e in magnetic - tu graz

Institute of Solid State PhysicsTechnische Universität Graz

Free Electrons in a Magnetic Field

Charged particle in a magnetic field

F ev B ma

2

zmvevB

R

x

y

kx

ky

2c

Rv RT

zc

eBm

Institute of Solid State PhysicsTechnische Universität Graz

Magnetron

zeB Rvm

solve for velocity

Why don't metals in a B field radiate?

x

y

kx

ky

zc

eBm

Magnetron

There are no lower lying states to fall into. We need a quantum description.

2 2

2 ( )2

d V x Em dx

Magnetic force depends on the velocity, not on the position.

Electrons in a magnetic field

Euler Lagrange equations

Lagrangian:

Kittel: Appendix Ghttp://lamp.tu-graz.ac.at/~hadley/ss2/IQHE/cpimf.php

F q E v B

0i i

d L Ldt x x

212 , ,L mv qV r t qv A r t

Lorentz force law

Lagrangian

,

x xx x

x

x x x x x

x x x x xx y z

dv dAd L d mv qA m qdt v dt dt dt

dv A A A Adx dy dzm qdt dt x dt y dt z t

dv A A A Am q v v vdt x y z t


yx zx y z

AA AL Vq q v v vx x x x x

yx x x xzy z

Adv A A AAVm q q v vdt x t x y x z

0i i

d L Ldt x x

kinetic momentum field momentum(inductance)

yx x x xzy z

Adv A A AAVm q q v vdt x t x y x z

Lagrangian


and AE V B At

xx x

dvm q E v Bdt

x x xx

Lp mv qAv

1x x xv p qA

m

conjugate variable:

Hamiltonian

Schrödinger equation

p i

Classical result


H v p L

Legendre transformation

212

H p qA qVm

212

i i qA qVt m

1x x xv p qA

m

Landau Levels

21 ( ) ( ).2

i qA r E rm

free particles in a magnetic field

ˆ.zA B xy

Landau gauge

V = 0

ˆ ˆ ˆ ˆ.y yx xz zz

dA dAdA dAdA dAB A B xy x y zdy dz dz dx dx dy

ˆ.zB B z

Landau Levels

21 ( ) ( ).2

i qA r E rm

free particles in a magnetic field

ˆ.zA B xy

Landau gauge

2 2 2 2 21 2 .2 z z

di qB x q B x Em dy

( ).y zik y ik ze e x

The solution has the form

V = 0

2

ˆ ˆz zi qA i qB xy i qB xy

ˆ ˆ ˆ ˆˆz z zd d d di qB xy i x y z qB xy i qB xdx dy dz dy

Landau Levels

2 2 2 2 21 2 .2 z z

di qB x q B x Em dy

( ).y zik y ik ze e x

The solution has the form

22 2 2 2 2 2 2 2

2

1 2 ( ) ( ).2 z y z y zk k qB k x q B x x E x

m x

Substitute this into the equation

The equation for (x) is

This is the equation for a harmonic oscillator

22 2 2 2 2 2 2 2

2

1 2 ( ) ( ).2 z y z y zk k qB k x q B x x E x

m x

2 2 222 2 2

2

1 ( ) ( ).2 2

y zz

z

k kq B x x E xm x qB m

2 2

202 ( ) ( ).

2 2K x x x E x

m x

Landau Levels

2

y zk qB x

This is the equation for a harmonic oscillator

2 22

22 2 202

1 ( ) ( ).2 2

zz

kq B x x x E xm x m

2 2

1, 22z

zk v c

kE vm

zc

qBm

Landau Levels

0y

z

kx

qB

2 2

2 2z

zc

q BKm

qBKm m

2 2

202 ( ) ( ).

2 2K x x x E x

m x

Charged particle in a magnetic field

x

y

Circular motion is harmonic motion. Harmonic motion is quantized.

2

2 212 2v c x yE v k k

m

2 R v

v labels the Landau level

kx

ky

Bohr - Sommerfeld quantization

Landau levels

The number of solutions is conserved

kx

ky

B = 0

kx

ky

B 0

22L

In 2-D, the k-volume per k state is:

0( ).yik ye x x 0y

z

kx

qB

Density of states 2D

12v cE v

The number of states between ring v-1 and ring v is

2 21

22v vk k

L

2 2

122

vc

k vm

2 2 1 11 2 2

2 21c cv v

m mk k v v

2

2cm L

kx

kykv

The number of states between ring v-1 and ring v is

The density of states per spin is2

cm

Spin

In a magnetic field, there is a shift of the energy of the electrons because of their spin.

Bohr magneton

2cm

2 BgE B B

D(E)

E0

D(E)

E0

2

m

c

2Be

em

1 12 4 2 4

0

( )2

g gcc c

v

mD E E v E v

B = 0 B 0

g-factor 2g 2c BeB Bm

Energy spectral density 2d

At T = 0

unfilled Landau level

analog to the Planck radiation law

Fermi energy 2d

Periodic in 1/BWhen there is only one Landau level, the Fermi energy rises linearly with field.

2 2c

FeBEm

Large field limit

( )FE

n D E dE

Internal energy 2d

At T = 0

2 2c eBu n n

m

Large field limit

( )FE

u ED E dE

Magnetization 2d

At T = 0

2du em ndB m

Large field limit

de Haas - van Alphen oscillations

Shubnikov-De Haas oscillations

Scattering at the Fermi surface

At room temperature, phonon energies are much less than the Fermi energy. The energy of electrons hardly changes as they scatter from phonons. Electrons scatter from a point close to the Fermi surface to another point close to the Fermi surface.

Changing the magnetic field changes the number of states at the Fermi energy.

There are oscillations in the electrical conductivity as a function of magnetic field.

e in magnetic - tu graz

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