 ·200 ·100 0
y (nm)
Classical trajectory of particle in crossed electric and magnetic field to illustrate the quenching field B. of Eq.(10). ky = - k, = -1.8· 108/m; B = B. = 1.4T.
strength B. the 'quenching field'. A particularly simple form for this quenching
field is obtained in the limit ~/E0-i0, i.e. in the case of a small well width:
-'Jd 2mE0 B. - e zw( L-zw,) (10)
The same expression is found for the special case that a = zw/L. Eq.(10) allows
a classical or geometric interpretation (see Fig.4): a particle having a transverse
momentum Pz = J 2mE0 at z = Zw will move along the curtate cycloid
y = iL(sinrp-cp) -!~~cp, z = !L(l-coscp) if B = B.. The general expression,
depending also on Er and a, lacks such a transparent interpretation.
In this section, we have demonstrated the lowering and broadening of the
current peak due to the application of a perpendicular magnetic field. We have
assumed that the resonance energy can be treated as a constant, a.nd that the
transmission peak has negligible width. We have only considered resonant
current a.nd charge build-up, and have ignored all demands of selfconsistency.
The quenching field, following from this analysis, is a consequence of the
transformation by the magnetic field of extended states into semi-extended ones.
152
Fig.5
Fig.6
Chapter 6
25
20 tr E 15 < b ::::. ..,w 10
5
0 0.0 0.2 0.4 0.6 0.8 1.0
v. (V)
1-V .curves for eight different values of the magnetic field strength B, resulting from selfconsistent calculations.
18
15 0.3V
tr 12 E < b 9 ::::. ..,w 6
3
10 20 30 40
magnetic field B (Tesla)
1-B curves for three values of the applied voltage Va, resulting from selfconsistent calculations.
Expressions for this field, calculated for the zero-temperature case; provide an
estimate for the length L over. which the magnetic field is effective. In the next
section, we present numerical calculations in which some of the above
mentioned restrictions are avoided.
Magneto-tunneling 153
IV. NUMERICAL RESULTS
To present 1-V curves for structures in a perpendicular magnetic field, numerical
calculations were done, assuming a GaAs/ AlGaAs structure characterized by an
effective mass of 0.067 times the free electron mass, and a band discontinuity of
0.44eV. Barrier widths are 5.6nm, the well is 5.0n.m wide. Details of the model
can be found in Refs.[11]. Here we only mention the adjustments to the magnetic
field situation. The Schrodinger equation (3,4) is quadratic in the coordinate z.
Its basic solutions can therefore be choosen to be parabolic cylinder functions
[10]. However, to avoid the difficulties inherent to working with these special
functions and to reduce computational time, we have approximated the potential
in each of the five layers (emitter, barrier, well, barrier, collector) by its average
value, thus obtaining plane waves at every position in the structure. This
apparently drastic approximation turns out to have little effect on the I-V
characteristics [12] while shortening calculations considerably. The length Lover
which the magnetic field is thought to be effective is taken to equal the structure
length [5]. The effect of the charge density in t.he well on the band bending is
taken into account selfconsistently. The main difference with the zero-field
ca.lculations is that the summation over the lateral momenta can now not be
done analytically, but necessitates an extra loop, enormously enlarging
computational times.
In Fig.5 a series of I-V curves is plotted for magnetic field strengths ranging
from ST to 30T. The lowering and broadening of the current peak, anticipated in
the previous section, are confirmed by the selfconsistent calculations. The
quenching field is more easily illustrated by the I-B curves of Fig.6. Here, the
effect of the charge build-up on the band bending is seen in the s:tn.all shift of the
quenching field to larger values for increasing applied voltage Va.· The value of
the quenching field, around 30T, may seem experimentally out of reach.
However, these unrealistically large fields are the. price we have to pay for the
simplification of equalling L with the structure length, thus greatly
underestimating L. Neglecting arguments of selfconsistency, we may say that the
quenching field is inversely proportional to L, so that a three times larger L
leads to fields that are accessible to experiment.
V. DISCUSSION
Nice agreement with the experimental findings of Ben Amor and coworkers [3,4]
154 Chapter 6
is found. Their I-V curves (Fig.1 in Ref.[4]) agree well with the ones we present
in Fig.5. Also, the observed linear increase of the width of the current peak with
B [1,2,4] is confirmed by our analysis. For B0 < B < B. a linear relation holds
for the voltage interval width and the magnetic field strength, corresponding to
a slope that equals 211.krzw/ma. Taking zw/L =a, this agrees well with the results of Ref.[4]. For B < B0 a quadratic increase is predicted by Eq.(9), whereas for B > B. a change from increase to decrease is expected. These two
regions, however, a.re not covered by the experimental data. Because of a bad
choice of structure parameters, Ancilotto does not find a substantial broadening
of the current peak [8].
The decrease of the peak current with increasing magnetic field is found in all
experiments [1,4]. However, this decrease does not continue as to be expected in
our model. In fact, the peak current reaches a minimum a.round 16T for the
AllnAs/GalnAs structure [4], and around 5T for the GaAs/ AlGaAs structure
(see Fig.9 in Ref.[1]). This is thought to be an indication that coherent tunneling
cannot be the whole story [2]. Sequential effects, or better: inelastic scattering
processes, seem to be more important in GaAs/ AlGaAs than in AllnAs/GainAs.
As a consequence, the quenching field is not directly obtainable from
experiments. It can only be extrapolated from the initial decrease of the peak
current. However, the linear increase of the current width with magnetic field
still provides a means of estimating the length L, over which B is to be taken
into account [1]. In the GaAs/AIGaAs structure, L is found to be "'35nm [1], corresponding to a quenching field of about lOT. In the AllnAs/GalnAs sample,
the quenching field is larger due to a much smaller effective length L (8.5nm [4]).
The decrease of. the turn-on voltage Va1{B), reported in Ref.[4], is not
expected from our analysis (see {9)), nor confirmed by the measurements of
Ref.[1] or the analysis in Ref.[6]. However, the accuracy of the data supporting
this result in Ref.[4] is probably such that they could also support Leadbeater's
or our analysis. This minor point of difference does not affect the overall agreement.
The difference between extended and. semi-extended states, introduced in
Sect. III, is also met in Eaves et al. [7]. There, the two types of state are
associated with 'traversing' and 'skipping' orbits. The latter, interacting with
the emitter barrier only, contribute to the density in the well. In a traversing
state an electron is repeatedly reflected off both barriers. In the semi-classical
picture, the important length scale is set by the well width. In the quantum
Magneto-tunneling 155
mechanical picture, this role is played by the effective length L. Although a
theory for L is missing, it is thought that L is related to the mean free path
rather than to the well width. Hence, both pictures may not be totally identical.
From the derivation of (9), it is clear that the expressions for Va1 and Va2
depend on the applied dispersion relation for the conduction band, in our model
a simply quadratic one. Inclusion of non-parabolicity or an energy-dependent
effective mass, or - in the case of holes - band mixing, will yield different
expressions for these quantities. Conversely, experimental determination of
quantities like Va1(B) and Va2(B} provides a powerful method for investigating
the dispersion curves [13].
Summarizing, we have presented a quantummechanical study of the effect of a transverse magnetic field on coherent resonant tunneling. The calculated I-V
curves agree nicely with experimental data. The quenching field, although
experimentally obscured by incoherent processes, is still a valuable quantity,
containing information about the resonance energy and the effective length L for the vector potential. The presented model provides a good description for
structures in whi.ch coherent tunneling is dominant.
Acknowledgements - We would like to thank H.P. Joosten for fruitful
discussions. This work is part of the research programme of the Foundation for
Fundamental Research on Matter (FOM), which is financially supported by the
'Nederlandse Organisatie voor Wetenschappelijk Onderzoek' (NWO).
References
[1]. P. Gueret, C. Rossel, E. Marclay and H. Meier, J. Appl Phys. 66 (1989} p.278
[2]. C. Rossel, P. Gueret and H.P. Meier, J. Appl. Phys. 67 (1990) p.900
(3]. S. Ben Amor, K.P. Martin, J.J.L. Rasco!, R.J. Higgins, A. Torabi, H.M.
Harris and C.J. Summers, Appl. Phys. Lett. 53 (1988) p.2540.
[4]. S. Ben Amor, J.J.L. Rasco!, K.P. Martin, R.J. Higgins, R.C. Potter and
H. Hier, Phys. Rev. B 41 (1990) p.7860
[5]. L. Eaves, K.W.H. Stevens and F.W. Sheard, in The physics and Fabrication of Microstructures and Microdevices, Springer Series in
. Physics 13 (1986) p.343.
156 Chapter 6
(6]. M.L. Leadbeater, L.Eaves, P.E. Simmonds, G.A. Toombs, F.W. Sheard,
P.A. Claxton, G. Hill and M.A. Pate, Solid State Electronics 31 (1988)
p.707.
[7]. L. Eaves, E.S. Alves, M. Henini, O.H. Hughes, M.L. Leadbeater, C.A.
Payling, F.W. Sheard, G.A. Toombs, A. Celeste, J.C. Portal, G. Hill and
M.A. Pate, in High Magnetic Fields in Semiconductor Physics II, Ed. G.
Landwehr, Springer Verlag Heidelberg (1989) p.324. [8]. F. Ancilotto, J. Phys. C: Solid State Phys. 21 (1988) p.4657.
[9]. G. Platero, P.A. Schulz, L. Brey and C. Tejedor, Surf. Sci. 228 (1990)
p.291, and references therein.
[10]. See e.g. A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions 2, McGraw-Hill, New York 1953, p.116;
E. Merzbacher, Quantum Mechanics, John Wiley & Sons,Inc. New York
1961, p.70; W. Greiner, Quantum Mechanics, Theoretical Physics 1, Springer-Verlag Berlin Heidelberg 1989, p.114.
[11]. H.P. Joosten, H.J.M.F. Noteborn and D. Lenstra, Thin Solid Films 184
(1990) p.199; H.J.M.F. Noteborn, H.P. Joosten and D. Lenstra, Phys.
Scripta T33 (1990) p.219; H.J.M.F. Noteborn, G.H.M. van Tartwijk,
H.P. Joosten and D. Lenstra, J. Phys.: Condens. Matter 3 (1991) p.4249.
[12]. See e.g. E.E. Mendez, in Physics and Applications of Quantum Wells and Superlattices, eds. E.E. Mendez and K. von Klitzing, Plenum Press New
York - London 1987, p.159.
[13]: L.A. Cury, A. Celeste, B. Goutiers, E. Ranz, J.C. Portal, D.L. Sivco and
A.Y. Cho, Superlattices and Microstructures 7 (1990) p.415; R.K.
Hayden, D.K. Maude, L. Eaves, E.C. Vala.dares, M. Henini, F.W. Sheard,
0.H. Hughes, J.C. Portal, and L. Cury, Phys. Rev. Lett. 66 (1991) p.1749.
EVALUATION AND OUTLOOK
The DBRT model developed in the chapters 3 and 4 ha.s proved meritorious in
describing the diode's intrinsic bista.bility (chapter 4) and its behaviour under ac
conditions (chapter 5) and in configurations with magnetic fields (chapter 6).
Charge storage in the well ha.s been taken into account, and its effects on the
diode impedance ha.s been studied; the concepts of fundamental field and
quenching field have been included. Thus the model ha.s coupled simplicity with
the ability to incorporate new concepts a.nd to a.llow necessary extensions.
Ultimately the usefulness of the model is decided on by the numerical
agreement of its predictions with the experimental findings. However, the
physics of quantum devices is not yet a.t the stage where numbers ca.n put a. veto. The difficulty (and cha.llenge!) of this research programme is in the
entangling involvement of many physical phenomena.. Experimental data. for
their pa.rt a.re refined with every technological development, and the
Pea.k-to-Va.lley Ratio, frequency limit and output power are still going up. In
this situation, a. numerica.l accordance between theory and experiment ma.y be as
questionable and disquieting a.s a. discordance. The scattering para.meter 7
(section 3:6) ma.y serve as an example. Although it allows to fit any P.V.R., its
value ha.s to be adjusted with every new insight, and must hence remain
insignificant. An example of opposite character is the use of different effective
masses for the contacts and barriers. Regardless of the fit, this reflects a. real
157
158 Evaluation and outlook
aspect of the physical situation.
The lines sketched in the first few chapters do not just support the simple
model calculations of the subsequent chapters, they also show ways how to go
beyond the present model. First of all, the band structures of the diode materials
are only poorly represented by parabolic conduction bands. Here the energy
dependent mass of chapter 2 is a first attempt, promising in the case of electron
tunneling, to incorporate nonparabolicity effects within the framework of the
present theory. A logical sequel would be the inclusion of the valence band using
the full Kane method1• Intervalley tunneling, however, is clearly beyond the
local Kane method and, consequently, outside the scope of our model.
A second problem is posed by the accumulation region at the emitter side.
For the calculation of the subband energies, the Transfer Matrix Approach is
very inefficient and· has to be replaced by numerical integration methods2. A
general draw-back of the TMA is the fact that it is limited to simple potentials
(piecewise constant, linear) with known fundamental solutions (plane waves,
Airy functions). The determination of the subband occupation draws once again
attention to the problem of the coupling between structure and reservoir:
non-equilibrium considerations can be valued only at the cost of major changes.
For many applications, on the other hand, the Thomas-Fermi approximation of
chapter 4 gives a fair description of the potential distribution, as is shown from
the voltage-dependence of the fundamental field3•
The possibilities to incorporate scattering effects (elastic• and inelastic5) are
being explored successfully. For computational reasons these developments a.re
limited to rather simple structures. Experimentally a shift is observed towards
more complex devices, in which one or more DBRT structures a.re integrated&.
To the description of these and other multi-barrier structures the presented
model is very useful. These application-oriented studies a.re expected to take an
increasingly important pa.rt in future research on quantum devices.
1See Calvin Yi-Ping Chao and Shun Lien Chuang, Phys. Rev. B 43 (1991) 7027.
2T.G. van de Roer, 0. Abu-Zeid, H.C. Heyker, J.J.M. Kwa.spen, H.P. Joosten, H. Notebom, D. Lenstra. and M. Henini, Procs. 15th annual semicond. conf CAS 1992, Sinaia Romania, 1992; p.557.
3See Fig. 4.2 on page 77. 4H.C. Liu and D.D. Coon, J. Appl. Phys. 64 (1988) 6785. 5R.J.P. Keijsers, H.J.M.F. Noteborn and D. Lenstra., unpublished, 1992. 6See section 1.1, and references therein.
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SUMMARY
Not only has the miniaturization penetrated the field of micro-electronics to the
point that the quantum character of matter has to be reckoned with, but also
there is nowadays a new class of semiconductor structures, the operation of
which is directly based on quantum mechanical phenomena such as tunneling.
These developments are interesting both from the viewpoint of applications and
for reasons of theoretical modeling. An important class of these nanostructures is
constituted by the heterojunction superlattices having two or more barriers, the
shortest of which - the Double-Barrier Resonant-Tunneling (DBRT) diode -
has been chosen for a model system in our study.
An interesting property of DBRT structures is the negative differential
resistance (NDR) in the current-voltage characteristics, making them useful
components in amplifiers, mixers and detectors. If the structure parameters a.re
chosen favourably, the charge build-up in the well between the barriers is
sufficiently substantial to give cause to a current bistability in the NDR region.
In this thesis, a DBRT model is presented, in which the electric current is
considered a tunnel current of coherently propagating electron w.aves between
two reservoirs. Varying the external bias voltage, one ca.n tune the resonance
energy in the well with respect to the Fermi seas in the reservoirs. The
167
I
168 Summary
electrostatic feed-back due to the charge displacements in the structure is taken
into account in the Hartree approximation. Both the NDR and the bistability
are within the domain of this model.
This simple DBRT model, containing in broad outline all the relevant
physics, has been the starting point for investigating more complex
configurations. It has been extended to situations with magnetic fields, parallel
or perpendicular to the barrier layers. In the parallel configuration, magneto
oscillations in the current at fixed bias arise as a function of the magnetic field.
From the spectrum of these oscillations, the charge densities in the well and the
accumulation layer can be determined. In the perpendicular configuration, a
shift of the current peak position is worked out, that is proportional to the field
strength. The peak height decreases to disappear at a so-called quenching field,
proportional to the square root of the resonance energy.
The stability of the static (de) solutions is investigated by using a small
signal analysis. In addition, this approach yields an impedance model and an
equivalent circuit for the DBRT diode, allowing us to study the dynamical
behaviour of the diode and the effect of the external circuit on the frequency
characteristic.
Numerical calculations based on the presented model agree with the
experimental findings on trends and order of magnitude. Quantitative agreement
can be improved by using non-parabolic bands and by enlarging the model to
include the contact regions. An outlook to further research developments
concludes the thesis.
SAMENV A 'ITING
Niet alleen is de miniaturisering in de micr~lectronica zover doorged.rongen
dat rekening gehouden moet worden met het quantummechaniscbe karakter van
de materie, maar ook worden er tegenwoordig halfgeleiderstructuren ontworpen en gef'abriceerd, waarvan de werking direct gebaseerd is op quantummecbaniscbe
verschijnselen zoals bijvoorbeeld tunneling. Deze ontwikkeling is interessant zo
wel vanuit bet oogpunt van de toepassingen, als vanuit de theoretische modellering. Een belangrijke klasse van deze nano-structuren vormen de zogenaamde
beterojunctie-superroosters met twee oi meer barrieres, waarvan de kortste, de
"Double-Barrier Resonant-Tunneling" (DBRT) diode, a1s modelsysteem in onze
studie is gekozen. Een interessante eigenschap van DBRT structuren is de negatieve
differentiele weerstand (NDR) in de stroom-spanningskarakteristiek, die ze
geschikt maa.kt als componenten van versterkers, mixers en detectoren. Bij
gunstig gekozen structuurparameters is de ladingsopbouw in de put tussen de
barrieres zo belangrijk, dat ze aanleiding geeit tot een bistabiliteit in de stroom in bet NDR interval.
In dit proeischriit wordt een model voorgelegd, waarin de electriscbe stroom
wordt beschreven als een tunnelstroom van coherent propagerende electron-
169
170 Samenvatting
golven tussen twee reservoirs. Door de uitwendig aa.ngelegde spanning te
va.rieren, kan de resonantie-energie in de put afgestemd worden ten opzichte va.n
de beide Fermi-zeeen in de reservoirs. De electrostatische terugkoppeling als
gevolg van de ladingsbewegingen in de structuur wordt in de Hartree-bena.dering
meegenomen. Zowel de NDR a.ls de bistabiliteit kunnen met dit model goed
beschreven worden.
Dit eenvoudige DBRT model dat de releva.nte fysica in eerste aa.nzet beva.t,
heeft als uitgangspunt gediend voor de bestudering va.n uitgebreider
configuraties. Zo is het geschikt gema.a.kt voor situa.ties met ma.gneetvelden,
parallel aa.n of loodrecht op de barriere-lagen. In het parallelle geval treden er
magneto--oscillaties op in de stroom bij vaste spanning a.ls functie van het
magneetveld. Uit het spectrum va.n deze oscillaties kunnen de ladingsdichtheden
in de put en in de accumula.tielaag worden bepa.ald. In de loodrechte configuratie
vindt men een verschuiving, evenredig met de veldsterkte, van de positie va.n de
stroompiek. De piekhoogte neemt af en verdwijnt geheel bij een zogenaamd
uitdoofveld, dat evenredig is met de wortel uit de resona.ntie-energie.
De sta.biliteit va.n de statische (de) oplossingen is onderzocht met behulp van
een kleine-signaala.nalyse. Bovendien levert deze aa.npa.k een impedantiemodel en
een verva.ngingsschema voor de DBRT-diode op. Hiermee is het tijda.fhankelijke
gedrag van de diode en de invloed van de toevoerkring op de frekwentie
ka.rakteristiek bestudeerd.
Numerieke berekeningen gebaseerd op het beschreven model komen goed
overeen met experimentele bevindingen in trends en orde van .grootte.
Qua.ntita.tieve overeenstemming kan worden verbeterd door gebruik te ma.ken
van niet-parabolische ban.den, en door uitbreiding va.n het model met een
beschrijving van de contactla.gen. Een a.a.ntal suggesties voor verder onderzoek
besluit het proefschrift.
LIST OF PUBLICATIONS
H.P. Joosten, H.J.M.F. Noteborn and D. Lenstra, Thin Solid Films 184 (1990).
H.P. Joosten, H.J.M.F. Noteborn, K. Kaski and D. Lenstra, J. Appl. Phys. 70
(1991) 3141.
H.P. Joosten, H.J.M.F. Noteborn, K. Kaski and D. Lenstra, Physica B 175
(1991) 297.
H.P. Joosten, H.J.M.F. Noteborn, K. Kaski and D. Lenstra, Superlattices and
Microstructures 12 (1992) 53.
R.J.P. Keijsers, H.J.M.F. Noteborn and D. Lenstra, unpublished (1992).
H.J.M.F. Noteborn, H.P. Joosten and D. Lenstra, Phys. Scripta T33 (1990) 219.
H.J.M.F. Noteborn, H.P. Joosten, D. Lenstra and K. Kaski, SPIE Vol. 1675
Quantum Well and Superlattice Physics IV, 1992; p.57.
H.J.M.F. Noteborn, G.H.M. van Tartwijk, H.P. Joosten and D. Lenstra, J.
Phys.: Condens. Matter 3 (1991) 4249.
H.J.M.F. Noteborn, G.H.M. van Tartwijk and D. Lenstra, J. Phys.: Condens.
Matter 4 (1992) 4125.
H.J.M.F. Noteborn, H.P. Joosten, K. Kaski and D. Lenstra, Superlattices and
Microstructures (1993), in print.
H.J.M.F. Notebom, D. Lenstra and W. van Haeringen, Bull. Am. Phys. Soc. 37
E22 10 (1992); p.245.
T.G. van de Roer, H.C. Heyker, L.M.F. Kaufmann, J.J.M. Kwaspen, M.
Schemmann, H.P. Joosten, D. Lenstra, H. Noteborn, M. Henini and
O.H. Hughes, Procs. 16th Int. Symp. on Ga.As and related compounds 1989, Bristol: IOP, 1990; p.831.
T.G. van de Roer, J.J.M. Kwaspen, H. Joosten, H. Noteborn, D. Lenstra and M.
Henini, Physica. B 175 (1991) 301.
T.G. van de Roer, 0. Abu-Zeid, H.C. Heyker, J.J.M. Kwaspen, H.P. Joosten,
H. Noteborn, D. Lenstra and M. Henini, Procs. 15th annual semicond. con/ CAS 1998, Sinaia Romania, 1992; p.557.
171
· STELLINGEN behorende bij het proefschrift van
Harry Noteborn
Eindhoven, 7 mei 1993
1. De foutievel vertaling in Richteren 4:21 van het hebreeuwse raqqah met
"slaap2 (van het hoofd)" verdoezelt de sexuele geladenheid van de betreffende
bijbeltekst.
1Mimi Deckers-Dijs, "Begeerte in bijbelse liefdespoezie", Kampen: Kok, 1991. M. Rozelaa.r, Amsterdamse cahiers voor exegese en bijbelse theologie 7 (1986) 123. 2Nederlands Bijbelgenootschap, "Bijbel; Nieuwe Vertaling", 1951.
2. Dat Deutero-Jesaja niet in maar pas na de ballingschap schreefl, betekent
slechts een kleine verandering van de chronologie, maar een aardverschuiving
in het verstaan van wat bijbelse profetie inhoudt.
1H. Leene, Amsterdamse cahiers voor exegese en bijbelse theologie 8 {1987) 28.
3. Uit de weergave van het Franse conception virginale - zes maal "onbevlekte
ontvangenis", een maal "maagdelijke geboorte" - in de Nederlandse versie
van Rene Girard, Des choses cachees depuis la fondation du monde, blijkt dat
de vertalers geen weet hebben van het verschil tussen het algemeen christe
lijk belijden aangaande de maagdelijke geboorte van Christus, en het puur
rooms-katholieke dogma van de onbevlekte ontvangenis dat op Maria betrek
king heeft.
R. Gira.rd, "Wat vanaf het begin der tijden verborgen was ... ", Kampen: Kok Agora, 1990; p.267vv.
4. De recente rehabilitatie1 van Galileo Galilei door het Vaticaan moet, gezien
de wetenschappeliJ1ce aanvechtbaarheid van zijn Copernicanisme2 en de theo
logische onhoudbaarheid van zijn atomisme3, veeleer gezien worden a1s een
kerkstrategisch gebaar in de richting van de katholieke intelligentsia.
5. Wie zijn boek1 over natuurkunde 11De bouwstenen van de schepping" noemt,
sticht een creationistische2 verwarring.
1G. •t Hooft, "De bou wstenen van de schepping", Amsterdam: Prometheus, 1992. 2c. Houtman, S. de Jong, A.W. Musschenga en W.J. van der Steen, "Schepping en evolutie:. het creationisme een alternatief?", Kampen: Kok, 1986.
6. Artikel VIlI.2 van de concept-kerkorde1 van de Verenigde Reformatorische
Kerk in Nederland laat met zijn retorische 11voor wie en door wie" de moge
lijkheid van de kinderdoop open, en miskent dus (in niet geringer mate dan
artikel XV van de Hervormde Kerkorde uit 19512) het beslissingskarakter
van de doop3• Aangezien het rechte verstaan van de doop van cruciaal belang
is voor het !even en werken van de gemeente, maakt het ontbreken hiervan in
de concept-kerkorde deze als 11kanaal voor het belijden en het besturen11 vol
strekt ondeugdelijk.
1Werkgroep Kerkorde SoW, "Concept-kerkorde van de Verenigde Reforma.torische Kerk in Nederland", Driebergen, oktober 1992. 211 Kew:e uit de kerorde", •s-Gravenha.ge: Boekencentrum, 1984.
3K. Barth, "Die Taufe a.ls Begriindung des Christlichen Lebens", Kirchliche Dogmatik
IV /4, Ziirich: EVZ-Verlag, 1967.
7. Tegen ouders die de autorijles van hun kind willen betalen, moet de mogelijk
heid van juridische dwang worden overwogen. Men kan bepleiten de daders
tijdelijk van de ouderlij1ce macht te ontheffen.
Vergelijk dagblad Trouw, 25 september 1992; p.4, waarin de mening van mr. J.E. Doek over polio-va.ccinatie wordt weergegeven. 1992 telde enkele tientallen poliogevallen; voor 1991 kwam het CBS uit op 48672 verkeerssla.chtoffers.
