ECE606: Solid State Devices Lecture 8 Temperature
Transcript of ECE606: Solid State Devices Lecture 8 Temperature
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
ECE606: Solid State Devices Lecture 8
Temperature Dependent Carrier Density Concepts of Recombination
Gerhard Klimeck [email protected]
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 4
Presentation Outline
•Reminder: »Basic concepts of donors and acceptors »Statistics of donors and acceptor levels »Intrinsic carrier concentration
•Temperature dependence of carrier concentration
•Multiple doping, co-doping, and heavy-doping •Conclusion
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Donor Atoms in Real and Energy Space
( ) 20
20
40
202 4
13
1
16
πε= −
= − ×
*host
s ,hos
*host
s ,host
tm
mm K
q mK
m
.
r0
ET=E1
( )4
1 202 4πε
= −
*host
s ,host
qK
E m
~10s meV
1/β~kBT~25meV at T=300K 4
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Summary …
0ADNp n dVN −+ − + + = ∫
0AD Nn Np + −− + + =
10
12 4F V B C F B
F D A F BB
( E E ) / k T ( E E ) / k T A( E E ) /V A
D( E E ) k/ k TTNe e
eN N N
e− − −
−−
−+ +− + − =
A bulk material must be charge neutral over all …
Further if the material is spatially homogenous
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 4
Presentation Outline
•Reminder »Basic concepts of donors and acceptors »Statistics of donors and acceptor levels »Intrinsic carrier concentration
•Temperature dependence of carrier concentration
•Multiple doping, co-doping, and heavy-doping •Conclusion
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Carrier-density with Uniform Doping
0D Ap n N N dV+ − − + + = ∫
0D Ap n N N+ −− + + =
01 2 1 4
− − − −− −− + − =
+ +V B C B
D B A
F F
F BF
( E ) / k T ( E ) / k TV A ( E ) / k T ( E
E EE E )
D/ k T
AN e N ee e
N N
A bulk material must be charge neutral over all …
Further if the doping is spatially homogenous
Once you know EF, you can calculate n, p, ND+, NA
-.
( ) ( )1 2 1 22 2 0
1 2 1 4β ββ βπ π − − − − − + − = + +F FD AF F E EV V A C ( E )
D)
A( E
NN F E E NN Fe e
E E
(approx.)
FD integral vs. FD function ?
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Intrinsic Concentration
0D Ap n N N+ −− + + =
01 2 1 4
− − − −− −− + − =
+ +F V B C F B
F D B A F B
( E E ) / k T ( E E ) / k T( E E ) / k T ( E E )V C
Ak T
D/
Ne
NN e Nee
( ) ( )0 β β− − + −− = ⇒ =c F v FE E EV
ECn p e eN N
12 2β
≡ = + VGF i
C
EE E l NN
n
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 4
Presentation Outline
•Reminder »Basic concepts of donors and acceptors »Statistics of donors and acceptor levels »Intrinsic carrier concentration
•Temperature dependence of carrier concentration
•Multiple doping, co-doping, and heavy-doping •Conclusion
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Carrier Density with Donors
0D Ap n N N+ −− + + =
01 2 1 4
− − − −− −− + − =
+ +F V B C F B
F D B A F B
( E E ) / k T ( E E ) / k T A( E E ) / k T ( E E ) / k T
DCVN Ne e
e eN N
In spatially homogenous field-free region …
Assume N-type doping …
n (will plot in next slide)
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Temperature-dependent Concentration
D
nN
Temperature
1
Freeze out Extrinsic
Intrinsic
i
D
nN
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Physical Interpretation
D
nN
Temperature
1
Freeze out Extrinsic
Intrinsic
i
D
nN
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Electron Concentration with Donors
1 2 β+
−=+ DFD E
D( E )N
eN
β β β− −= ⇒ =FC FC( E )E E
C
EC
nN
n e e eN
1 2 β −
=
+
c D
D
( E E )
C
eN
Nn 1
ξ
≡+
DNn
N
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Electron concentration with Donors
0+− + =Dp n N
01 2
F V B C F B
F D B
( E E ) / k T ( E E ) / k T DV C ( E E ) / k T
NN e N ee
− − − −−− + =
+
2
01
ξ
− + =+
i Dnn
N
N
nn
No approximation so far ….
2× = ip n n
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
High Donor density/Freeze out T
2
01
ξ
− + =+
i Dnn
N
N
nn
D iN n
2
01
0ξ
ξ ξ
⇒ − + ≈+
⇒ + − =
D
D
N
N
N N
n
N
n
n n1 2
41 12ξ
ξ
= + −
DNn N
N
D
nN
Temperature
1
Freeze out Extrinsic
Intrinsic
i
D
nN
( )
2ξβ− −≡ C DE ECNN e
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Extrinsic T
( )
2C DE E / kTC
DNN e Nξ
− −≡
1 241 1
2ξ
ξ
= + −
DNn
NN
Electron concentration equals donor density hole concentration by nxp=ni
2
D
nN
Temperature
1
Freeze out Extrinsic
Intrinsic
i
D
nN
412
1 12ξ
ξ
≈ + −
DNNN
≈ DN
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Intrinsic T
2
01
ξ
− + =+
i Dnn
N
Nnn
12
22
2 4
= + +
iD D nn N N
2 for
1+
−= ≈ <+ F D B( E E ) /
DDk TD F D
N N Ee
N E
D
nN
Temperature
1
Freeze out Extrinsic
Intrinsic
i
D
nN
0
2
0− + ≈iD
n n Nn
2 2 0⇒ − + − =i Dn n N n
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Extrinsic/Intrinsic T
For i Dn N
1 222
2 4
= + + ≈
i iD Dn nN N n
2 1 22
2 4
= + + ≈
iD D
Dn nN N N
For D iN n
D
nN
Temperature
1
Freeze out Extrinsic
Intrinsic
i
D
nN
What will happen if you use silicon circuits at very high temperatures ? Bandgap determines the intrinsic carrier density.
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Determination of Fermi-level
( ) 1β
β− −
= ⇒ = +
FcEC C
EF
C
N e ENnlnn E
0Dp n N +− + =
01 2
− − − −−− + =
+F FV B C B
D BF
( E ) / k T ( E ) / k T DV
E E( kEC E ) / T
NN e N ee
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reference: Vol. 6, Ch. 4
Presentation Outline
•Reminder »Basic concepts of donors and acceptors »Statistics of donors and acceptor levels »Intrinsic carrier concentration
•Temperature dependence of carrier concentration
•Multiple doping, co-doping, and heavy-doping •Conclusion
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Multiple Donor Levels
210
1 2 1 2 1 4− − −− + + − =+ + +DF B F B A F BD
D D A( E ) / k T ( E ) / k T ( E EE kE ) / T
N N Np ne e e
Multiple levels of same donor …
1 2
1 2 01 2 1 2 1 4− − −− + + − =+ + +DF B F B AD F B
D A( E ) / k T ( E ) / k T ( E
DE E ) / k TE
N Np ne e
Ne
Codoping…
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Heavy Doping Effects: Bandtail States
E
k
k ?
E
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Heavy Doping Effects: Hopping Conduction
E
k
Bandgap narrowing
β−× =*
GC V
Ep n N N e
Band transport vs.
hopping-transport
e.g. Base of HBTs K?
E
e.g. a-silicon, OLED
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Arrangement of Atoms
Poly-crystalline Thin Film Transistors
Crystalline
Amorphous Oxides
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Poly-crystalline material
Isotropic bandgap and increase in scattering
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Band-structure and Periodicity
Periodicity is sufficient, but not necessary for bandgap. Many amorphous material show full isotropic bandgap
PR
B, 4
, 250
8, 1
971
Eda
gaw
a, P
RL,
100
,013
901,
200
8
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Conclusions
1. Charge neutrality condition and law of mass-action allows calculation of Fermi-level and all carrier concentration.
2. For semiconductors with field, charge neutrality will not hold and we will need to use Poisson equation.
3. Heaving doping effects play an important role in carrier transport.
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Outline
29
1) Non-equilibrium systems
2) Recombination generation events
3) Steady-state and transient response
4) Derivation of R-G formula
5) Conclusion
Ref. Chapter 5, pp. 134-146
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Current Flow Through Semiconductors
30
I
V
Depends on chemical composition, crystal structure, temperature, doping, etc.
Carrier Density
velocity
I G Vq n Av
= ×= × × ×
Transport with scattering, non-equilibrium Statistical Mechanics Encapsulated into drift-diffusion equation with recombination-generation (Ch. 5 & 6)
Quantum Mechanics + Equilibrium Statistical Mechanics Encapsulated into concepts of effective masses and occupation factors (Ch. 1-4)
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Non-equilibrium Systems
31
vs.
Chapter 6 Chapter 5
I
V
How does the system go BACK to equilibrium?
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Direct Band-to-band Recombination
32
Photon
GaAs, InP, InSb (3D)
Lasers, LEDs, etc.
In real space … In energy space …
Photon Direct transistion – direct gap material
e and h must have same wavelength
1 in 1,000,000 encounters
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Direct Excitonic Recombination
33
Photon (wavelength reduced from bulk)
CNT, InP, ID-systems Transistors, Lasers, Solar cells, etc.
In energy space …
In real space …
Mostly in 1D systems Requires strong coulomb interactions
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Indirect Recombination (Trap-assisted)
34
Phonon
Ge, Si, …. Transistors, Solar cells, etc.
Trap needs to be mid-gap to be effective. Cu or Au in Si
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Auger Recombination
35
Phonon (heat)
InP, GaAs, … Lasers, etc.
1 2
3
1 2
4
3
4
Requires very high electron density
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Impact Ionization – A Generation Mechanism
36
Si, Ge, InP Lasers, Transistors, etc.
4 3
1
2
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Indirect vs. Direct Bandgap
37
The top & bottom of bands do not align at same wavevector k for indirect bandgap material
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Photon Energy and Wavevector
38
21 21 in eV
π=
photonE. / 4
2 25 10 m
π πµ−<< =
×a
photon
photonkE
=
=
+
+ V
V phot
photon
n C
C
o
k kE E
kE
Photon has large energy for excitation through bandgap, but its wavevector is negligible compared to size of BZ
2πa
2 in mπ
λ µ=photonk
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Phonon Energy and Wavevector
39
2 2
phonphon
souo
nn
d on
kE/
= =
π πλ υ
=
=
+
+ V
V phon
phonon
n C
C
o
k kE E
kE
4
2 25 10 m
π πµ−≈ =
×a
Phonon has large wavevector comparable to BZ, but negligible energy compared to bandgap
vsound ~ 103 m/s << vlight=c ~ 106m/s λsound >> λlight
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Localized Traps and Wavevector
40
4
2 25 10trap ~k
a mπ π
µ−≈×
Trap provides the wavevector necessary for indirect transition
a