Lecture IX dr hab. Ewa popko Electron dynamics

1. Lecture IXdr hab. Ewa popkoElectron dynamics • Band structure calculations give E(k) • E(k) determines the dynamics of the electrons

2. Semi-classical model of electron dynamics • E(k), 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 example, 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

3. Classical free electrons F = -e (E + v  B) = dp/dt and p =mev. Quantum free electrons the eigenfunctions are ψ(r) = V-1/2 exp[i(k.r-wt) ] The wavefunction extends throughout the conductor. Can construct localise wavefunction i.e. a wave packets The velocity of the wave packet is the group velocity of the waves The expectation value of the momentum of the wave packet responds to a force according to F = d<p>/dt (Ehrenfest’s Theorem) Dynamics of free quantum electrons for E = 2k2/2me Free quantum electrons have free electron dynamics

4. Allowed wavefunctions are The wavefunctions extend throughout the conductor. Can construct localise wavefunctions The electron velocity is the group velocity in 3D This can be proved from the general form of the Bloch functions (Kittel p205 ). In the presence of the lattice potential the electrons have well defined velocities. Dynamics of free Bloch electrons

5. Consider an electron moving in 1D with velocity vx acted on by a force Fx for a time interval dt. The work, dE, done on the electron is and so In 3D the presence of electric and magnetic fields since Note: Momentum of an electron in a Bloch state is not hk and so the ! Because the electron is subject to forces from the crystal lattice as well as external forces Response to external forces

6. In considering the response of electrons in a band to external forces it is useful to introduce an effective electron mass, m*. Consider an electron in a band subject to an external force Fx differentiating Gives and So where An electron in a band behaves as if it has an effective mass m*. Note magnitude of m* can depend on direction of force Electron effective mass

7. 0 0 0 0 0 Dynamics of band electrons Consider, for example, a 1D tight-binding model: E(kx) = -a- 2gcos(kxa) In a filled band the sum over all the vg values equals zero. A filled band can carry no current For electrons in states near the bottom of the band a force in the positive x-direction increases k and increases vx . For electrons in states near the bottom of the band a force in the positive x-direction increases k but decreases vx .

8. 0 0 Effective Mass Consider, for example, a 1D tight-binding model: E(k) = -a- 2gcos(ka) Near the bottom of the band i.e. |k|<<p/a cos(ka) ~ 1 So m* ~ h2/2a2g States near the top of the band have negative effective masses. Equivalently we can consider the mass to be positive and the electron charge to be positive As before. For a = 2 x 10-10 andg = 4 eV m* =0.24 x me

9. -p/a p/a k(t) 0 0 0 0 0 -p/a Bloch Oscillations Consider a conductor subject to an electric –Ex Consider an electron at k = 0 at t = 0 When the electron reaches k = p/a it is Bragg reflected to k = -p/a. It them moves from -p/a to p/a again. Period of motion Expect “Bloch oscillations” in the current current of period T Not observed due to scattering since T >> tp

10. (i)  tpmomentum relaxation time at the Fermi surface as before (ii)   m is replaced by m* at the Fermi surface (iii)  Each part filled band contributes independently to conductivity,s (iv) Filled band have zero conductivity Conductivity • Conductivity is now given by • s =ne2 tp/m*

11. Free electrons The electrons move in circles in real space and in k-space. Bloch electrons In both cases the Lorentz force does not change the energy of the electrons. The electrons move on contours of constant E. y ky x kx Motion in a magnetic field

12. Electron and Hole orbits Filled states are indicated in grey. • Electron like orbit centred on k = 0. Electrons move anti-clockwise. • Hole like orbit. Electrons move clockwise as if they have positive charge

13. Grad E E E1 -p/a kx p/a 0 Electron like orbits Periodic zone picture of Fermi contour ( E1 ) near bottom of a band. E1

14. Grad E Hole like orbits Periodic zone picture of the Fermi contour at the top of a band E2 E E2 -p/a kx p/a 0

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

16. A full band has If one electron, of k-vector ke, is missing the total wavevector of the band is –ke. A hole has k-vector kh = -ke Energy kh Eh Hole k E = 0 ke Ee Vacancy Energy & k-vector of a hole Choose E = 0 to be at the top of the band. If we remove one electron from a state of energy –Ee the total energy of the band is increased by Eh = -Ee This is the energy of the hole and it is positive.

17. In an electric field the electron wavevector would respond as since kh = -ke So the hole behaves as a positively charged particle. Charge of a hole The group velocity of the missing electron is . The sign of both the energy and the wave vector of the hole is the opposite of that of the missing electron. Therefore the hole has the same velocity as the missing electron. vh = ve

18. The effective mass is given by 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. The electron mass near the top of the band is usually negative so the hole mass is usually positive. Holes - positive charge and usually positive mass. Can measure effective masses by cyclotron resonance. Effective mass of a hole