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Nanoscience and Nanotechnology Lecture 19
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Review of traditional microelectronics
Course notes are at:http://www.phys.appstate.edu/nanotech/NanoCourseLecturePPTs2008/
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Review of basic semiconductor Physics
When considering the Physics and Engineering of traditional semiconductors, we are mainly concerned with ‘bulk’ properties; and consider surfaces as defects.
This approach is the opposite of the nanotechnology approach, where the goal is to use the special properties of materials and devices at the nm scale, where surface atoms control properties.
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Material based on content of:
Principles of Electronic Materials and Devices,
by S. O. Kasap Third Edition
Text WEB site:
http://materials.usask.ca/textbook/
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E l e c t r o n e n e r g y
V a l e n c e B a n d ( V B )
F u l l o f e l e c t r o n s a t 0 K .
E c
E v
0
E c+ χ
(c)
B a n d g a p = E g
(b)
ψB
C o n d u c t i o n B a n d (C B )
E m p ty o f el ec t r o n s a t 0 K .
ψh yb
o r b i t a l s
S i i o n c o r e ( + 4 e )
V a l en c e
el ec t r o n(a)
Si crystal in 2-D
. 5.1: ( ) Fig a A simplified two dimensional illustration of a Si atom with four , hybrid orbitalsψhyb. . ( ) Each orbital has one electron b A simplified two
. ( ) dimensional view of a region of the Si crystal showing covalent bonds c The .energy band diagram at absolute zero of temperature
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Recall the Si band structure
P. E. Russell: [email protected] Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.2: A two dimensional pictorial view of the Si crystal showingcovalent bonds as two lines where each line is a valence electron.
Simplified 2 dim Si showing 4 bonds per atom
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e–
h o l e
C B
V B
Ec
Ev
0
Ec+ χ
Eg
F r ee e≠h υ > Eg
H o le
E lec t r o n en er gy
h υ
( a) (b )
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.3: (a) A photon with an energy greater than Eg can excite anelectron from the VB to the CB. (b) When a photon breaks a Si-Sibond, a free electron and a hole in the Si-Si bond is created.
Electron excitation across the forbidden gap
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h+e–
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.4: Thermal vibrations of atoms can break bonds and therebycreate electron-hole pairs.
Thermal carrier (electron-hole pair) generation
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h+ E g
V B
C Bh
+
e–
(a)
h+
h+
h+
e–
h+
h+
h+ (d)
h+
e– e
–
h+
(f)
(b)
(c)
(e)
Fig. 5.5: A pictorial illustration of a hole in the valence bandwandering around the crystal due to the tunneling of electrons fromneighboring bonds.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Concept of hole motion in valence band and annihilation
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.
V B
C B
V B
C B
x
x = 0 x = L
V ( x )E l e c t r o s t a t i c P E ( x )
( a ) ( b )
Ex
Ex
Fig. 5.6: When an electric field is applied, electrons in the CB andholes in the VB can drift and contribute to the conductivity. (a) Asimplified illustration of drift in Ex. (b) Applied field bends the energybands since the electrostatic PE of the electron is -eV(x) and V(x)decreases in the direction of Ex whereas PE increases.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Both electrons and holescan drift in an electric field
P. E. Russell: [email protected] Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
E
g(E)
g (E ) ∝ (E –Ec) 1 /2
f(E)
EF
nE(E) orpE(E)
E E
F or
electr ons
F or holes
[ 1 – f(E ) ]
nE(E )
pE(E )
A r ea = p
A r ea = ∫ nE(E )dE = n
Ec
Ev
Ev
Ec
0
Ec+ χ
EF
V B
C B
(a) (b ) (c) (d )
. 5.7: ( ) . ( ) ( Fig a Energy band diagram b Density of states number of states ). ( ) - per unit energy per unit volume c Fermi Dirac probability function
( ). ( ) probability of occupancy of a state d The product ofg(E) andf(E) is ( the energy density of electrons in the CB number of electrons per unit
). energy per unit volume The area undernE(E) . vsE is the electron .concentration in the conduction band
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Ec
Ev
EF i
C B
EF p
EF n
Ec
Ev
Ec
Ev
( a ) ( c )( b )
V B
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.8. Energy band diagrams for (a) intrinsic (b) n-type and (c) p-typesemiconductors. In all cases, np = ni
2
Mass action law np=ni2
Mass action law np=ni2 = NcNvexp(-Egap/kT)
Where Nc is the density of states near conduction band edgeNv is the density of states near valence band edge
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A s+
e–
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.9: Arsenic doped Si crystal. The four valence electrons of Asallow it to bond just like Si but the fifth electron is left orbiting theAs site. The energy required to release to free fifth-electron into theCB is very small.
Adding impurities or dopants
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x
As+ As+ As+ As+
Ec
Ed
Ev
As atom sites every 106 Si atoms
Distanceintocrystal
~0.03 eV
CB
Electron Energy
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.10: Energy band diagram for an n-type Si doped with 1 ppmAs. There are donor energy levels just below Ec around As+ sites.
Dopant levels for As impurities (valence V, n-type))
Each as provides an extra loosely bound electron
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B –
h +
F r e e
B –
(a )
(b )
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.11: Boron doped Si crystal. B has only three valence electrons.When it substitutes for a Si atom one of its bonds has an electronmissing and therefore a hole as shown in (a). The hole orbits aroundthe B- site by the tunneling of electrons from neighboring bonds asshown in (b). Eventually, thermally vibrating Si atoms providesenough energy to free the hole from the B- site into the VB as shown.
Boron as an example of n type doping
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x
B–
Ev
Ea
B a t o m s i t e s e v e r y 1 06
S i a t o m s
D i s t a n c e
i n t o c r y s t a l
~ 0 .0 5 e V
B– B
–B
–
h+
V B
Ec
E l e c t r o n e n e r g y
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.12: Energy band diagram for a p-type Si doped with 1 ppm B.There are acceptor energy levels just above Ev around B- sites. Theseacceptor levels accept electrons from the VB and therefore create holesin the VB.
Band diagram for b doped (p-type) Si
Small ionization energy allows thermal ionization at room temperature
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x
B–
Ev
Ea
B a t o m s i t e s e v e r y 1 06
S i a t o m s
D i s t a n c e
i n t o c r y s t a l
~ 0 .0 5 e V
B– B
–B
–
h+
V B
Ec
E l e c t r o n e n e r g y
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.12: Energy band diagram for a p-type Si doped with 1 ppm B.There are acceptor energy levels just above Ev around B- sites. Theseacceptor levels accept electrons from the VB and therefore create holesin the VB.
Band diagram for b doped (p-type) Si
Small ionization energy allows thermal ionization at room temperature
Note that the electron from the valance band becomes fixed charge (in energy and position)
Dopants are far apart
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.
V
n - T y p e S e m i c o n d u c t o r
Ec
EF
A
B
V(x)
x
Electrostatic PE(x) = -eV
Ex
EFi
Ed
Ev
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.13: Energy band diagram of an n-type semiconductor connectedto a voltage supply of V volts. The whole energy diagram tilts becausethe electron now has an electrostatic potential energy as well
For an n-type semiconductor (e.g. Arsenic doped) only the electrons are mobile. The ionized dopant atoms do not move.
Thus we have added both ‘mobile” and ‘fixed’ or ‘localized’ charge!
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we have both ‘mobile’ and ‘fixed’ or ‘localized’ charge!
For the Case of an extrinsic (doped) semiconductor:
this is a very Important Concept!!!
Fixed charge: the ionized dopant atomsMobile charge: the extra electron or hole contributed by ionization
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V B
C B
As As AsAs+E
F
Eg
T < Ts
Ts < T < T
i
(b ) T = T2
(a ) T = T1
T > Ti
( c ) T = T3
As+EF
As+ As+ As+ As+E
F
As+ As+ As+
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.14: (a) Below Ts, the electron concentration is controlled bythe ionization of the donors. (b) Between Ts and Ti, the electronconcentration is equal to the concentration of donors since theywould all have ionized. (c) At high temperatures, thermally generatedelectrons from the VB exceed the number of electrons from ionizeddonors and the semiconductor behaves as if intrinsic.
Example of n-type material versus temperature
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ln(n)
1/T
Ti
Ts
I N T R I N S I C
E X T R I N S I CI O N I Z A T I O N
ni( T )
l n ( Nd
) s l o p e = – Δ E / 2 k
sl o p e = – E g / 2 k
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.15: The temperature dependence of the electron concentrationin an n-type semiconductor.
Temperature dependency of conductivity: critical concept!
From this data we can measure:Band gap energy Egap, and
ln(n)
1/T
Ti
Ts
I N T R I N S I C
E X T R I N S I CI O N I Z A T I O N
ni( T )
l n ( Nd
) s l o p e = – Δ E / 2 k
sl o p e = – E g / 2 k
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.15: The temperature dependence of the electron concentrationin an n-type semiconductor.
Dopant ionization energy ΔEionization
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Conductivity of a Semiconductor
= conductivity, e = electronic charge,
n = electron concentration in the CB, e = electron drift mobility,
p = hole concentration in the VB, h = hole drift mobility
= ene + eph
Note: this expression is valid for both intrinsic and extrinsic semiconductors!
We now see that n≠p for extrinsic semiconductors.
P. E. Russell: [email protected] Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
E
g(E)
g (E ) ∝ (E –Ec) 1 /2
f(E)
EF
nE(E) orpE(E)
E E
F or
electr ons
F or holes
[ 1 – f(E ) ]
nE(E )
pE(E )
A r ea = p
A r ea = ∫ nE(E )dE = n
Ec
Ev
Ev
Ec
0
Ec+ χ
EF
V B
C B
(a) (b ) (c) (d )
. 5.7: ( ) . ( ) ( Fig a Energy band diagram b Density of states number of states ). ( ) - per unit energy per unit volume c Fermi Dirac probability function
( ). ( ) probability of occupancy of a state d The product ofg(E) andf(E) is ( the energy density of electrons in the CB number of electrons per unit
). energy per unit volume The area undernE(E) . vsE is the electron .concentration in the conduction band
(The intrinsic case is shown here.)
We can determine n and p (the electron and hole concentrations) if we know g(E) and f(E) and…
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Ec
Ev
EF i
C B
EF p
EF n
Ec
Ev
Ec
Ev
( a ) ( c )( b )
V B
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.8. Energy band diagrams for (a) intrinsic (b) n-type and (c) p-typesemiconductors. In all cases, np = ni
2
Mass action law np=ni2: in equilibrium
Valid for both intrinsic and extrinsic semiconductors!
Mass action law np=ni2 = NcNvexp(-Egap/kT)
Where Nc is the density of states near conduction band edgeNv is the density of states near valence band edgeIf more electrons are added by doping, then fewer holes will exist due to recombination, keeping the product np constant!
Ec
Ev
EF i
C B
EF p
EF n
Ec
Ev
Ec
Ev
( a ) ( c )( b )
V B
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.8. Energy band diagrams for (a) intrinsic (b) n-type and (c) p-typesemiconductors. In all cases, np = ni
2
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Carrier recombination
In addition to carrier creation, we will have carrier recombination, i.e. if an electron excited to the CB is located (spatially) near a hole in the VB, the electron will reduce its energy by filling the hole (allowed electron state).
The electron and hole annihilate each other (as free charge carriers) via recombination.
.
Ev
Ec
CB
VB
ψcb(k
cb)
ψvb(k
vb)
h υ = Eg
Distance
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig.5.22: Direct recombination in GaAs. kcb = kvb so that momentumconservation is satisfied
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Carrier RecombinationThe recombination rate R will be proportional to both the electron and hole concentrations (and hence to their product) Rnp
What about the carrier generation rate G?G will depend on the number of electrons available for excitation at Ev i.e. Nv and empty states at Ec i.e. Nc.And the probability of an electron making the transition i.e. exp[-Eg/kT] so: G NvNc exp[-Eg/kT] In equilibrium G = R, generation and recombination rates are equal. (more on this later)
Next: Where is Ef within the band gap?
.
Ev
Ec
CB
VB
ψcb(k
cb)
ψvb(k
vb)
h υ = Eg
Distance
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig.5.22: Direct recombination in GaAs. kcb = kvb so that momentumconservation is satisfied
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Eg
(eV)
χ ( )eV Nc (cm–3) N
v (cm–3) n
i
(cm–3)
e
(cm2 V–1 s–1)
h
(cm2 V–1 s–1)
me*/m
e m
h*/m
e ε
r
Ge 0.66 4.13 1.04×1019 6.0×1018 2.4×1013 3900 1900 0.12 a
0.56 b
0.23 a
0.40 b
16
Si 1.10 4.01 2.8×1019 1.04×1019 1.45×1010 1350 450 0.26 a
1.08 b
0.38 a
0.56 b
11.9
GaAs 1.42 4.07 4.7×1017 7×1018 1.8×106 8500 400 0.067 ,a b 0.40 a
0.50 b
13.1
Use effective mass related to conductivity, rather than for density of states.
Table 5.1 Selected typical properties of Ge, Si and GaAs at 300 K. Effective mass related to conductivity (labeled a) is different than that for density of states (labeled b).
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.
Fig. 5.20: Temperature dependence of electrical conductivity for adoped (n-type) semiconductor.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
log(n)
Intrinsic
Extrinsic
Ionization
log()
log() ∝ T - 3 / 2 ∝ T 3 / 2
L a t t i c e
sc a t t e r i n g
I m p u r i t y
sc a t t e r i n g
1/TL o w T e m p e r a t u r eH i g h T e m p e r a t u r e
T
M e t a l
S e m i c o n d u c t o r
The combined temperature effects of carrier creation and carrier scattering.
KEY CONCEPT!!
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We have been assuming that there are many more empty states than there are electrons in the conduction band, and/or many more filled states than holes in the valence band; i.e. that Boltzmann statistics rather than Fermi-Dirac apply.
This is the non-degenerate semiconductor case;
i.e when n<<Nc and p<<Nv
What is the other possibility? Degeneracy!Occurs when doping concentration is very large; usually 1019 or 1020 cm-3 (out of 1023 cm-3)
Non-degenerate semiconductors (the normal case)
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C B
g ( E )
E
I m p u r i t i e s
f o r m i n g a b an d
( a ) ( b )
E Fp
E v
E c
EF n
E v
E c
C B
V B
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.21: (a) Degenerate n-type semiconductor. Large number of donorsform a band that overlaps the CB. (b) Degenerate p-type semiconductor.
What happens with very high doping?: Degeneracy
Degenerate semiconductors behave as metals; this is very useful!
P. E. Russell: [email protected]
.
Ev
Ec
CB
VB
ψcb(k
cb)
ψvb(k
vb)
h υ = Eg
Distance
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig.5.22: Direct recombination in GaAs. kcb = kvb so that momentumconservation is satisfied
Direct gap recombination; are there other options?
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Ek
kπ/a–π/a
Ec
Ev
CB
VB
Ec
Ev
The E-k Diagram The Energy BandDiagram
Empty ψk
Occupied ψk
h+
e–
Eg
e–
h+
hυ
VB
hυ
CB
Fig. 5.49: The E-k diagram of a direct bandgap semiconductor such asGaAs. The E-k curve consists of many discrete points each pointcorresponding to a possible state, wavefunction ψ k(x), that is allowedto exist in the crystal. The points are so close that we normally draw theE-k relationship as a continuous curve. In the energy range Ev to Ecthere are no points (ψk(x) solutions).From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
P. E. Russell: [email protected]
E
CB
VB
k–k
Direct Band Gap
(a) GaAs
Eg PhotonEc
Ev
E
CB
VB
Indirect Band Gap, Eg
k–k
kcb
(b) Si
Ec
Ev
kvb
E
k–k
Phonons
(c) Si with a recombination center
VB
CB
Er Ec
Ev
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.50: (a) In GaAs the minimum of the CB is directly above the maximumof the VB. GaAs is therefore a direct band gap semiconductor. (b) In Si, theminimum of the CB is displaced from the maximum of the VB and Si is anindirect band gap semiconductor. (c) Recombination of an electron and a holein Si involves a recombination center.
Si is not direct gap!This is most important in optoelectronic applications (laser, LED, photo…)More on this later.
P. E. Russell: [email protected]
Ec
Recombinationcenter
C B
V B
Phonons
( a ) R e c o m b i n a t i o n
Trappingcenter
Et
C B
V B
Ec
Ev
Et
Et
Ev
Er
Er
Er
( b ) T r a p p i n g
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.23: Recombination and trapping. (a) Recombination in Si via arecombination center which has a localized energy level at Er in thebandgap, usually near the middle. (b) Trapping and detrapping ofelectrons by trapping centers. A trapping center has a localizedenergy level in the band gap.
Cases where midgap states exist (due to impurities and/or defects)
Be sure you understand that midgap states can act as Trap sites or Recombinationsites
In this caseEemission
is less than Egap
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How can there be allowed states in the forbidden gap?
Allowed states (such as the charge carrier trap states and recombination centers of figure 5.23) are localized states within the forbidden band gap energy range of a defect-free material. The states are created by the perturbation of the normal periodic lattice by a point (or other) defect.
The trap states are localized in space to the area of the defect, and in energy to a specific energy within the band gap; determined by the type of defect creating the state.
Examples of defects creating midgap states include vacancies, interstitial and substitutional impurities, dislocations, stacking faults, etc.
P. E. Russell: [email protected]
How can there be allowed states in the forbidden gap?
These allowed individual states created by the perturbation of the normal periodic lattice act as:
Recombination sites if near midgap,Or asTrapping sites for electrons if near the CB edge, Or asTrapping sites for holes if near the CB edge.
Note that trapping just removes the charge carrier from the free carrier pool for a brief time; with the overall effect of having many traps being a reduction in charge carrier density available for conduction at any given time.
P. E. Russell: [email protected]
Ec
Recombinationcenter
C B
V B
Phonons
( a ) R e c o m b i n a t i o n
Trappingcenter
Et
C B
V B
Ec
Ev
Et
Et
Ev
Er
Er
Er
( b ) T r a p p i n g
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.23: Recombination and trapping. (a) Recombination in Si via arecombination center which has a localized energy level at Er in thebandgap, usually near the middle. (b) Trapping and detrapping ofelectrons by trapping centers. A trapping center has a localizedenergy level in the band gap.
Charge carrier pair is lost permanently due to recombination.
Trapping causes a temporary loss of a single charge carrier.
Note: phonon assistance is usually required!
P. E. Russell: [email protected]
How long do free carriers exist?
How long do free carriers exist before being trapped or recombined?
This depends on the number of traps and recombination centers; which depends on the number (actually density) of defects.
If we measure carrier lifetime, we get an idea of the perfection of the semiconductor: which is very useful in developing processing.
How can we measure carrier lifetime?
P. E. Russell: [email protected]
Ec
Ev
Ed
C B
V B
Fig. 5.24: Low level photoinjection into an n-type semiconductor inwhich Δnn < nno
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Carrier creation: Photoinjected charge carriers
If we shine light on a semiconductor, we will generate new charge carriers (in addition to those thermally generated) if Ephoton>Egap.
If the light is always on and of constant intensity, some steady state concentration of electrons and holes will result.
Ec
Ev
Ed
C B
V B
Fig. 5.24: Low level photoinjection into an n-type semiconductor inwhich Δnn < nno
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
P. E. Russell: [email protected]
Carrier creation: Photoinjected charge carriersLet’s consider the case of n-type material
Consider an n-type semiconductor with a doping concentration of 5 x 1016 cm-3.
What are the carrier concentrations?Let’s define some terms;nno majority carrier concentration in the n-type semiconductor in the dark (only thermally ionized carriers) (i.e. the electron concentration in n-type)
pno minority carrier concentration in the n-type semiconductor in the dark (only thermally ionized carriers) (i.e. the hole concentration in n-type)
Note: the no subscript implies that mass action law is valid!
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When we have light:
With light of Ephoton>Egap hitting the semiconductor, we get photogeneration of excess charge carriers.
Δnn excess electron concentration such that::Δnn = nn-nn0
And Δpn excess hole concentration such that::Δpn = pn-pn0Note that photogenerated carriers excited across the gap can only be created in pairs i.e. Δpn = Δnn and now (in light) nnpn≠ni
2 i.e. mass action not valid!
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If the temperature is constant, nn0 and pn0 are not time dependent, so
and
Consider the case of ‘weak’ illumination, which creates a 10% change in nn0 i.e. Δnn = 0.1nn0
Or if the doping level is nno=5 x 1016cm-3, then
Δnn = 0.1nn0= 0.5 x 1016cm-3
And Δpn =Δnn = 0.5 x 1016cm-3
Which change is more important? Majority or minority?
€
dnn
dt=
dΔnn
dt
€
dpn
dt=
dΔpn
dt
Carrier density change under illumination
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.
S i
G e
G a A s
0 ° C2 0 0 ° C4 0 0 ° C6 0 0 ° C 2 7 ° C
LLLL
1018
1 1.5 2 2.5 3 3.5 4
1000/T (1/K )
1015
1012
103
106
1091.45×1 0 1 0 cm -3
2 .4×1 0 1 3 cm -3
2 .1×1 0 6 cm -3
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.16: The temperature dependence of the intrinsic concentration.
Recall the intrinsic carrier concentration
For Sini is roughly 1.5x1010cm-3
At room temperature
Since pno=ni2/nn0
= (1.5x1010)2/5x1016
pno =4500 cm-3
P. E. Russell: [email protected]
.
n n o
n i
p n o
5 × 1 01 6
1 . 5 × 1 01 0
4 . 5 × 1 03
( a ) I n th e d a r k : n p = ni
2
nn = n
n o+ ∆ n
n
p n o
0 . 5 × 1 01 6
pn = p
n o+ ∆ p
n
5 . 5 × 1 01 6
∆ pn = 0 . 5 × 1 0 1 6
n i
( b ) I n l i g h t : n p ≠ ni
2
. 5.25: Fig Low level injection in ann- type semiconductor does not affectnn but drastically affects the minority carrier
concentrationpn.
From , Principles of Electronic Materials and Devices Second Edition, . . (© - , 2002)S O Kasap McGraw Hill:// . .http Materials Usask Ca
An extremely important concept!
Minority carrier concentration can be controlled over many orders of magnitude with only a small change in majority concn.
Δpn =Δnn = 0.5 x 1016cm-3
P. E. Russell: [email protected]
–
–
–
–
–
–
–
–
–
–
– ––
–
–
– –––
–
–
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–
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–+
+
–
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– ––
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–+
+
+
+
++
+ +
++
+
+
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++
+
+
–
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–
– ––
–
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–
–
– ––
–
–
–
–
–
–
–
++
+ +
+
+
++
++
n - t y p e s e m i c o n d u c t o r i n
t h e d a r k .
pn
= pn o
< < nn o
A
I l l u m i n a t i o n w i t h h υ > Eg
c reates ex cess ho les :
pn = p
n o+ Δ p
n = Δ n
n
B
I l l u m in a t i o n
I n dark af ter
i l lum inatio n . E x cess
ho les are d isappearing
by reco m binatio n .
C
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.26: Illumination of an n-type semiconductor results in excesselectron and hole concentrations. After the illumination, therecombination process restores equilibrium; the excess electronsand holes simply recombine.
Carrier creation followed by recombination
P. E. Russell: [email protected]
–
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–
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– ––
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–+
+
–
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– ––
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– ––
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–+
+
+
+
++
+ +
++
+
+
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++
+
+
–
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–
–
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– ––
–
–
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–
–
– ––
–
–
–
–
–
–
–
++
+ +
+
+
++
++
n - t y p e s e m i c o n d u c t o r i n
t h e d a r k .
pn
= pn o
< < nn o
A
I l l u m i n a t i o n w i t h h υ > Eg
c reates ex cess ho les :
pn = p
n o+ Δ p
n = Δ n
n
B
I l l u m in a t i o n
I n dark af ter
i l lum inatio n . E x cess
ho les are d isappearing
by reco m binatio n .
C
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.26: Illumination of an n-type semiconductor results in excesselectron and hole concentrations. After the illumination, therecombination process restores equilibrium; the excess electronsand holes simply recombine.
Carrier creation followed by recombination
Mostly majority carriers in the dark
Almost equalCarrier concnsIn light
The extra minority carriers recombine once the generation source is removed.
How quickly do the carriers recombine?
P. E. Russell: [email protected]
Minority carrier lifetime h for n-type
h = average time a hole exists in the valance band from its generation until its recombination
And so 1/ h is the average probability (per unit time) that a hole will recombine with an electron.
h depends on impurities, defects and temperature.
The recombination process in a real semiconductor usually involves a carrier being localized at a recombination center.
can be short (nanoseconds) allowing fast response (e.g. switch)or slow (seconds) for a photoconductor or solar cell
P. E. Russell: [email protected]
Excess Minority Carrier Concentration
Δpn = excess hole (minority carrier) concentration,
t = time,
Gph = rate of photogeneration,
h = minority carrier lifetime (mean recombination time)
h = average time a hole exists in the valance band from its generation until its recombination
dΔpn
dt=Gph −
Δpn
τh
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I l l u m i n a t i o n
0Time, t
p no
G ph
t off
pno+ Δ pn(∞)
t'
hG ph
0
G and pn(t)
Δ p n( t' ) = Δ p n( 0 ) e x p ( - t' / h)
Fig. 5.27: Illumination is switched on at time t = 0 and then off at t = toff.
The excess minority carrier concentration, Δ pn(t) rises exponentially to itssteady state value with a time constant h. From toff, the excess minoritycarrier concentration decays exponentially to its equilibrium value.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Carrier concentration versus time with pulsed illumination
t’ is time after illumination is removed
P. E. Russell: [email protected]
W
D
L
V Ip h
L i g h t
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.28: A semiconductor slab of length L, width W and depth D isilluminated with light of wavelength λ. Iph is the steady statephotocurrent.
Continuous illumination provides increased conductivity
Often used as a switchor motion detector
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Carrier motion in a concentration gradient
How can we describe how charge carriers move is response to a concentration gradient?
It is just simple diffusion; i.e Ficks first law; but of course it is with charged ‘particles’
Diffusion current density = (charge) x fluxWhere Flux = ΔN/A Δt or #of charge carriers passing unit area per unit time
Ficks first law relates flux to concentration gradientFlux =e= -De dn/dx for electronsWhere De = diffusion coefficient for electrons
P. E. Russell: [email protected]
Carrier motion in a concentration gradient
Diffusion current density = (charge) x fluxWhere Flux = ΔN/A Δt or #of charge carriers passing unit area per unit time
Ficks first law relates Flux to concentration gradientFlux =e= -De dn/dx for electrons
So the current density (due to diffusion) becomes: Diffusion current density = (charge) x flux
note: diffusion is away from high concentrationJDe= -ee for charge carriers due to diffusion, orJDe= eDe dn/dx for electrons and JDh= -eDe dp/dx for holes
( is gamma)
P. E. Russell: [email protected]
Carrier motion in a concentration gradient
So, now for the total current we have both drift and diffusion currents for both types of charge carriers!
Jtotal = Jdrift + Jdiffusion Jtotal = Jdrift + Jdiffusion
Je= eneEx + eDe dn/dx for electrons, and
Jh= ephEx - eDh dp/dx for holes.
We relate the diffusion coefficient to temperature and mobility by the Einstein relationship:De = (kT/e) e or De/e = (kT/e) and Dh/h = (kT/e)
We also find that D=l2/ where l is the mean free path, and is the mean time between scattering events.
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Definition of Particle Flux
= particle flux, ΔN = number of particles crossing A in Δt, A = area, Δt = time interval
Γ =ΔNAΔt
Definition of Current Density
J = electric current density,
Q = charge of the particle, = particle flux
J =QΓ
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n1
n2
xo
xo
+ lxo
- l
n ( x , t )
x
n(x,t)
x
N e t e l e c t r o n d i f f u s i o n f l u x
E l e c t r i c c u r r e n t
xo+ lx
o- l
xo
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.29: (a) Arbitrary electron concentration n(x,t) profile in asemiconductor. There is a net diffusion (flux) of electrons fromhigher to lower concentrations. (b) Expanded view of twoadjacent sections at xo. There are more electrons crossing xocoming from left (xo-l ) than coming from right (xo+l ).
(a) (b)
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Fick’s First Law: relating diffusion current to carrier concentration gradient.
e = electron flux, De = diffusion coefficient of electrons, dn/dx = electron concentration gradient
Γe =−De
dndx
JD, e = electric current density due to electron diffusion,
e = electron flux, e = electronic charge,
De = diffusion coefficient of electrons,
dn/dx = electron concentration gradient
Electron Diffusion Current Density
J D,e =−eΓe =eDe
dndx
Where:
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p(x,t)
x
N e t h o l e d i f f u s i o n f l u x
E l e c t r i c c u r r e n t
xo
+ lxo- l
xo
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.30: Arbitrary hole concentration p(x,t) profile in asemiconductor. There is a net diffusion (flux) of holes from higherto lower concentrations. There are more holes crossing xo comingfrom left (xo-l ) than coming from right (xo+l ).
Carrier diffusion away from high concentration
holes in this p-type example
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Ex
n - T y p e S e m i c o n d u c t o r
L i g h t x
H o l e D r i f t
H o l e D i f f u s i o n
E l e c t r o n D r i f t
E l e c t r o n D i f f u s i o n
S e m i t r a n s p a r e n t e l e c t r o d e
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.31: When there is an electric field and also a concentrationgradient, charge carriers move both by diffusion and drift. (Ex is theelectric field.)
Carrier motion: via diffusion (due to concn gradient) and drift (due to electric field)
Both diffusion and drift occur in semiconductors.
Note here that holes (minority carriers) drift and diffuse in the same direction; but electrons (majority carriers) do not!
With light we alter minority carrier concn
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Hole Diffusion Current Density
JD, h = electric current density due to hole diffusion, e = electronic charge, h = hole flux, Dh = diffusion coefficient of holes, dp/dx = hole concentration gradient€
JD,h = eΓh = −eDh
dp
dx
Total Electron Current Due to Drift and Diffusion
Je = electron current due to drift and diffusion, n = electron concentration, e = electron drift mobility, Ex = electric field in the x direction, De = diffusion coefficient of electrons, dn/dx = electron concentration gradient
J e =enμeEx +eDe
dndx
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Total Currents Due to Drift and Diffusion
Jh = hole current due to drift and diffusion, p = hole concentration,
h = hole drift mobility, Ex = electric field in the x direction,
Dh = diffusion coefficient of holes,
dp/dx = hole concentration gradient
J h =epμhEx −eDh
dpdx
J e =enμeEx +eDe
dndx
Je = electron current due to drift and diffusion, n = electron concentration
e = electron drift mobility, Ex = electric field in the x direction,
De = diffusion coefficient of electrons,
dn/dx = electron concentration gradient Jtotal = Jh+Je
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De = diffusion coefficient of electrons,
e = electron drift mobility,
Dh = diffusion coefficient of the holes,
h = hole drift mobility
De
μe
=kTe
andDh
μh
=kTe
Einstein Relation: diffusion coefficient and mobility are related!
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E x p o s e d
A s+
D o n o r
n2
n1
D i f f u s i o n F l u x
D r i f t
N e t c u r r e n t = 0
Ex
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.32: Non-uniform doping profile results in electron diffusiontowards the less concentrated regions. This exposes positively chargeddonors and sets up a built-in field Ex . In the steady state, the diffusion ofelectrons towards the right is balanced by their drift towards the left.
Vo
Carrier diffusion due to doping level gradient.This is a common device fabrication step.
Diffusion occurs until an electric field builds up!
We call this the built-in potential.
Note: the As+
are fixed,non-mobile charges!
· represents electrons
(majority carriers in this case)
P. E. Russell: [email protected]
Built-In Potential and Concentration
V2 = potential at point 2, V1 = potential at point 1,
n2 = electron concentration at point 2,
n1 = electron concentration at point 1
V2 −V1 =kTe
lnn2
n1
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
Built-In Field in Nonuniform Doping
Ex = electric field in the x direction,
b = characteristic of the exponential doping profile,
e = electronic charge .
Ex =
kTbe
E x p o s e d
A s+
D o n o r
n2
n1
D i f f u s i o n F l u x
D r i f t
N e t c u r r e n t = 0
Ex
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca
Fig. 5.32: Non-uniform doping profile results in electron diffusiontowards the less concentrated regions. This exposes positively chargeddonors and sets up a built-in field Ex . In the steady state, the diffusion ofelectrons towards the right is balanced by their drift towards the left.
Vo
P. E. Russell: [email protected]
The SUNY buffalo Applets on conventional semiconductors:http://jas.eng.buffalo.edu/
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
http://jas.eng.buffalo.edu/education/system/cdrw/index.html
P. E. Russell: [email protected]
What’s next
Optoelectronics as examples of nanoelectronics Light emission from quantum well devices Light absorption in solar cells/photovoltaics