Lecture 22: Ionized Impurity Scattering

Transcription

1 ECE-656: Fall 20 Lecture 22: Ionized Impurity Scattering Mark Lundstrom Purdue University West Lafayette, IN USA 0/9/ scattering of plane waves ψ i = Ω ei p r U S ( r,t) incident plane wave ( ) = 2π H S p, p H H p, p = + p, p + ψ * f p, p = Ω e i p r 2 δ ( E E ΔE ) U S ( r )ψ i d r U S ( r )e i p r d r ψ f = Ω ei p r weak scattering infrequent scattering 2

2 examples short range potential oscillating, propagating potential ( ) = U a,e β U S r,t ( ) = 2π S p, p Ω e±i U β a,e 2 β r ωt ( ) Ω δ ( E E ω )δ p, p± β τ ( E) = τ m E ( ) D E f ( ) τ ( E) = τ m E ( ) D E ± ω f ( ) τ E p ( ) = ω E τ p ( ) 3 static potential summary (isotropic) (elastic) 4

3 oscillating potential summary ψ i = Ω ei p r ( ) ~ D E ± ω f ( ) 2 ( p ) = τ ( p ) (isotropic) τ p τ m τ E p ( ) = ω E ( ) (inelastic) τ p U S r,t ψ f = Ω ei p r ( ) = U β a,e Ω e±i β r ωt ( ) 5 acoustic vs. optical phonon scattering ( ) ~ D E f ( ) τ p ( ) ~ n D E + ω ω f ( 0 ) τ p LA (ABS + EMS) ( ) ~ ( n + ω ) D f ( E ω 0 ) τ p LO (EMS) LO (ABS) 6

4 summary ) 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) 7 covalent vs. polar semiconductors covalent polar τ τ 8

5 i) electrons in P-type material II scattering potential impact parameter ii) electrons in N-type material According to FGR, the transition rate is independent of the sign of the scattering potential. 9 outline ) Review 2) Screening 3) Brooks-Herring approach 4) Conwell-Weisskopf approach 5) Discussion 6) Summary / Questions (Reference: Chapter 2, Lundstrom, FCT) This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License. 0

6 screening Bare Coulomb potential. Screened Coulomb potential:?? Mobile charges attracted to fixed charges screen out the fixed charge. screening in 3D 2

7 screening in 3D U S (r) = qδv (r) = q 2 4πκ S ε 0 r e r L D 3 Debye length in 3D n 0 = N 3D F / 2 ( η F ) n 0 η F = N 3D F / 2 η F Debye length (non-degenerate) F ( ) = n / 2 ( η F ) 0 F / 2 ( η F ) = n 0 (non-degenerate) 4

8 comments on screening ) 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 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, 4 th Ed. and Chapter 0 in Ridley, Electrons and Phonons in Semiconductor Multilayers.) 4) Screening is generally less effective in 2D and in D. (See J.H. Davies, The Physics of Low-Dimensional Structures, pp outline ) Review 2) Screening 3) Brooks-Herring approach 4) Conwell-Weisskopf approach 5) Discussion 6) Summary / Questions (Reference: Chapter 2, Lundstrom, FCT) This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License. 6

9 transition rate and scattering potential 7 II scattering (Brooks-Herring) 8

10 Fourier transform of the screened Coulomb potential choose z-axis along β: r θ 9 Fourier transform (ii) U S ( β) = q2 κ S ε 0 β L D U S ( β) = q2 κ S ε 0 4 p ( ) 2 sin 2 2 ( α 2) + L D small angle scattering preferred!! 20

11 small angle scattering impact parameter 2 II scattering of high energy carriers For a given deflection angle, higher energies scatter less. Random charges introduce random fluctuations in E C, which act a scattering centers. High energy electrons don t see these fluctuations and are not scattered as strongly. 22

12 II scattering: recap ( ) = 2πq4 N I S p, p S p, p κ S 2 ε 0 2 Ω ( ) = 2πq4 N I κ 2 S ε 2 S Ω 4 p 2 2 H p, p δ ( E E) β 2 2 ( + L D ) 2 = Ω Need to multiple by the total number of ionized impurities in the volume, Ω. δ ( E E) sin 2 2 α 2 + L D U S ( β) 2 U S ( β) = q2 κ S ε 0 β L D ( ) 23 examine result ( ) ~ N I ) S p, p ( ) ~ q 4 2) S p, p 3) ( ) ~ E 2 S p, p 4) favors small angle scattering 24

13 4) angular dependence examine result 25 momentum relaxation time favors small angles expect: 26

14 = S p, p τ m p momentum relaxation time ( )( cosα ) τ m (E) = 6 2m* πκ 2 2 S ε 0 ln + γ 2 N I q 4 ( ) γ 2 E 3/ 2 + γ 2 γ 2 = 8m * EL D 2 2 See Lundstrom, pp τ m (E) ~ E 3/ 2 τ m (E) τ 0 ( E k B T L ) 3/ 2 3/ 2 τ 0 ~ T L s = 3 / 2 27 outline ) Review 2) Screening 3) Brooks-Herring approach 4) Conwell-Weisskopf approach 5) Discussion 6) Summary / Questions (Reference: Chapter 2, Lundstrom, FCT) This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License. 28

15 BH vs.cw Brook-Herring means screened Coulomb scattering. Conwell-Weisskopf means unscreened Coulomb scattering. 29 Conwell-Weiskopf approach unscreened Coulomb potential Can we specify a minimum angle, so that the integral does not blow up? 30

16 Conwell-Weiskopf approach As the impact parameter increases, the deflection angle decreases. But there is a maximum impact parameter? 3 Conwell-Weisskopf approach (Rutherford) 32

17 Conwell-Weisskopf approach τ m (E) τ 0 ( E k B T L ) 3/ 2 3/ 2 τ 0 ~ T L Much like the Brooks-Herring result. 33 CW vs. BH Compare b MAX to L D b max = 2 N /3 I Use BH if: b max > L D L D = κ S ε 0 k B T q 2 n 0 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. 0, p. 589 doi:0.088/ /0/0/003,

18 outline ) Review 2) Screening 3) Brooks-Herring approach 4) Conwell-Weisskopf approach 5) Discussion 6) Summary / Questions (Reference: Chapter 2, Lundstrom, FCT) This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License mobility s = 3 / 2 ~ T L 3/ 2 µ n = qτ 0 m * 3 π 4 ~ T 3/ 2 L T 3/2 temperature dependence is the signature of charged impurity scattering. 36

19 PN junction screened screened unscreened 37 screening 2D modulation-doped layers AlGaAs delta doped layer centroid of electron wavefunction GaAs 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. > 0 6 cm 2 /V-s) can be achieved at about T = K. 38

20 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, outline ) Review 2) Screening 3) Brooks-Herring approach 4) Conwell-Weisskopf approach 5) Discussion 6) Summary / Questions (Reference: Chapter 2, Lundstrom, FCT) This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License. 40

21 summary ) 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. 4 questions ) Review 2) Screening 3) Brooks-Herring approach 4) Conwell-Weisskopf approach 5) Discussion 6) Summary / Questions 42