A Physical Si 1-x Ge x /Si Heterojunction Bipolar Transistor Model for Device and Circuit Simulation M. Andersson, Z. Xia, P. Kuivalainen and H. Pohjonen VTT Electronics, Electronic Materials and Components, Integrated Circuits Olarinluoma 9, FIN-02200 Espoo, Finland Teknillinen Korkeakoulu Sähköteknillinen osasto Elektronifysiikan laboratorio Helsinki University of Technology Faculty of Electrical Engineering Electron Physics Laboratory Reports in Electron Physics 1994 8 Otaniemi 1994
Transcript
-2
A Physical Si1-xGex/Si Heterojunction Bipolar Transistor Modelfor Device and Circuit Simulation
M. Andersson, Z. Xia, P. Kuivalainen and H. PohjonenVTT Electronics, Electronic Materials and Components, Integrated CircuitsOlarinluoma 9, FIN-02200 Espoo, Finland
Helsinki University of TechnologyFaculty of Electrical EngineeringElectron Physics Laboratory
Reports in Electron Physics 1994 8
Otaniemi 1994
-1
A Physical Si1-xGex/Si Heterojunction Bipolar Transistor Modelfor Device and Circuit Simulation
M. Andersson, Z. Xia, P. Kuivalainen and H. PohjonenVTT Electronics, Electronic Materials and Components, Integrated Circuits
Olarinluoma 9, FIN-02200 Espoo, Finland
IMPORTANT NOTE:
The current SiGe HBT model in APLAC is based on the ideas and concepts in this report, butit has been modified and developed, and so does not correspond exactly to the model presentedhere.
A physical but compact Si1-xGex/Si heterojunction bipolar transistor (HBT) model suited fo
device design and circuit simulation is presented. The model is based on the de G
Kloosterman formalism for the modeling of the bipolar transistors, but adds impor
heterostructure device physics as well as physical properties of SiGe material. The m
implemented in the APLAC circuit simulator, shows how currents and charges depen
minority carrier concentrations, which in turn are functions of the heterojunction voltages
this way the influence of the built-in electric fields due to Ge concentration and doping de
gradients, the bias-dependent transit times and the Early effect can be incorporated natu
Comparisons between the model prediction and the experimental data for the DC cu
voltage characteristics and cutoff frequencies in Si1-xGex/Si HBT’s are included to demonstrate
the model utility and accuracy.
1. INTRODUCTION
The application of strained silicon-germanium alloys to heterojunction bipolar transistor (H
technology provides many advantages over the conventional silicon homojunction bi
transistors. The SiGe-base HBT’s allow extreme vertical scaling without excessive
resistance for high speed performance. In addition to having already established a record
frequency performance of 75 GHz [1], SiGe-base HBT’s can also offer analog designers a
high current gain through dramatically improved Early voltages. As a result, HBT’s
prominent devices for both analog, digital and mixed signal applications.
IC design based on SiGe HBT devices will require a physical device model for cir
simulation. The model should properly account for the effect of the heterojunctions and
uniform Ge and doping densities on the device speed performance. Empirical bipolar tran
models in SPICE [2] may describe the I-V characteristics of the HBT’s adequately, but the
not simulate accurately the stored charge dynamics, which determines high speed perfor
Since the device physics in AlGaAs/GaAs heterojunctions differs a lot from that in Si1-xGex/Si
heterojunctions, the physical models developed for III-V compound devices [3] canno
applied as such to the SiGe-based HBT’s. The first physical model for SiGe HBT’s was rec
developed by Yuan [4]. The model was developed having the device design and optimizat
mind, and it was not intended for circuit simulation. The first circuit simulator level model
4
n the
d in
Aner, are
ations,enceratedctureunt for
[5],t. We
n beLAC
ly theopingmodelrs are
d the
tant ofceultinginkage.in theange, built-tion.
SiGe HBT’s was recently briefly presented by Hong et al. [5]. This model was based o
previous bipolar transistor modeling work [6] by the authors. The model, implemente
SPICE2, was physical, requiring iterative (semi-numerical) solution.
In this paper, we present a compact but physical model for Si1-xGex/Si HBT’s, based on thebipolar transistor modeling originally developed by de Graaff and Kloosterman [7].advantage of their approach is that currents and charges, independently from each othrelated to the injected minority carriers at the emitter and collector edges. These concentrin turn, are functions of the base-emitter and base collector voltages. In this way the influof the built-in electric fields, the bias-dependent transit times and Early effect are incorponaturally. We have modified the de Graaff-Kloosterman model by adding the heterostrudevice physics and the physical properties of the SiGe base. Our model equations accovalence band discontinuity, heavy doping effects and high collector current effects. As inalso the additional built-in field due to the Ge concentration gradient is taken into accounderive the basic current and charge equations for the Si1-xGex/Si HBT’s and summarize thephysical model functions for SiGe material. The resulting analytical HBT model, which caused both for predictive device design and circuit simulation, is implemented in the APcircuit simulator [8], and the model prediction is compared to measured data.
2. SiGe/Si HETEROSTRUCTURE PHYSICS
An advantage of a physics-based semiconductor device model is that it requires onknowledge of the material parameters and device processing information, such as dconcentrations, device geometry and layer thicknesses, thereby reducing the number ofparameters. However, this necessitates that the model functions for material parametestrictly based on the material physics.
The incorporation of germanium significantly changes the physics of the base region anbase-emitter and base-collector junctions in a Si1-xGex/Ge HBT [9,10]. Addition of Ge reducesthe bandgap of Si, leading to the narrow bandgap SiGe base of the HBT. The lattice consthe Si1-xGex alloy differs considerably from that of Si, but in thin SiGe layers the lattimismatch is accommodated elastically and no misfit dislocations are formed. The resSiGe pseudomorphic layers are considerably strained, which enhances the bandgap shrThe incorporation of Ge also modify the energy band structure and density of statesconduction and valence bands. In addition, charge carrier mobilities and diffusivities chdue to changes in the effective masses and alloy scattering. Finally, the dielectric constantin potentials and depletion widths in the pn-heterojunctions depend on the Ge concentra
5
s andd Ge
in a
Next we derive and summarize the new modeling functions for the material parameterheterojunctions in the SiGe base. Fig. 1 shows the band diagrams, typical doping anconcentration profiles in a Si1-xGex/Si HBT.
Figure 1. Qualitative energy band diagrams and doping and Ge concentration profilestypical Si1-xGex/Si HBT. dEB(dBE) is the depletion layer width on the emitter(base) side of the e-b junction, and dBC(dCB) is the depletion layer width on thebase (collector) side of the b-c junction.
emitter base collector
EcSi
n-Si p-Si1-xGex n-Si
∆Ev
∆Ep
∆En
EvSi
∆Ec
EFn
EcSiGe
EvSiGe
EFp
EgSiqVbi
Impu
rity/
Ge
Con
cent
ratio
n (a
.u.)
yWB WBM
Ge
dEB dBE dBC dCB
6
ity oftrainedand
and intwo-the]
ive
band
ergy
In a BJT model the intrinsic carrier concentration, which depends on the effective densstates in the conduction and valence bands and the band gap, plays an important role. In sSi1-xGex alloys, with increasing Ge content, the energies of the X and L conduction bminima approach each other. Furthermore, the six-fold degeneracy of the conduction bthe X-valley will be split into an energy-lowered fourfold quadruplet and an energy-raisedfold pair [11]. In the derivation of the expressions for the effective density of states forconduction and valence bands,NC andNV, respectively, we follow the treatment given in [12for GaAs. When the splitting of the X-valley is taken into account, NC is given by
(1)
wherem*nX = 0.33 m0 and m*nL = 0.22 m0 are the conduction band density of states effectmasses for X and L minima [13].EcX and EcL are the bottoms of X- and L-valleys, thesuperscripts 4- and 2- referring to the four-fold and two-fold degeneracy due to thesplitting. The energy differences in (1) depend on the Ge contentx as follows [11,13]
(2)
(3)
given in units of eV.
The same analysis in the case of valence band in Si1-xGex yields for the effective density ofstates
(4)
where the light-hole (LH) and heavy-hole (HH) bands were taken into account. The endifference for these bands isELH - EHH = -0.155x (eV), and according to [14] the Ge fractiondependence of the effective masses is given by
(5)
(6)
NC 22πmnX
* kBT
h2
---------------------------- 3 2⁄
4 2e
EcX4 –
EcX2 –
–
kBT---------------------------------
+ 82πmnL
* kBT
h2
---------------------------- 3 2⁄
e
EcX4 –
EcL–
kBT----------------------------
+=
EcX4 –
EcL– 1.0709– 1.3660x 0.1615x2
–+=
EcX4 –
EcX2 –
– 0.6x–=
NV 22πmHH
* kBT
h2
------------------------------ 3 2⁄
22πmLH
* kBT
h2
----------------------------- 3 2⁄
e
ELH EHH–
kBT----------------------------
+=
m0
mHH------------ 1 x–
0.336------------- x
0.210-------------+=
m0
mLH----------- 1 x–
0.234------------- x
0.141-------------+=
7
e Ge
S
hargeportfield
ility
d
in
By using (1) and (4) we can calculate the intrinsic carrier concentration as a function of thcontent
(7)
Temperature, Ge fraction, and impurity concentration dependences of the band gapEg can beexpressed as follows [15,16,17]
(8)
whereEg(0) = 1.17 eV stands for the silicon bandgap at T=0 K,α = 4.73 10-4 eV/K, β = 636 K,V1 = 9 mV,N0 = 1017cm-3, B = 0.5 andN is the doping concentration. The third term on the RHof (8) takes into account the bandgap narrowing.
One of the most important model parameters in any semiconductor device model is the ccarrier mobilityµ. The incorporation of Ge introduces alloy scattering into the base transand affectsµ also through the changes in the effective masses. In our model the total lowmobility µ is described by using a semiempirical expression for the impurity-limited mobµimp and a simplified formula for the alloy scattering
(9)
where
(10)
(11)
(12)
The values for the parametersµmin, µ0, Nref, α, A anda for silicon and germanium can be founfrom refs.[18 - 21].
Finally, HBT modeling requires bias dependent expressions for the depletion widthsd andcapacitancesC of the Si1-xGex/Si-heterojunctions. Here we can follow the treatment given[12] for AlGaAs/GaAs-heterojunctions yielding for a Si1-xGex/Si pn-junction the followingexpressions (see Fig. 1)
whereε0 is the vacuum permittivity,εSiGe(x) = 11.9 + 4.1x, VBE the voltage across the baseemitter heterojunction,NA is the acceptor dopant concentration in the SiGe base, andND is thedonor concentration in the emitter region. The depletion capacitanceCBE of the base-emitterheterojunction is given by
(15)
whereABE is the junction area. Similar expressions can also be derived in the case of thecollector junction biased byVCB, ND now representing the donor concentration in the collecepilayer. The built-in voltage Vbi in (13) to (15) can be estimated from Fig.1.
(16)
where∆En = EcSi-EFn and∆Ep = EFp-Ev
SiGe. In the nondegenerate case these energy differencan be estimated by using Boltzmann statistics for the electron and hole concentrations
(17)
(18)
The conduction band offset is∆Ec = 0.12x eV [16]. The built-in potential can now be calculateby inserting (1), (4), (8), (17) and (18) into (16).
3. HBT MODEL
Next we describe analytically the base transport in a HBT structure shown in Fig. 1. Herfollow the modeling methodology for BJT’s developed by de Graaff et al. [7,22]. However,Ge profile in the base modifies the treatment in many respects, e.g. by introducing aninternal drift field in the base. Also the modified model parameters summarized in Sec. 2to be taken into account.
The electron current density in the quasi-neutral regiondBE < y < Wb (see Fig. 1) consists of thediffusion and drift contributions
where < > refer to the average diffusivity and mobility weighted by the germanium and accconcentrations in the base region
(20)
whereCGe is the germanium concentration. For givenNA(y) andCGe(y), <µn> and <Dn> canbe estimated by using (9), (10), (11), (12) and (20), and the Einstein relationDn = kTµn/q. Inthe base region both the acceptor doping profileNA(y) and the Ge concentration gradient creaa built-in electric fieldF given by
(21)
Next we assume a linear position dependence for the Ge concentration,x(y) = x0 + ∆x y/WBM,and an exponential doping profile for the acceptor concentration in the base (see Fig. 1)
(22)
whereη is a model parameter andNA0 the peak concentration close to the base-emitter juncti
From (8) we get a constant built-in field Fg related to the bandgap reduction ∆Eg
(23)
Notice that the bandgap narrowing reduces the drift field caused by the Ge concentgradient. The electron current density (19) can now be written as
(24)
Assuming quasi-neutrality in the base,p = n + NA, we get further
(25)
Jn has a negative value and it is independent of positiony. In general (25) has no analyticasolution, but it can be solved in two asymptotic cases (low and high injection) and then,[7,22], interpolation formulas can be constructed for the general case.
Let us first consider the low injection case, n <<NA, when (25) simplifies to
(26)
This equation can be solved by integrating across the base, and we obtain
(27)
where
(28)
The current densityJn can be solved from (27) in terms of the minority carrier densitiesn(dBE)andn(Wb) at the base-emitter and base-collector junctions, respectively, whereWb = WBM - dBC(see Fig. 1). MultiplyingJn by the emitter areaAe gives the collector current
(29)
where the subscriptL refers to low injection, and superscriptsf andr refer to forward and reversecurrents, respectively. The minority carrier concentrations in (29) are related to the apjunction voltages through the pn products at y = dBE and y = Wb
(30)
(31)
Also the total stored base charge Qb can now be estimated from (27) and (29)
(32)
or
(33)
In the case of high injection,n >> NA, (25) becomes
exp γ1Wb( ) 1–-------------------------------------– I CL
fI CLr
–= =
n dBE( ) NA dBE( )+[ ]n dBE( ) niSiGe( )
2e
qVbe kBT⁄=
n Wb( ) NA Wb( )+[ ]n Wb( ) niSiGe( )
2e
qVbc kBT⁄–=
Qb qAe n y( )dy
0
Wb
∫=
QbL qAe
γ1---------
γ1Wbexp γ1Wb( )exp γ1Wb( ) 1–
------------------------------------------ 1– n dBE( )qAe
γ1--------- 1
γ1Wb
exp γ1Wb( ) 1–-------------------------------------– n Wb( )+=
Jn 2q Dn⟨ ⟩ dndy------
q2n Dn⟨ ⟩Fg
kBT-----------------------------+=
11
for
Jeongs by
ircuitif theon theby deas a
and
The solution to (34) is given by
(35)
with
(36)
The collector current can be solved from (35) yielding
(37)
where the subscriptH refers to high injection. An expression similar to (33) can be derivedthe total base chargeQb
H from (35) and (37), where only the parameterγ1 in QbL is replaced by
γ2.
In the case of intermediate injection various interpolation methods [6,7,22] can be used.and Fossum [6] introduced a weighted linear sum of the low- and high-injection solutionusing empirical weighting constants. This is acceptable, if the model is used only for csimulations, where also a lot of other empirical parameters must be optimized. However,same model is used also for the device design, the interpolation formulas must be basedphysical parameters. Therefore we have chosen the interpolation formulation developedGraaff et al. [7,22]. Their approach allows one to express the interpolation formulasfunction of, e.g., the built-in field dependent parameters derived above.
Let us introduce the following notations
(38)
(39)
Then from (29) and (37) the forward part of the current can be expressed as
(40)
(41)
Following the interpolation method introduced by de Graaff et al. [7,22] we get from (40)(41) a general formula for the forward current
where a small factorK is introduced in order to remove the singularity atVj = Vbi.
The base current in an injection-efficiency-limited bipolar transistor is governed by propeof the emitter [22]. Therefore the two base currents in a homo- and heterojunction bitransistors should be identical, if the transistors have identical emitters. Howeverexperimental results [23] show that the base current has a tendency to grow when changina homojunction transistor to a heterojunction transistor. The enhancement of the base cindicates that charge carrier recombination mechanisms may be important the SiGeHBT’s. Therefore a complete model for the base current in a npn HBT should include theback-injection into the wide gap emitter, recombination in the base-emitter junction deplregion, recombination in the quasi-neutral base, recombination in the epitaxial collectoemitter surface recombination and emitter-base tunneling current. Since most ofmechanisms typically are negligible, we have taken in to account only the hole back-injeand recombination current in the base-emitter junction. The other mechanisms can eaadded if necessary [24]. Then we can express
(62)
where
(63)
(64)
and Ge is the emitter Gummel number defined as
(65)
andND is the doping density in the emitter having a widthWe. The model parameters Ibf and Vlfcan be related to the recombination physics, i.e., the recombination timeτ and the energy of thetrap level [22].
In the present version of our model we have assumed that all the series resistancesRE, RB, andRC for emitter, base and collector region, respectively, are constants. The non-constant pthe internal base and collector resistances can be added to the model in a standard mannif more accuracy is needed. The voltagesVbeandVcb in the equations above denote the internvoltages, which are obtained when the ohmic drops due to the series resistancesRE, RB, andRCare subtracted from the terminal voltagesVBE andVCB.
Cutoff frequencyfT is one of the most important parameters in HBT’s, since these devicestypically aimed at high speed applications.fT can be calculated by using the charge contrprinciple [22]
where <Dp> is the average diffusivity of the holes in the emitter. The total base chargeQb aswell as the depletion chargesQTe andQTc have already been estimated in (51), (60) and (6respectively. When modeling the collector current dependence of the cutoff frequency, theof the current to the depletion width of the base-collector junction has to be taken into accHere we have adopted the simple expression fordBC(IC) given in [24].
4. MODEL VERIFICATION
The present HBT model has been implemented in APLAC circuit simulator [8]. An advanof APLAC is that it offers a set of data structures and subroutines, while all the standard feaof the C-language are available. Therefore it is rather straightforward to implement new min APLAC, including the tedious implementation of the physical device models.
The present model has been verified through comparisons between the modeled resumeasured data both for Si BJT’s made in our laboratory and SiGe-base HBT’s made in vlaboratories. Here we present the simulation results for the experimental data from [5], sincallows us to compare not only the measured and simulated results but also the present mthat developed by Hong and Fossum [5].
A problem typically encountered in the standard Gummel-Poon model [2] is a ratheragreement of the model with the measured high frequency properties, when the mparameters first have been found through a fitting to the DC characteristics. An advantagepresent physical model is that it simultaneously gives a good fit both for DC and AC propeThis is shown in Figs. 2 to 4. The accuracy of our model is comparable to that developeHong and Fossum [5]. The model parameters needed and their values used in the simuhave been listed in Tables 1 and 2. The model parameters compare well - as they shouldphysical device models - to the reported values of the real SiGe-base HBT structure [5].
1f T------
2π Q∑I C
------------------=
Q∑ Qe Qb QTe QTc+ + +=
Qe
We2I B
2 Dp⟨ ⟩----------------=
16
17
JT
BT
BT
Figure 2. Measured (squares)[5] and simulated (solid) Gummel plots for a pure silicon BatVCE=2.0 V. The model parameters have been listed in Table 1.
Figure 3. Measured (squares) [5] and simulated (solid) Gummel plots for a SiGe-base H(x=0.08) atVCE=2.0 V. The model parameters have been listed in Table 2.
Figure 4. Measured (squares) [5] and simulated (solid) Gummel plots for a SiGe-base H(x=0.08) atVCE=2.0 V. The model parameters have been listed in Table 2.
Table 1. Values of the model parameters for Si BJT.
19
aseddhavee
profiled fromctionto III.
ns.
Ge
A one-dimensional device model is useful even in the device design, if the model is bstrictly on device physics. In the case of Si1-xGex/Si HBT’s an interesting question is, what kinof Ge concentration profile maximizes the current gain and the cutoff frequency. Wecompared the simulation results forIC vs. VBE andfT vs. IC in the cases of three different Gconcentration profiles shown in Fig. 5.
The first profile (case I) is the same as in the HBT device discussed above. In the second(case II) the maximum Ge concentration at the base-collector junction has been increase8 % to 13 %, and in the third case additionally the Ge concentration in the base-emitter junhas been increased from 0 % to 5 %. Figs. 6 and 7 show the simulation results in cases I
0.0 0.5⋅WBM WBM
0
5
10
15
Distance in base
Ge
Con
cent
ratio
n [%
]
III
II
I
Figure 5. Ge concentration profiles (cases I, II and III) used in the comparative simulatio
Figure 6. Collector currents vs. base-emitter voltage in the cases of three differentconcentration profiles shown in Fig. 4.
0.4 0.55 0.7 0.85 1.0-12.0
-9.5
-7.0
-4.5
-2.0
log(IC)
VBE [V]
IIIIII
20
se III)
mize.
theSiGehightiones the
circuitpared
ith
istor
Ge
Fig. 6 indicates that the increase of the Ge concentration at the base-emitter junction (camost clearly enhances the collector current and consequently the DC current gainIC /IB, sincethe base currentIB does not depend on the Ge concentration. However, if one wants to optithe cutoff frequency, then the profile of the case II with a steep Ge gradient is preferable
5. CONCLUSIONS
A comprehensive HBT model for SiGe-base transistors is developed by combininganalytical de Graaf-Kloosterman model for BJT’s and physics of heterostructure andmaterial. Our model equations account for valence band discontinuity, heavy doping andcollector current effects as well as the additional built-in field due to the Ge concentragradient. The model parameters are technology and geometry dependent, which relaxneed for a parameter-optimization scheme, and allows the model to be used both forsimulation and device design. Good agreement is found when the modeled results are comto measured data.
Figure 7. Cutoff frequencies vs. collector current in the cases of three differentconcentration profiles shown in Fig. 4.
-6.0 -5.0 -4.0 -3.0 -2.00.0
12.5
25.0
37.5
50.0fT
log(IC)
[GHz]
I
IIIII
21
for
el for
nd
-sign,
r)
.:
ron
ncy
n Si
d
in
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