Lecture 21: Scattering and FGR

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Φωκάς Στεφανόπουλος
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1 ECE-656: Fall 009 Lecture : Scattering and FGR Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA

2 Review: characteristic times τ ( p), (, ) == S p p p( t = 0) (τ, single particle lifetime) τ m ( p) = S( p, ) ( p0 ) cosα, t = 0 t τ τ E ( p) (, ) = S p p E E, 0 α p z t τ m τ t τ E > τ m τ

3 outline ) Fermi s Golden Rule ) Example: static potential 3) Example: oscillating potential 4) Discussion 5) Summary (Reference: Chapter, Lundstrom) 3

4 FGR p US (,) rt π S( p, ) = Hpp δ ( E E E) E = E0 + E E = 0 for a static U S E =± ω for an oscillating U S (See Sec..7 of Lundstrom for a derivation of FGR) 4

5 FGR ψ i ( r ) p p US (,) rt π S p, p H, p E E E ( ) = δ ( ) ψ f ( r ) + *, p= ψ f S i H U r ψ dr matrix element (note the order of the subscripts) ) Identifiy the scattering potential ) Specify the wavefunction (e.g. D, D, 3D) 5

6 scattering of Bloch electrons ipr ψ i = e u r k p US (,) rt ψ f i r = e u r k π S p, H, p E E E ( ) = δ ( ) + *, p= ψ f S i H U r ψ dr + i r ip r H, p= I k, k e U S( r ) e dr I k, k u ( r ) u ( r ) dr = I k, k parabolic bands * unit cell k k 6

7 overlap integrals B.K. Ridley, Quantum Processes in Semiconductors, 4 th Ed., pp. 8-86, Cambridge, 997 B.K. Ridley, Electrons and Phonons in Semiconductor Multilayers, pp , Cambridge, 997 D.K. Ferry, Semiconductors, pp. 4, , Macmillan, 99 7

8 scattering of plane waves i ipr ψ = e p US (,) rt i r ψ f = e π S p, H, p E E E ( ) = δ ( ) + *, p= ψ f S i H U r ψ dr + i r ipr H, p= e U S( r ) e dr 8

9 FGR: assumptions p US (,) r t p ) Weak scattering ) Infrequent scattering: E t Need a long time between scattering events so that the energy is shaoprpy defined ( collisional broadening ). 9

10 outline ) Fermi s Golden Rule ) Example: static potential 3) Example: oscillating potential 4) Discussion 5) Summary (Reference: Chapter, Lundstrom) 0

11 FGR: static scattering potential i ipr ψ = e p π S p, p H, p E E E ( ) = δ ( ) US ( r ) i r ψ f = e p = p + q + i r ipr S H, p= e U r e dr + i( p) r, p= S = S H U r e d ru q H, = U ( q) p p S short range: neutral impurity long range: charged impurity q = p = k k The matrix element is the Fourier transform of the scattering potential.

12 FGR: delta-function example i ipr ψ = e p US (,) rt U r Cδ S i r ψ f = e = ( 0) π S p, p H, p E E E ( ) = δ ( ) + i r ipr H, p= e Cδ (0) e dr = π C C S p p E E K E E (, ) = δ ( ) = δ ( ) short range potential

13 S p p K E E (, ) = δ ( ) ( p) scattering rate = S p p = K E E (, ) δ ( ) τ δ ( E E) = D 3D ( E) * * m me δ ( E E) = 3 π (for parabolic energy bands) τ ( p) Df ( E) For an incident electron with energy, E, the scattering rate is proportional to the density of final states at energy, E (D, D, 3D) 3

14 momentum relaxation rate S p p K E E (, ) = δ ( ) τ m ( p) = S p p p p, z, z0 p z α = θ p x τ m ( p) = S p p, (, )( cosα ) p y 4

15 momentum relaxation = K δ E E τ m ( p), ( )( cosα) α = θ in our coordinate system π π ( π ) ( cos ) sin δ E E = dφ θ θdθ d δpe E extra term = π 0 ( cosθ)sinθ dθ π = cos 0 = 0 τ p = p m τ isotropic 5

16 energy relaxation ipr ψ = e i p US (,) rt S i r ψ f = e U r Cδ = ( 0) S p p K E E (, ) = δ ( ) τ E ( p) = S p p, (, ) E E τ E p = 0 6

17 static potential summary ipr ψ = e i p US (,) rt S i r ψ f = e U r Cδ = ( 0) τ ( p) D E τ p = p m (isotropic) τ τ E p = 0 (elastic) 7

18 outline ) Fermi s Golden Rule ) Example: static potential 3) Example: oscillating potential 4) Discussion 5) Summary (Reference: Chapter, Lundstrom) 8

19 example: oscillating potential i ipr ψ = e p E US (,) rt E i r ψ f = e ae, U β ± i( r t) U rt e β = ω S (, ) π S p, p H, p E E E ( ) = δ ( ) E =±ω + ae, + U i r β ± iβ r ipr a, e i( p ± β ) r H, p= e e e dr = U β e dr 3/ H U ae,, p= β δ, p± β 9

20 momentum conservation i ipr ψ = e p E US (,) rt E i r ψ f = e ae, U β ± i( r t) U rt e β = ω S (, ) H U ae,, p= β δ, p± β = p+ β (ABS) = p β (EMS) (momentum conservation) β p p β 0

21 energy-momentum conservation i ipr ψ = e p E US (,) rt E i r ψ f = e ae, U β ± i( r t) U rt e β = ω S (, ) π S p, p H, p E E E ( ) = δ ( + ) H U = δ β, p, p± β E =±ω ae, π U S p p E E β (, ) = ( ) δ ω δ p, p ± β

22 energy-momentum conservation i ipr ψ = e p E US (,) rt E i r ψ f = e ae, U β ± i( r t) U rt e β = ω S (, ) ae, π U S p p E E β (, ) = ( ) δ ω δ p, p ± β E = E+ ω = p+ β E = E ω = p β ABS EMS

23 scattering rate Assume (for now) that for any transition from E i to E f, we can find a vibration that conserves momentum. ae, π U S p p E E β (, ) = δ ( ω) ABS EMS = S p p = K E E ( p) (, ) δ ( ω) τ τ ( p) ( E± ω) Df = K Df E± ( ω) Scattering rate is proportional to the density of final states. 3

24 momentum relaxation: oscillating potential ae, π U S p p E E β (, ) = ( ) δ ω δ p, p ± β (isotropic) p (, ) z = S p p = τ p p τ p m z 4

25 energy relaxation: oscillating potential ae, π U S p p E E β (, ) = δ ( ω) τ E E ω ω = S( p, ) = S( p, ) = p E E E τ p < = τ τ τ E ( p) ( p) ( p) E m τ τ τ ( p) > ( p) = ( p) m 5

26 oscillating potential summary i ipr ψ = e p US (,) rt i r ψ f = e ae, U β ± i( r t) U rt e β = ω S (, ) τ ( p) Df τ p = p m ( E± ω) (isotropic) τ ω = τe p E p (inelastic) τ 6

27 an aside on phonon scattering small β intravalley scattering ω ω 0 LO LA intervalley scattering ω = υ S β β g f large β 7

28 phonon scattering τ ( p) ~ D f ( E± ω ) 0 ω ω 0 ω = ω 0 LO LA ω = υ S β τ β ( p) ~ D f ( E± ω) 8

29 intravalley phonon scattering ω ω 0 τ ( p ) τ ω = ω 0 LO LA (ABS + EMS) D E ~ f LO (EMS) LA LO (ABS) ω = υ S β τ ( p) ~ D f ( E± ω ) 0 β ω 0 E 9

30 outline ) Fermi s Golden Rule ) Example: static potential 3) Example: oscillating potential 4) Discussion 5) Summary 30

31 constraints on scattering Cf ˆ = S( p, p ) f ( ) f ( p) S( p, p ) f ( p) f ( ) in equilibrium: Cf ˆ 0 = 0 = S( p, p ) f0( ) f0( p) S( p, p ) f0( p) f0( ) S(, p) f0( p) f0( ) = = e S( p, ) f0( ) f0( p) EkT B E(p) E = E( p) E E p 3

32 constraints on scattering S p p (, ) (, ) S p p = e EkT B E = E( p) E ) elastic scattering: E = 0 S( p, ) = S(, p) ) phonon scattering: E =ω 0 E(p) S S ABS EMS (, p) ( p, ) = e ω 0 kt B ω 0 p 3

33 constraints on phonon scattering S S ABS EMS (, p) ω0 kt ABS B = e (, ) ( p, ) S p N ω ω0 EMS kt B S p, p e Nω N ω = e ω k T B E(p) ω kbt ω kbt e e Nω = = N ω kbt ω + e (, ) ABS S p p N ω EMS S ( p, ) N ω + p ω 0 33

34 outline ) Fermi s Golden Rule ) Example: static potential 3) Example: oscillating potential 4) Discussion 5) Summary 34

35 summary ) Characteristic times are derived from the transition rate, S(p,p ) ) 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) 35

36 covalent vs. polar semiconductors covalent E E polar x 00 x Γ Γ E E 36

37 questions ) Fermi s Golden Rule ) Example: static potential 3) Example: oscillating potential 4) Discussion 5) Summary 37

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