Post on 17-Oct-2021
ECE-656: Fall 2011
Lecture 22:
Ionized Impurity Scattering
Mark Lundstrom Purdue University
West Lafayette, IN USA
1 10/19/11
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scattering of plane waves
S p, ′p( ) = 2π
H ′p , p
2δ ′E − E − ΔE( )
US (r ,t)
H ′p , p = ψ f
*
−∞
+∞
∫ US (r )ψ idr
ψ i =
1Ω
ei p• r ψ f =
1Ω
ei ′p • r
H ′p , p =
1Ω
e− i ′p •r
−∞
+∞
∫ US (r )ei p• r dr
incident plane wave
weak scattering
infrequent scattering
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examples
1τ E( ) =
1τm E( ) ∝ Df E( )
“short range potential” “oscillating, propagating potential”
USr ,t( ) = Uβ
a,e
Ωe± i
β • r −ω t( )
S p, ′p( ) = 2π
Uβa,e 2
Ωδ ′E − E ω( )δ ′p , p±
β
1τ E( ) =
1τm E( ) ∝ Df E ± ω( )
1τ Ep( ) =
ωE
⎛⎝⎜
⎞⎠⎟
1τ p( )
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static potential summary
(isotropic)
(elastic)
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oscillating potential summary
1τ p( ) ~
Df E ± ω( )2
1τmp( ) =
1τ p( ) (isotropic)
1τ Ep( ) =
ωE
1τ p( ) (inelastic)
ψ i =
1Ω
ei p• r
ψ f =
1Ω
ei ′p • r
USr ,t( ) = Uβ
a,e
Ωe± i
β • r −ω t( )
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acoustic vs. optical phonon scattering
LO (ABS)
1τ p( ) ~ nω Df E + ω0( )
LA (ABS + EMS)
1τ p( ) ~ Df E( )
LO (EMS)
1τ p( ) ~ nω +1( )Df E − ω0( )
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summary
1) Characteristic times are derived from the transition rate, S(p,p’)
2) S(p,p’) is obtained from Fermi’s Golden Rule
3) The scattering rate is proportional to the final DOS
4) Static potentials lead to elastic scattering
5) Time varying potentials lead to inelastic scattering
6) General features of scattering in common semiconductors can now be understood (almost)
8
covalent vs. polar semiconductors covalent polar
1τ
1τ
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II scattering potential
“impact parameter”
i) electrons in P-type material
According to FGR, the transition rate is independent of the sign of the scattering potential.
ii) electrons in N-type material
10
outline
Lundstrom ECE-656 F11
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/
(Reference: Chapter 2, Lundstrom, FCT)
1) Review 2) Screening 3) Brooks-Herring approach 4) Conwell-Weisskopf approach 5) Discussion 6) Summary / Questions
Lundstrom ECE-656 F11 11
screening
Bare Coulomb potential.
Mobile charges attracted to fixed charges “screen” out the fixed charge.
Screened Coulomb potential: ??
12
screening in 3D
13
screening in 3D
US (r) = −qδV (r) =
q2
4πκ Sε0re−r LD
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14
Debye length in 3D
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n0 = N3DF 1/ 2 ηF( )
∂n0
∂ηF
= N3DF −1/ 2 ηF( ) = n0
F −1/ 2 ηF( )F 1/ 2 ηF( ) = n0 (non-degenerate)
(non-degenerate)
Debye length
15
comments on screening
Lundstrom ECE-656 F11
1) Our semi-classical approach assumes that the potential is slowly varying on the scale of the electron’s wavelength. For rapidly varying potentials, a more sophisticated approach is needed. (See Ashcroft and Mermin, pp. 340-343 for a discussion of the Lindhard theory.)
2) Our semi-classical approach also assumes that the potential is slowly in time. (See Ashcroft and Mermin, p. 344 for a brief discussion.)
3) For potentials that vary rapidly in space and time, a “dynamic screening” treatment is needed. (See chapter 9 in Ridley, Quantum Processes in Semiconductors, 4th Ed. and Chapter 10 in Ridley, Electrons and Phonons in Semiconductor Multilayers.)
4) Screening is generally less effective in 2D and in 1D. (See J.H. Davies, The Physics of Low-Dimensional Structures, pp. 350-356
16
outline
Lundstrom ECE-656 F11
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/
(Reference: Chapter 2, Lundstrom, FCT)
1) Review 2) Screening 3) Brooks-Herring approach 4) Conwell-Weisskopf approach 5) Discussion 6) Summary / Questions
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transition rate and scattering potential
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II scattering (Brooks-Herring)
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Fourier transform of the screened Coulomb potential
choose z-axis along β: r
θ
small angle scattering preferred!! Lundstrom ECE-656 F11 20
Fourier transform (ii)
US β( ) = q2
κ Sε0
1β 2 +1 LD
2
⎛
⎝⎜
⎞
⎠⎟
US β( ) = q2
κ Sε0
1
4 p ( )2sin2 α 2( ) +1 LD
2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
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small angle scattering
impact parameter
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II scattering of high energy carriers For a given deflection angle, higher energies scatter less.
High energy electrons don’t “see” these fluctuations and are not scattered as strongly.
Random charges introduce random fluctuations in EC, which act a scattering centers.
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II scattering: recap
S p, ′p( ) = 2πq4 N I
κ S2ε0
2 Ω
δ ′E − E( )β 2 +1 LD
2( )2
S p, ′p( ) = 2πq4 N I
κ S2εS
2Ω
δ ′E − E( )4 p2
2 sin2α 2 +1 LD
2⎛⎝⎜
⎞⎠⎟
2
US β( ) = q2
κ Sε0
1β 2 +1 LD
2( ) H p, ′p =
1ΩUS β( )
Need to multiple by the total number of ionized impurities in the volume, Ω.
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examine result
1) S p, ′p( ) ~ N I
2) S p, ′p( ) ~ q4
S p, ′p( ) ~ 1 E23)
4) favors small angle scattering
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examine result 4) angular dependence
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momentum relaxation time
favors small angles
expect:
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momentum relaxation time
τm(E) ~ E3/ 2
τm(E) ≈ τ 0 E kBTL( )3/ 2
s = 3 / 2 τ 0 ~ TL3/ 2
1τm
= S p, ′p( ) 1− cosα( )′p∑
τm(E) =
16 2m*πκ S2ε0
2
N Iq4 ln 1+ γ 2( ) − γ 2
1+ γ 2
⎡
⎣⎢
⎤
⎦⎥E3/ 2
γ2 = 8m* ELD
2
2 See Lundstrom, pp. 69-70
28
outline
Lundstrom ECE-656 F11
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/
(Reference: Chapter 2, Lundstrom, FCT)
1) Review 2) Screening 3) Brooks-Herring approach 4) Conwell-Weisskopf approach 5) Discussion 6) Summary / Questions
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BH vs.CW
Brook-Herring means “screened Coulomb scattering.”
Conwell-Weisskopf means “unscreened Coulomb scattering.”
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Conwell-Weiskopf approach
unscreened Coulomb potential
Can we specify a minimum angle, so that the integral does not blow up?
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Conwell-Weiskopf approach
As the impact parameter increases, the deflection angle decreases.
But there is a maximum impact parameter?
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Conwell-Weisskopf approach
(Rutherford)
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Conwell-Weisskopf approach
τm(E) ≈ τ 0 E kBTL( )3/ 2
τ 0 ~ TL3/ 2
Much like the Brooks-Herring result.
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CW vs. BH
bmax =
12
N I−1/3
LD =
κ Sε0kBTq2n0
Compare bMAX to LD
Use BH if:
bmax > LD
B. K. Ridley, “Reconciliation of the Conwell-Weisskopf and Brooks-Herring formulae for charged-impurity scattering in semiconductors: Third-body interference,” J. Phys. C: Solid State Phys. 10, p. 1589 doi:10.1088/0022-3719/10/10/003, 1977.
35
outline
Lundstrom ECE-656 F11
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/
(Reference: Chapter 2, Lundstrom, FCT)
1) Review 2) Screening 3) Brooks-Herring approach 4) Conwell-Weisskopf approach 5) Discussion 6) Summary / Questions
Lundstrom ECE-656 F11 36
mobility
s = 3 / 2
µn =
qτ 0
m*
3 π4
~ TL3/ 2
~ TL3/ 2
T3/2 temperature dependence is the “signature” of charged impurity scattering.
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PN junction
screened screened
unscreened
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screening 2D modulation-doped layers
+ delta doped layer + + + + + + + + + + + + + + + + + + + + +
GaAs
AlGaAs
centroid of electron wavefunction
The heterojunction interface can be atomically smooth and at low temperatures, phonon scattering is absent, so scattering by remote impurities dominates. Extraordinarily high mobilities (e.g. > 106 cm2/V-s) can be achieved at about T = 1K.
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modulation-doped structures
For a discussion of modulation doping, screening in 2D, and remote impurity scattering in 2D, see:
J.H. Davies, The Physics of Low-Dimensional Semiconductors, Chapter 8, Cambridge Univ. Press, 1998.
40
outline
Lundstrom ECE-656 F11
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/
(Reference: Chapter 2, Lundstrom, FCT)
1) Review 2) Screening 3) Brooks-Herring approach 4) Conwell-Weisskopf approach 5) Discussion 6) Summary / Questions
41
summary
Lundstrom ECE-656 F11
1) The two classic treatments of II scattering are Brooks-Herring and Conwell-Weiskopf
2) II scattering is actually difficult to treat properly because:
FGR does not account for the difference in sign of the scattering potential
“multiple scattering” occurs at heavy doping.
Lundstrom ECE-656 F11 42
questions
1) Review 2) Screening 3) Brooks-Herring approach 4) Conwell-Weisskopf approach 5) Discussion 6) Summary / Questions