Solid State TheorySolid State TheoryPhysics 545Physics 545

Fermi SurfacesFermi Surfaces

F i f d El tFermi surfaces and Electron dynamicsdynamics

Band structure calculations give E(k)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 importantForm of Fermi surface is important

Fermi surface can be complicated due to overlapping bandsoverlapping bands.

Constructing Brillouin Zones2D Square lattice. BZ constructed from the perpendicular bisectors of the vectors joining a reciprocal lattice point t i hb i l tti i tto neighbouring lattice points 2π/a

1st B. Z.

2ndB. Z.

The Fermi Metals have a Fermi energy E

SurfaceMetals have a Fermi energy, EF.

The Fermi Temperature,TF, is thetemperature at which kBTF = EFtemperature at which kBTF = EF.

All the free electron states withina Fermi sphere in k-space area Fermi sphere in k space arefilled up to a Fermi wavevector,kF.

The surface of this sphere isThe surface of this sphere iscalled the Fermi surface.

On the Fermi surface the freeelectrons have a Fermi velocity vF= hkF/me.

A Fermi surface still exists when the states are not freeA Fermi surface still exists when the states are not free electron states but it need not be a sphere.

Brillouin Zones and Fermi SurfacesEmpty Lattice model (limit of weakEmpty Lattice model (limit of weak lattice potential):

States are Bloch states.IndependentE E2

States are Bloch states.Independent states have k-vectors in first BZ.

No energy gaps at the BZ boundaries.

E1

kx π/a−π/a 0ky[100]

kx = kyE1E2

y

E21

E1

1st B Z k−21/2π/a 0 21/2π/a[110]2st B. Z.

1st B. Z.

Fermi Contours in reduced Zone

E2

PLUSPLUS

Parts of Fermi circle moved into 1st BZ1st B. Z. moved into 1st BZ

from 2nd BZ2st B. Z.. .

Reduced Zone schemeExtended Zone scheme

Fermi Contours i i di Zin periodic Zone

E2

1st B Z2st B. Z.1st B. Z.

E = -α –γ( Cos[kx x] - Cos[ky y]),

2D simple square

Lattice tight bindingLattice tight binding

model.

Changing FermiChanging Fermi

Contour with

I i F iIncreasing Fermi

Energy.

http://dept.physics.upenn.edu/~mele/phys518/anims/Kronig/FermiSurf1.gif

BZs and Fermi Surfaces with gaps

E2

E1

E2

/ /E1

Energy gaps make the Fermi contours

0−π/a −π/a kx

1st B Z Energy gaps make the Fermi contours appear discontinuous at the BZ boundaries.

dE/dk = 0 at BZ boundaries. Fermi contour 2st B. Z.1st B. Z.

perpendicular to BZ boundary.

BZs and Fermi Surfaces with gaps

EkyNo gaps With gaps

E2

E1E1

E2E1

1st B Z2st B. Z.1st B. Z.

Energy gaps: Fermi contours appear discontinuous at the BZ boundaries.

dE/dk = 0 at BZ boundaries. Fermi contour perpendicular to BZ boundary.

Fermi Surfaces with gaps “HoleFermi Surfaces with gaps “Hole like” orbitslike orbits

Periodic zone picture of part of the Fermi contour at energy E1.1On this part of the Fermi contour electrons behave lik iti l h dlike positively charged “holes”. See later

Fermi Surfaces with gaps:Fermi Surfaces with gaps:“Electron like” orbits

Periodic zone picture ofPeriodic zone picture of part of the Fermi contour at energy E2.

On this part of the Fermi contour electrons behave like negatively chargedlike negatively charged “electrons”. See later

Motion in a magnetic fieldMotion in a magnetic fieldFree electrons BkBvF ×−=×−= )/( mee

The electrons move in circles in real space and in k-space.

Bloch electrons BkBvk k ×∇−=×−= )(2 Eee

dtd

In both cases the Lorentz force does not change the energy of the electrons. The electrons move on contours of constant E.

y ky ky

x kx

Electron and Hole orbitsdkdkdkdk

B)k(Eedtkd

k2 ×∇−=Filled states are indicated in grey.

dEdkdkdE

dtdEdk

dkdE

dtdEdk dEdk

dkdE

dt

dkdE

dt

dkdt dk

Bz

kyky dkdt

dkdkdt dkdt dkdk

BzBz

kyky

(a) (b)(a) (b)

kxkx kxkx

(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

Electron like orbits

Periodic zone picture of Fermi contour ( E ) nearcontour ( E1 ) near bottom of a band.

E1

Grad E

E

k // 0

E1 kx π/a−π/a 0

Hole like orbits

Periodic zone picture of the Fermi contour atFermi contour at the top of a band

Grad EGrad E

E

E2

E

E2

k /π/a 0 kx π/a−π/a 0

Tight binding simple cubic d l F i S fmodel:Fermi Surfaces

-α – γ(Cos[kx x] - Cos[ky y] - Cos[kz z]

Increasing Fermi Energy

h //h i b / l l /f i i l h lhttp://home.cc.umanitoba.ca/~loly/fermiarticle.html

The Fermi Metals have a Fermi energy E

SurfaceMetals have a Fermi energy, EF.

The Fermi Temperature,TF, is thetemperature at which kBTF = EFtemperature at which kBTF = EF.

All the free electron states withina Fermi sphere in k-space area Fermi sphere in k space arefilled up to a Fermi wavevector,kF.

The surface of this sphere isThe surface of this sphere iscalled the Fermi surface.

On the Fermi surface the freeelectrons have a Fermi velocity vF= hkF/me.

A Fermi surface still exists when the states are not freeA Fermi surface still exists when the states are not free electron states but it need not be a sphere.

Sodium Copper

http://www.phys.ufl.ed /f i f /hdu/fermisurface/http

Strontium

Lead

Palladium

Tungsteng

YttriumY

ThoriumThorium

RheniumRe