L22 Ionized Impurity Scattering - nanoHUB

ECE-656: Fall 2011

Lecture 22:

Ionized Impurity Scattering

Mark Lundstrom Purdue University

West Lafayette, IN USA

1 10/19/11

Lundstrom ECE-656 F11 2

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


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( )


static potential summary

(isotropic)

(elastic)


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( )


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( )


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


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


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


14

Debye length in 3D


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


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






transition rate and scattering potential


II scattering (Brooks-Herring)


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

⎛

⎝⎜⎜

⎞

⎠⎟⎟


small angle scattering

impact parameter


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.


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.


examine result

1) S p, ′p( ) ~ N I

2) S p, ′p( ) ~ q4

S p, ′p( ) ~ 1 E23)

4) favors small angle scattering


examine result 4) angular dependence


momentum relaxation time

favors small angles

expect:


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






BH vs.CW

Brook-Herring means “screened Coulomb scattering.”

Conwell-Weisskopf means “unscreened Coulomb scattering.”


Conwell-Weiskopf approach

unscreened Coulomb potential

Can we specify a minimum angle, so that the integral does not blow up?


Conwell-Weiskopf approach

As the impact parameter increases, the deflection angle decreases.

But there is a maximum impact parameter?


Conwell-Weisskopf approach

(Rutherford)


Conwell-Weisskopf approach

τm(E) ≈ τ 0 E kBTL( )3/ 2

τ 0 ~ TL3/ 2

Much like the Brooks-Herring result.


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






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.


PN junction

screened screened

unscreened


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.


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.

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outline





41

summary


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.


questions


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