1 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 is important  Form of Fermi surface is important  Fermi surface can be complicated due to overlapping bands.

2 Constructing Brillouin Zones 1 st B. Z. 2 nd B. Z. 2D Square lattice. BZ constructed from the perpendicular bisectors of the vectors joining a reciprocal lattice point to neighbouring lattice points 2  /a

3 Brillouin Zones and Fermi Surfaces Empty Lattice model (limit of weak lattice potential): States are Bloch states.Independent states have k-vectors in first BZ. No energy gaps at the BZ boundaries. kxkx  a  a E 0 E1E1 E2E2 [100] k  2 1/2  a 0 2 1/2  a k x = k y E1E1 E2E2 [110] 2 st B. Z. 1 st B. Z. E1E1 E2E2 kyky

4 Reduced Zone scheme PLUS Fermi Contours in reduced Zone Parts of Fermi circle moved into 1 st BZ from 2 nd BZ E2E2 Extended Zone scheme 2 st B. Z. 1 st B. Z.

5 Fermi Contours in periodic Zone E2E2 2 st B. Z. 1 st B. Z.

6 E = -  –  Cos[k x x] - Cos[k y y]), 2D simple square Lattice tight binding model. Changing Fermi Contour with Increasing Fermi Energy.

7 BZs and Fermi Surfaces with gaps Energy gaps make the Fermi contours appear discontinuous at the BZ boundaries. dE/dk = 0 at BZ boundaries. Fermi contour perpendicular to BZ boundary. 0  a E1E1 E2E2 kxkx E2E2 E1E1 2 st B. Z. 1 st B. Z.

8 E2E2 E1E1 Energy gaps: Fermi contours appear discontinuous at the BZ boundaries. dE/dk = 0 at BZ boundaries. Fermi contour perpendicular to BZ boundary. BZs and Fermi Surfaces with gaps 2 st B. Z. 1 st B. Z. E1E1 E2E2 kyky No gapsWith gaps

9 Fermi Surfaces with gaps “Hole like” orbits Periodic zone picture of part of the Fermi contour at energy E 1. On this part of the Fermi contour electrons behave like positively charged “holes”. See later

10 Fermi Surfaces with gaps: “Electron like” orbits Periodic zone picture of part of the Fermi contour at energy E 2. On this part of the Fermi contour electrons behave like negatively charged “electrons”. See later

11 Tight binding simple cubic model:Fermi Surfaces -  –  Cos[k x x] - Cos[k y y] - Cos[k z z] Increasing Fermi Energy

12 du/fermisurface/http SodiumCopper Strontium

13 Lead

14 Palladium

15 Tungsten

16 Y Yttrium

17 Thorium

18 Re Rhenium

19 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

20 Dynamics of free quantum electrons Classical free electrons F = -e (E + v  B) = dp/dt and p =m e v. Quantum free electrons the eigenfunctions are ψ(r) = V -1/2 exp[i(k.r-  t) ] 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 /dt(Ehrenfest’s Theorem) for E =  2 k 2 /2m e Free quantum electrons have free electron dynamics

21 Dynamics of free Bloch electrons 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 and appendix E). In the presence of the lattice potential the electrons have well defined velocities.

22 Response to external forces Consider an electron moving in 1D with velocity v x acted on by a force by a force F x for a time interval  t. The work,  E, 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  k and so the! Because the electron is subject to forces from the crystal lattice as well as external forces

23 Electron effective mass 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 F x differentiating Givesand Sowhere An electron in a band behaves as if it has an effective mass m *. Note magnitude of m * can depend on direction of force

24 Dynamics of band electrons Consider, for example, a 1D tight-binding model: E(k x ) =  2  cos(k x a) In a filled band the sum over all the v g 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 v x. For electrons in states near the bottom of the band a force in the positive x-direction increases k but decreases v x.

25 Effective Mass Consider, for example, a 1D tight-binding model: E(k) =  2  cos(ka) 0 0 Near the bottom of the band i.e. |k|<<  /a cos(ka) ~ 1 So m* ~  2 /2a 2  As before. For a = 2 x and  = 4 eV m* =0.24 x m e 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

26 Bloch Oscillations Consider an electron at k = 0 at t = 0 When the electron reaches k =  /a it is Bragg reflected to k = -  /a. It them moves from -  /a to  /a again. Period of motion Consider a conductor subject to an electric –E x Expect “Bloch oscillations” in the current current of period T Note observed due to scattering since T >>  p k(t)  /a  /a

27 Conductivity (i)  p momentum 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,  (iv) Filled band have zero conductivity Conductivity is now given by  ne 2  p /m *

28 Motion in a magnetic field 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. xkxkx ykyky

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

30 Periodic zone picture of Fermi contour ( E 1 ) near bottom of a band. Electron like orbits Grad E kxkx  a  a E 0 E1E1 E1E1

31 Periodic zone picture of the Fermi contour at the top of a band Hole like orbits Grad E kxkx  a  a E 0 E2E2 E2E2

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

33 Energy & k-vector of a hole Vacancy Energy k Hole keke khkh EeEe EhEh E = 0 Choose E = 0 to be at the top of the band. If we remove one electron from a state of energy –E e the total energy of the band is increased by E h = -E e This is the energy of the hole and it is positive. A full band has If one electron, of k-vector k e, is missing the total wavevector of the band is –k e. A hole has k-vector k h = -k e

34 Charge of a hole In an electric field the electron wavevector would respond as since k h = -k e So the hole behaves as a positively charged particle. 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. v h = v e

35 Effective mass of a hole 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.