Fermi surfaces and Electron

Transcription

1 Solid State Theory Physics 545 Fermi Surfaces

2 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.

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

4 The Fermi Surface Metals have a Fermi energy, E F. The Fermi Temperature,T F, is the temperature at which k B T F = E F. All the free electron states within a Fermi sphere in k-space are filled up to a Fermi wavevector,k F. The surface of this sphere is called the Fermi surface. On the Fermi surface the free electrons have a Fermi velocity v F =hk F /m e. A Fermi surface still exists when the states are not free A Fermi surface still exists when the states are not free electron states but it need not be a sphere.

5 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. E E 2 E 1 π/a 0 k y [100] k x π/a E 1 E 2 1 k x = k y E 2 E 1 1 st BZ B. Z. 2 st B. Z. 2 1/2 π/a 0 k 2 1/2 π/a [110]

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

7 Fermi Contours in periodic Zone E 2 1 st BZ B. Z. 2 st B. Z.

8 E = -α γ( Cos[k x x] - Cos[k y y]), 2D simple square Lattice tight binding model. Changing Fermi Contour with Increasing Fermi Energy.

9 BZs and Fermi Surfaces with gaps E 2 E 1 E2 E 1 π/a 0 π/a k x 1 st BZ B. Z. 2 st B. Z. Energy gaps make the Fermi contours appear discontinuous at the BZ boundaries. de/dk = 0 at BZ boundaries. Fermi contour perpendicular to BZ boundary.

10 BZs and Fermi Surfaces with gaps No gaps k y With gaps E 2 E 2 E 1 E 1 1 st BZ B. Z. 2 st B. Z. Energy gaps: Fermi contours appear discontinuous at the BZ boundaries. de/dk = 0 at BZ boundaries. Fermi contour perpendicular to BZ boundary.

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

12 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

13 Motion in a magnetic field Free electrons F = ev B = ( e / m) k B The electrons move in circles in real space and in k-space. Bloch electrons dk dt e e = v B = k E( k) B 2 In both cases the Lorentz force does not change the energy of the electrons. The electrons move on contours of constant E. y k k y x k x

14 Electron and Hole orbits dk dt e = E(k) B Filled states are indicated in grey. 2 k k y de dk de dk dt dk k y B z B z dk dt (a) (b) k x k x (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

15 Electron like orbits Periodic zone picture of Fermi contour ( E 1 ) near bottom of a band. E 1 E Grad E / 0 k kx π/a E 1 π/a 0

16 Hole like orbits Periodic zone picture of the Fermi contour at the top of a band Grad E E 2 E E 2 π/a 0 k x π/a

17 Tight binding simple cubic model:fermi Surfaces -α γ(cos[k x x] - Cos[k y y] - Cos[k z z] Increasing Fermi Energy / l /f i i l h l

18 The Fermi Surface Metals have a Fermi energy, E F. The Fermi Temperature,T F, is the temperature at which k B T F = E F. All the free electron states within a Fermi sphere in k-space are filled up to a Fermi wavevector,k F. The surface of this sphere is called the Fermi surface. On the Fermi surface the free electrons have a Fermi velocity v F =hk F /m e. A Fermi surface still exists when the states are not free A Fermi surface still exists when the states are not free electron states but it need not be a sphere.

19 Sodium Copper d/f du/fermisurface/http f /h Strontium

20 Lead

21 Palladium

22 Tungsten

23 Yttrium Y

24 Thorium

25 Re Rhenium