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

1st B. Z.

2ndB. 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 SurfacesEmpty 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.

kx aa

E

0

E1

E2

[100]

k21/2a 0 21/2a

kx = ky

E1

E2

[110]2st B. Z. 1st B. Z.

E1

E2

ky

4

Reduced Zone scheme

PLUS

Fermi Contours in reduced Zone

Parts of Fermi circle moved into 1st BZ

from 2nd BZ

E2

Extended Zone scheme

2st B. Z. 1st B. Z.

5

Fermi Contours in periodic Zone

E2

2st B. Z. 1st B. Z.

6http://dept.physics.upenn.edu/~mele/phys518/anims/Kronig/FermiSurf1.gif

E = - – Cos[kx x] - Cos[ky 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.

0a a

E1

E2

kx

E2

E1

2st B. Z. 1st B. Z.

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E2

E1

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

2st B. Z. 1st B. Z.

E1

E2

ky

No gaps With gaps

9

Fermi Surfaces with gaps “Hole like” orbits

Periodic zone picture of part of the Fermi contour at energy E1.

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 E2.

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

11

Tight binding simple cubic model:Fermi Surfaces

- – Cos[kx x] - Cos[ky y] - Cos[kz z]

Increasing Fermi Energy

http://home.cc.umanitoba.ca/~loly/fermiarticle.html

12

http://www.phys.ufl.edu/fermisurface/http

Sodium Copper

Strontium

13

Lead

14

Palladium

15

Tungsten

16

YYttrium

17

Thorium

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ReRhenium

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 =mev .

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<p>/dt (Ehrenfest’s Theorem)

for E = 2k2/2me

)]t -exp[i(k.rA (r)k k

kkv

d

dE

d

d

1

ee mm

pkv

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.

)(A (r)k k rk

kkv

d

dE

d

d

1

)()( . rr rk ueik

)(1

kv kE

22

Response to external forcesConsider an electron moving in 1D with velocity vx acted on by a force by a force Fx 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

tvFE xx

dt

d xx

kF

))(1

()( BkEBvEFk

k Eee

dt

d

xx dk

dE

1

v

)(1

kv kE

tdk

dE

dk

dE

x

xx

x

F

k

dt

dpF

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Electron effective massIn 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

dt

dk

dk

Ed1

dtdk

Ed1

dt

dv x2x

2

x

2x

xx

dt

d Fk

1

2x

22*

dk

Edm

xx dk

dE

1

v

dt

dvx*mx F

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Dynamics of band electronsConsider, for example, a 1D tight-binding model: E(kx) = 2cos(kxa)

0 0

0

x

xx e

dt

d FE

k

aka

dk

dEx

x

sin21

vg

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 .

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Effective MassConsider, for example, a 1D tight-binding model: E(k) = 2cos(ka)

0

0kacos

1

a2dk

Edm

2

21

2x

22*

Near the bottom of the band i.e. |k|<</a cos(ka) ~ 1 So m* ~ 2/2a2

As before.

For a = 2 x 10-10 and = 4 eV

m* =0.24 x me

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

dt

dv)( x*mex BvEF

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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

x

xx F

Ee

dt

kd

te

t xx Ek

)(

Consider a conductor subject to an electric –Ex

Expect “Bloch oscillations” in the current current of period T

Note observed due to scattering since T >> p

aeT

x

2

E

0 1 2

0

v(t

)

t/T

F1

k(t)

/a

/a

/a

0

0

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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

ne2p/m*

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Motion in a magnetic fieldFree 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.

BkBvF )/( mee

x kx

y ky

BkBvk

k )(2

Eee

dt

d

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Electron and Hole orbits

dk

dE

dt

dkdk

dEdt

dk

(a) (b)

Bz

kxkx

kyky dk

dE

dt

dkdk

dEdt

dk

dk

dE

dt

dk

dk

dE

dt

dkdk

dEdt

dk

dk

dEdt

dk

(a) (b)

BzBz

kxkx

kyky

(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

B)k(Ee

dt

kdk2

Filled states are indicated in grey.

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

Electron like orbits

Grad E

kx aa

E

0

E1

E1

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

Hole like orbits

Grad E

kx aa

E

0

E2

E2

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HolesCan 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 Electronsdt

d xx

kF

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Energy & k-vector of a hole

Vacancy

Energy

k

Hole

ke

kh

Ee

Eh

E = 0

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.

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

0 k

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Charge of a holeIn an electric field the electron wavevector would respond as

 

 

since kh = -ke

 

So the hole behaves as a positively charged particle.

F

Ek

e

dt

d h

F

Ek

e

dt

d e

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

)(1

kv kE

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.

1

2x

22*

dk

Edm

*e

*h mm