Electron-phonon scattering (Finish Lundstrom Chapter 2)

Transcription

1 Electron-phonon scattering (Finish Lundstrom Chapter ) Deformation potentials The mechanism of electron-phonon coupling is treated as a perturbation of the band energies due to the lattice vibration. Equilibrium atomic positions: R 1, R, R3, Lattice vibrations cause perturbations about equilibrium: locally due to perturbation. In a linear approximation: R + δr. Energy bands shift δe c δr D c a δe v δr D v a D c, D v are constants known as deformation potentials. Lattice vibration waves lead to sinusoidal variations δe. These mix Bloch waves for the electrons which leads to scattering. We will use δe as a perturbation and then apply Fermi s Golden Rule to get the scattering rates. Energy and momentum conservation We consider inelastic scattering where an electron makes a transition from momentum p to p involving emission or absorption of a phonon q: Energy conservation: + sign phonon absorption - sign phonon emission Assuming spherical, parabolic bands, and intra-valley scattering (i.e. both initial and final electron states are within the same parabolic band): Momentum conservation tells us that: (1) Electron phonon scattering Handout.fm

2 p' p ± h _ q take p + h _ q ± h _ pqcosθ () compare (1) & () to obtain: simplifying: _ ω h q p + cos θ ± qv e where v e p. m ± m h _ ω h _ q ± h _ pqcosθ (3) (3) sets min, max values for phonon wave vectors via 1 cosθ 1 Intravalley acoustic phonon scattering. ω qv s v s : sound velocity _ h q v p + cos θ ± ---- s v e ω optical q max : θ π absorption θ 0 emission acoustic q π/a Typically, 10 5 cm/sec and v e 10 7 cm/sec (300K). So, v s

3 p' p nearly elastic _ h q p magnitude of q max m v e _ h For m 0.5m o, with v e 10 7 cm/sec, we find where a is taken as 5 Å. Thus, for intravalley acoustic phonon scattering, the participating phonons are near zone center. Energy transfer ΔE max h _ ω max h _ q max v s 10 3 ev, which is small, so we find that intravalley acoustic phonon scattering is nearly elastic Intravalley optical phonon scattering Optic phonon energies ω ο are in the range of 30-50meV, so optic phonon scattering is definitely inelastic. Once again applying the conservation law (3): Solving for q: h _ q + ω pq cos θ ± p o 0 ± pcosθ 4p cosθ ± 8h _ ω q o m h _ (only the - sqrt term is physical, since q is a magnitude, hence must be positive. So: v e

4 _ h q p θ cos θ h _ m + cos + ± ω o p p[ + cosθ + cos θ± hω o E] for room temp E ( hω o ) hq max.7p, which again involves phonons near zone center since the electron energy is usually thermal, hence the electron momentum is small. Scattering rates Fermi golden rule: Spp' (, ) For deformation potential scattering π H h pp' δ( Ep' ( ) Ep ( ) + hω q ) - abs + emiss In the case of optic phonons, δa u, since the atoms are moving in opposite directions. u On the other hand, for acoustic phonons, δa -----, since, for acoustic phonons, the atoms x are moving in nominally the same direction. So we have: D A - acoustic deformation potential δe o D 0 u D o - optic deformation potential Acoustic Since H p'p ( D A qa q ) δ( p' p hg ± hq) Where the parameter A q is derived from the quantum mechanics of phonons. Recall the phonon operator:

5 δr q _ h ê Mω q q [ a q e iq r + a q e iq r ] For phonon absorption: where N q phonon occupation for mode q. For phonon emission we get: A q h _ N Mω q + 1 a q N q ( N q + 1)h _ q Mωq At room temperature, N q is typically >> 1, so N q N q + 1 kt _. We can therefore add the h ω emission and absorption processess. If we neglect G 0 processes, which are generally weaker, we obtain the acoustic phonon scattering rate: Sp' (, p) π _ D A kt q δ( p' p + hq) δ( E' E + hω) h M ω After integration over all allowed p', and use of ω qv s (see text for full details), we obtain: Sp ( ) 1 -- τ π _-- V cell D A kt DE ( ) h Mv s density of electron states where ρ M is the mass density. Again a power law in E: unit V for acoustic phonon scattering

6 Optic deformation potential scattering In this case, the perturbation of the band energy goes as so inherently inelastic, we can t combine absorption and emission. The matrix element is similar to the acoustic phonon case. For phonon absorption: δe o D o u. Since the process is For phonon emission: hd o H pp' ( N Mω q + 1) δ( p' p + h _ q o ) o The scattering rate is a sum of emission and absorption scattering: Sp ( ) D o ( m ) πρh _3 N q ( E+ _ h ω o ) 1 + ( N q + 1) ( E _ h ω o ) 1 u o ( E _ h ω o ) ω o Heaviside or unit-step function Phonon Scattering Rates S S Abs + Emission ABS Acoustic E _ h ωo Optic E

7 Polar optic phonon scattering This is the process that usually dominates in compound semiconductors. In this case, since we are considering optic phonons, the induced dipole is directly proportional to the phonon amplitude: where e* is the effective charge on the dipole. A simple analysis leads to an expression for the perturbation energy: Here, V c is the unit cell volume. The resulting scattering rate for the unscreened case is (see text): Sp ( ) The screened result for this case is a complex expression (see Ferry if interested). Intervalley scattering In Si, zone edge phonons (optical or high energy acoustic) can move carriers from one equivalent valley to another. Optical deformation potential scattering rate applies, modified by Z f, the number of equivalent final valleys, and using D if, which is defined as the intervalley deformation potential. In GaAs e ω o πh _ E E m ε ε( o) Nq sinh hω o 1 + ( N q + 1) sinh E 1 hω o L ΔE Γ L 0.9eV Γ only very high energy carriers in Γ valley can scatter to L valley. This only occurs in high fields, or certain laser excitation expts

8 Summary - Scattering in intrinsic Si, GaAs (Lundstrom.11) 1 -- (see) τ ADP in Si equiv intervalley in Si at 300K Polar optic phonon in GaAs at 300k Γ - L scatt in GaAs τ τ m E(eV - ADP scattering dominates for E< for both Si, GaAs hω op - Si has equivalent intervalley scattering for low energies involving optic phonons. - In GaAs, no intervalley scattering for E < 0.9eV. Low field behavior in GaAs dominated by POP scattering. This is weaker and ~ const with energy