Semiclassical Electron Transport

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1 Semiclassical Electron Transport Branislav K. Niolić Department of Physics and Astronomy, University of Delaware, U.S.A. PHYS 64: Introduction to Solid State Physics

2 Quasiparticle Propagation: Bloch Wavepacets Wavepacets represent quasiparticle localized in space: In the wea external field we can neglect the transitions between different bands fix the zone index n=const. Ψ ( r ) = C ( t) Φ ( r )

3 Semiclassical Dynamics of Bloch Wavepacets Wavepacet in external field described by potential ħ Ψ + U ( r ) + Wext ( r ) Ψ ( r ) = i m ħ ε t Introduce new operator ε( i ) Electron in a crystal represented as wavepacet of Bloch states propagates as a free quasiparticle with charge e, energy ( ), and Hamiltonian ε( i ) : P ˆ, ε ( i ) = 0 W ( r ) Ψ C ( t) ( ) + Wext ( r) Φ ( r) = iħ t ε( ) εe R R R ir ε( i ) Φ ( r) = εe Φ ( r) = εφ ( r + R) = e εφ ( r) = ε( ) Φ( r) Band energy is periodic function in the reciprocal space ext = ir R R R R R R Ψ Ψ( r) C ( t) [ ε( i ) + W ( )] ( ) [ ( ) ( )] ext r Φ r = iħ ε i + Wext r Ψ ( r) = iħ t t ε

4 Hamiltonian, Velocity, and Effective Mass of Bloch Electron ħ * m = d ε( ) d v Ehrenfest theorem: Schrödinger evolution of the center of wavepacet can be obtained from the trajectory of the corresponding classical particle: p correspondence principle: i ħ p H ( r, p ) = E + W ext ( r ) ħ H ( r, p ) 1 ε ( ) = = p ħ tensor of inverse effective mass: d 1 ε ( ) 1 ε( ) ħ a = = dt ħ ħ t ħ H = = W ext ( r ) t r 1 1 ε ( ) = m αβ α ħ β

5 Limitations of Semiclassical Dynamics The spatial scale of all external potentials must be much larger than interatomic spacing, maing it possible to construct wave pacets spanning many unit cells, but seeing the external potential as very slowly varying The magnitude of the electric field cannot be too large, or else they induce Zenner tunneling between bands (electrons are quite effective in screening external fields so this strengths are hard to achieve in metals): ee F ε gap ε ε gap F The magnitude of magnetic fields cannot be too large: ħeb ε m ħω = ε gap The fields must be slowly varying to avoid excitations across the gap. gap ε ε gap F

6 Currents in Bands: Insulators Add Pauli principle to Bloch electron propagation under the influence of an applied electric field Full band of states is insulating! e ( ) j = ε d 8 πħ ( π ) First BZ time reversal invariance: ε ( ) = ε ( ) ε ( ) = ε ( ) 1 v = ε ( ) = ε ( ) = v j 0 for insulator ħ 1

7 Currents in a Band: Metals Fermi sea of a partially filled band will shift under the influence of an applied electric field this destroys inversion symmetry of the Fermi sea therefore causing a net current: e ε j = ( ) d 8 πħ ( π) Occupied 1 e ε( ) j = d 8 πħ ( π) Empty e 1 e 1 j = ε( ) d ε( ) d 8 π ( π) 8 π ( π) ħ ħ First BZ Empty 1

8 Properties of Holes A nearly full simple band has states near the Fermi surface that can be thermally or electrically excited of a negative mass Density of states with holes at the top of the band which have positive charge and positive mass 1 d ε * d ε ħ m * top ħ ħ = ± π / a 1 1 ( ) ( ) = = = < 0 m d d ta ħ 1 d ee ε top = ε 0 t + ta = ε 0 t + vɺ = ( ) * ε = * m ħ dt m

9 Currents in Many Bands: Insulators at Finite Temperature An insulator forms when Fermi energy falls within the gap of the density of states as the temperature is raised, electrons are - Eg / BT promoted over the gap n e and both electrons and holes contribute to the conductivity which increases with temperature.

10 Sources of Electric Resistivity Bloch states are stationary states that describe unperturbed propagation of electrons perfect lattice yields no resistivity! Resistivity is dominated by the scattering off the deviations from a perfect lattice: defects and lattice vibrations=phonons Resistivity arises also due to electron-electron interactions (from simple order of magnitude arguments based on relative strength of the interactions, their contribution should dominate but due to the Pauli principle it does not).

11 Distribution Function In and Out of Equilibrium equilibrium: f ( ) = f ( r,, t) = eq E = 0 ( ( ) F ) e β ε ε no scattering: (,, ) (, ee f r t = f r vdt + dt, t dt) ħ ee f f ( r,, t) = f ( r vdt, + dt, t dt) + dt ħ t scattering e n( r ) = d f ( r,, t) 8π e j( r ) = d v( ) f ( r,, t) 8π

12 Boltzmann Equation ee f f ( r,, t ) = f ( r,, t ) v + f f + r ħ t t ee f f + v f + f = t r ħ t scattering scattering If the phonon and defect perturbations are small, time-independent, and described by Hamiltonian Ĥ, then the scattering rate from a Bloch state to (occupied to unoccupied) is w = π Hˆ ħ f V = d (1 f ( )) w f ( ) (1 f ( )) w f ( ) t ( π ) scattering The Boltzmann equation is valid under assumptions of semi-classical transport: Effective mass approximation (which incorporates the quantum effects due to periodicity of the crystal); Born approximation for the collisions, in the limit of small perturbation for the electron-phonon interaction and instantaneous collisions; no memory effects, i.e. no dependence on initial condition terms. The phonons are usually treated as in equilibrium, although the condition of non-equilibrium phonons may be included through an additional equation.

13 Relaxation Time Approximation Ansatz: The rate at which a system returns to equilibrium is proportional to its deviation from equilibrium (i.e., we mae the assumption that scattering merely acts to drive a non-equilibrium system bac to equilibrium): f t = f ( ) feq( ) τ( ) scattering If E 0, t < 0 and then at t 0, E 0 the external electric field is switched off, then for a homogeneous system we find: f f f feq = = f feq = f ( t = 0) f eq e t t scattering τ f f e f f f ( ) f ( ) = 0, = 0 E = = t r ħ t τ( ) In the steady state transport regime induced by a time-independent external electric field: e f ( ) = f eq ( ) + τ ( ) E ħ f ( ) scattering t / τ

14 Linear Semiclassical Response For small electric field (Ohmic regime), the relaxation time approximation solution can be linearized: e f ( ) feq ( ) + τ ( ) E feq ( ) ħ e E = E ˆ xx f ( ) feq + τ ( ) Ex ħ According to the linear Boltzmann equation, the effect of the electric field E x is to shift the Fermi surface by δ x = eτ E / ħ x Note that elastic scattering cannot restore equilibrium; rather they would cause Fermi surface to expand inelastic scattering (i.e., from phonons) is needed to explain relaxation.

15 Drude Conductivity: Naïve Derivation Drude (1900) assumptions: all electrons participate and electron-lattice -1 scattering yields a scattering rate τ which introduces the mean free time (or relaxation time or collision time which electrons travel, on average, between collisions): τ m mvɺ + ( v vtherm ) = ee τ drift velocity: v v = v friction: eτ m v τ D therm ne τ steady state: vd = E j = envd = E m m ne τ Ohmic conductivity: j = σ E σ = m eτ Mobility: σ = µ ne µ = m Drift

16 Drude Conductivity: Bloch-Boltzmann Derivation Quantum Mechanics (197) maes Drude reasoning problematic not all electrons can participate in the conduction due to the Pauli principle! e e eτ ( ) eq j = d v( ) f ( ) d v( ) f ( ) eq + Ex 8π 8π ħ x isotropic material: jy = jz = 0 ( ) v = v v f 0 eq d = feq ( ) f E feq = = ħvx δ ( E EF ) ħvx E E x x e f j E d v E 8π E eq x ( ) x xτ = σ x f

17 Drude-Boltzmann Conductivity as a Fermi Surface Property d = ds d = ds E E de ħv( ) j e v ( ) e v ( ) = = ds de ( ) ( E E ) = ds ( ) E v v ħ spherical Fermi surface: ds ( ) ( E ) v( E ) ( E ) m x x x E F E x 8π ħ ( ) 8π ħ E= E ( ) F vx ( ) 4π 4π E τ = Fτ F F = Fτ F ( ) E= E v σ τ δ τ F 4 BT E e π ħ F F e τ E F ( ) Fτ F * * π F = π n ( ) σ = E = n 8 ħ m m F *

18 Temperature Dependence of Drude-Boltzmann Conductivity Mattheiesen Rule: the phonon and defect scattering mechanism are independent = + ρ = ρ ρ phonon + τ τ τ phonon defect defect T T ρ = at + Debye ρ defect

19 Drift-Diffusion approximation to Boltzmann Equation for Semiconductors * The drift-diffusion equations are derived introducing the mobility µ = eτ / m and * replacing v = BT / m with its average equilibrium value, therefore neglecting thermal effects. The diffusion coefficient D = µ (Einstein's relation) is BT / e also introduced, and the resulting drift-diffusion current is: dn( x) n 1 jn = eµ nn( x) E( x) + ed = jn + U n n dx t e C ontin uity: dp( x) p 1 j p = eµ p p( x) E( x) ed = j p + U p dx t e + q( n p + N A N D ) Poisson: V = ε The choice of equilibrium (thermal) velocity means that the drift-diffusion equations are only valid for very small perturbations of the equilibrium state (low fields). The validity of the drift-diffusion equations is empirically extended by introducing field-dependent mobility µ ( E) and diffusion coefficient D( E), obtained from empirical models or detailed calculations. 0 p

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