The semiclassical semiclassical model model of Electron dynamics

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Meredith Strickland
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1 Solid State Theory Physics 545 The semiclassical model of The semiclassical model of lectron dynamics

2 Fermi surfaces and lectron dynamics Band structure calculations give () () determines the dynamics of the electrons It is () at the Fermi Surface that is important Form of Fermi surface is important Fermi surface can be complicated due to overlapping bands.

3 Semi-classical model of electron dynamics (), which is obtained from quantum mechanical band structure calculations, determines the electron dynamics It is possible to move between bands but this requires a discontinuous change in the electron s energy that can be supplied, for eample, by the absorption of a photon. In the following we will not consider such processes and will only consider the behaviour of an electron within a particular band. The wavefunctions are eigenfunctions of the lattice potential. The lattice potential does not lead to scattering but does determine the dynamics. Scattering due to defects in and distortions of the lattice

4 Dynamics of free quantum electrons Classical free electrons F -e ( + v B) dp/ and p m e v. Quantum free electrons the eigenfunctions are ψ(r) V -1/ ep[i(.r-ωt) ] The wavefunction etends throughout the conductor. Can construct localise wavefunction i.e. a wave pacets Ψ(r) Aep[i(.r -ωt )] The velocity of the wave pacet is the group velocity of the waves d 1 d v ω p for /m e v m e me The epectation value of the momentum of the wave pacet responds to a force according to F d<p>/ (hrenfest s Theorem) Free quantum electrons have free electron dynamics

5 Dynamics of free Bloch electrons Allowed wavefunctions are ψ ( r ). r e i u ( r ) The wavefunctions etend throughout the conductor. Can construct localise wavefunctions Ψ(r) A ( ) The electron velocity is the group velocity d 1 d v ω 1 in 3D v ( ) ψ r This can be proved from the general form of the Bloch functions (Kittel p05 and appendi ). In the presence of the lattice potential the electrons have well defined velocities.

6 Response to eternal forces Consider an electron moving in 1D with velocity v acted on by a force by a force F for a time interval δt. The wor, δ, done on the electron is and δ F v t δ 1 d v so d F d δ δt F In 3D the presence of electric and magnetic fields F e e 1 since ( + v B) ( + ( ) B) Note: Momentum of an electron in a Bloch state is not v 1 ( ) and so the dp! F Because the electron is subject to forces from the crystal lattice as well as eternal forces

7 lectron effective mass In considering the response of electrons in a band to eternal forces it is useful to introduce an effective electron mass, m *. Consider an electron in a band subject to an eternal force F differentiating Gives dv 1 v d 1 d 1 d and d F So F m * dv where m * d 1 An electron in a band behaves as if it has an effective mass m *. Note magnitude of m * can depend on direction of force

8 Dynamics of band electrons Consider, for eample, a 1D tight-binding model: ( ) α γcos( a) v g 1 d aγ sin In a filled band the sum over all the v g values equals zero. A filled band can carry no current d e F a v g 0 0 π/a 0 π/a For electrons in states near the bottom of the band a force in the positive -direction increases and increases v. For electrons in states near the bottom of the band a force in the positive -direction increases but decreases v.

9 ffective Mass Consider, for eample, a 1D tight-binding model: () α γcos(a) v * g m 1 * d 1 m a γ cos a 0 π/a π/a 0 π/a Near the bottom of the band i.e. <<π/a cos(a) ~ 1 So m* ~ /a γ States near the top of the band have negative effective masses. F * dv e( + v B) m quivalently we can consider the mass to be positive and the electron charge to be positive As before. For a and γ 4 ev m* 0.4 m e

10 Bloch Oscillations Consider a conductor subject to an electric e F Consider an electron at 0 at t 0 e ( t) t π/a π/a 0 F1 π/a When the electron reaches π/a it is Bragg reflected to -π/a. It 0 them moves from -π/a to π/a again. Period of motion T e π a pect Bloch oscillations in the current current of period T (t) v Note observed due to scattering since T >> τ p π/a 0 t/t 1

11 Conductivity Conductivity is now given by σ ne τ p/m * (i) τ p momentum relaation time at the Fermi surface as before (ii) m is replaced by m * at the Fermi surface (iii) ach part filled band contributes independently to conductivity, σ (iv) Filled band have zero conductivity

12 Motion in a magnetic field Free electrons F ev B ( e / m) B The electrons move in circles in real space and in -space. Bloch electrons e e v B ( ) B In both cases the Lorentz force does not change the energy of the electrons. The electrons move on contours of constant. y y

13 lectron and Hole orbits e () B Filled states are indicated in grey. y d d y B z B z (a) (b) (a) lectron lie orbit centred on 0. lectrons move anti-clocwise. (b) Hole lie orbit. lectrons move clocwise as if they have positive charge

14 lectron lie orbits Periodic zone picture of Fermi contour ( 1 ) near bottom of a band. 1 Grad / 0 π/a 1 π/a 0

15 Hole lie orbits Periodic zone picture of the Fermi contour at the top of a band Grad π/a 0 π/a

16 Holes Can consider the dynamical properties of a band in terms of the filled electron states or in terms of the empty hole states F nergy d Force on lectrons Consider an empty state (vacancy) in a band moving due to a force. The electrons and vacancy move in the same direction.

17 nergy egy& -vector of a hole Choose 0 to be at the top of the band. If we remove one electron from a state of energy e the total energy of the band is nergy increased by h - e Hole h h This is the energy of the hole and it is positive. 0 Af full llband dhas 0 e e Vacancy If one electron, of -vector e, is missing the total wavevector of the band is e. A hole has -vector h - e

18 Charge of a hole In an electric field the electron wavevector would respond as since h - e d e d h e So the hole behaves as a positively charged particle. + e The group velocity of the missing electron is. v ( ) The sign of both the energy and the wave vector of the hole is the opposite of that of the missing electron. F F 1 Therefore the hole has the same velocity as the missing electron. v h v e

19 ffective mass of a hole The effective mass is given by m * d 1 Since the sign of both the energy and the wave vector of the hole is the opposite of that of the missing electron the sign of the effective mass is also opposite. m * h m The electron mass near the top of the band is usually negative so the hole mass is usually positive. * e Holes positive charge and usually positive mass. Can measure effective masses by cyclotron resonance.

Fermi surfaces and Electron

Solid State Theory Physics 545 Fermi Surfaces Fermi surfaces and Electron dynamics Band structure calculations give E(k) E(k) determines the dynamics of the electrons It is E(k) at the Fermi Surface that