Guest Lecture Stephen Hill University of Florida – Department of Physics

Guest Lecture Stephen HillGuest Lecture Stephen HillUniversity of Florida – Department of University of Florida – Department of

PhysicsPhysics

•Reminder about HO and cyclotron motion

•Schrodinger equation

•Wave functions and quantized energies

•Landau quantization

•Some consequences of Landau quantization in metals

Reading: Reading:

My web page: My web page: http://www.phys.ufl.edu/~hill/http://www.phys.ufl.edu/~hill/

Cyclotron motion and the Cyclotron motion and the Quantum Harmonic OscillatorQuantum Harmonic Oscillator

E

0 +AA x

V(x)

The harmonic oscillatorThe harmonic oscillator

2 21

2m x

2 2 2

212

1 1

2 2

when 0,

/

E K U

E mv m x

v

Ex A

m

k m

Classical turning points at x = ±A, when kinetic energy = 0, i.e. v = 0

x

y

B out of page

Cyclotron motion: classical resultsCyclotron motion: classical results

e, m

dpF q v B

dt

Lorentz force:

2

c

mv mvevB R

R eBv eB

R m

RF

x

y

B out of page

What does this have to do with today’s lecture?What does this have to do with today’s lecture?Cyclotron motion....Cyclotron motion....

e, m

ˆ

ˆ ˆ

ˆ ˆx y

x y

B Bk

v v i v j

p p i p j

Restrict the problem to 2D:

dpF q v B

dt

Lorentz force:

It looks just like the Harmonic OscillatorIt looks just like the Harmonic Oscillator

2 2 22 2 2 21 1

2 22 2x

x c

p e BE x mv m x

m m

2 22 21

222 c

dm x E

m dx

c

eB

m

E

0 +AA x

V(x)

The harmonic oscillatorThe harmonic oscillator

Classical turning points at x = ±A, when kinetic energy = 0, i.e. v = 0

2 22 21

222 c

dm x E

m dx

2

2 2122 c'' m x E

m

'' '' ''

20as ; 1x dx

Constraints:

Due to symmetry, one expects:Due to symmetry, one expects:

Thus, the solutions must be either Thus, the solutions must be either symmetric, symmetric, ((xx) = ) = xx), or ), or antisymmetric, antisymmetric, ((xx) = ) = xx).).

2 2( ) ( )x x

The quantum harmonic oscillator solutionsThe quantum harmonic oscillator solutions

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html

*,n m n m

OrthogonalityOrthogonality


Due to symmetry, one expects:Due to symmetry, one expects:

Thus, the solutions must be either Thus, the solutions must be either symmetric, symmetric, ((xx) = ) = xx), or ), or antisymmetric, antisymmetric, ((xx) = ) = xx).).

Further discussion regarding the Further discussion regarding the symmetry of symmetry of can be found in the can be found in the Exploring section on page 268 of Exploring section on page 268 of Tippler and Llewellyn. Tippler and Llewellyn.

2 2( ) ( )x x

The quantum harmonic oscillator solutionsThe quantum harmonic oscillator solutions



The correspondence principleThe correspondence principle

The harmonic oscillator wave functionsThe harmonic oscillator wave functions2 2

2 22

( ) 1( ) ( )

2 2

xm x x E x

m x

2 22 2 2 21 1( ) ; ''( ) 4 ( ) 4 ( )

2 2x xx Ae x A x e x

Solutions will have a form such that Solutions will have a form such that '' '' (A (Axx22 + B) + B) A function that A function that works is the Gaussian:works is the Gaussian:

For higher order solutions, things get a little more complicated.For higher order solutions, things get a little more complicated.

where where HHnn((xx) is a polynomial of order ) is a polynomial of order nn called a Hermite polynomial. called a Hermite polynomial.

2 / 2( ) ( )m xn n nx C e H x

The first three wave functionsThe first three wave functions2

2

2

/ 20 0

/ 21 1

2/ 2

2 2

( )

( )

2( ) 1

m x

m x

m x

x A e

mx A xe

m xx A e

This leads to a selection rule for electric dipole radiation emitted or This leads to a selection rule for electric dipole radiation emitted or absorbed by a harmonic oscillator. The selection rule is absorbed by a harmonic oscillator. The selection rule is nn = = ±1. Thus, ±1. Thus, a harmonic oscillator only ever emits or absorbs radiation at the a harmonic oscillator only ever emits or absorbs radiation at the classical oscillator frequency classical oscillator frequency cc = = eBeB//mm..

* *, ; also 0 unless 1n m m n n mdx x dx n m

A property of these wavefunctions is that:A property of these wavefunctions is that:

12

1,2,3....

n cE n

n

The quantum harmonic oscillatorThe quantum harmonic oscillator

12 c

32 c

52 c

72 c

92 c

Landau levels (after Lev Landau)

c

2 c

eBf

m 28 GHz/T

2 e

e

m

What happens if we have lots of electrons?What happens if we have lots of electrons?

Filled

EmptyEEFF

# states per LL= 2eB/h

Landau levels

What happens if we have lots of electrons?What happens if we have lots of electrons?

Filled

EmptyEEFF

Cyclotronresonance

Landau levels


c

cyclotron resonance

1

| cos |CRB

Electrons in an ‘effectively’ 2D metalElectrons in an ‘effectively’ 2D metal

Width of resonance a measure of scattering time (lifetime/uncertainty)

f = 62 GHzT = 1.5 K

But these are crystals – electrons experience lattice potentialBut these are crystals – electrons experience lattice potential

12 c

32 c

52 c

72 c

92 c

c

2 22 21

222 c

dm x E

m dx

crystal+ V ( )

x

2 c3 c

Anharmonic oscillatorAnharmonic oscillator•See harmonic resonancesSee harmonic resonances•wwcc depends on depends on EE

• nn no longer no longer strictly a good strictly a good quantum numberquantum number• no longer form no longer form

an orthogonal an orthogonal basis setbasis set

Electrons in an ‘effectively’ 2D metalElectrons in an ‘effectively’ 2D metalLook more carefully – harmonic resonances: measure of the lattice potential

Electrons in an ‘effectively’ 2D metalElectrons in an ‘effectively’ 2D metalEven stronger anharmonic effects

f = 54 GHzT = 1.5 K

Harmonic cyclotron frequenciesHarmonic cyclotron frequencies2

eBf

m Heavy masses: m = 9me

What if we vary the magnetic field?What if we vary the magnetic field?

Filled

EmptyEEFF

Landau levels

# LLs below EF

= mEF/eB # states per LL

= 2eB/h

EEFF

Filled

Empty

Landau levels



# LLs below EF

= mEF/eB

EEFF

Filled

Empty

Landau levels

eV

kBT ~ meV


Properties oscillate as LLs pop through Properties oscillate as LLs pop through EEFF


Properties oscillate as LLs pop through Properties oscillate as LLs pop through EEFF

Period 1/B

10 15 20 25 30

4.2 K 2 K 1.34 K

Con

duct

ivity

(ar

b. u

nits

- o

ffse

t)

Magnetic field (tesla)

Microwave surface impedance an for organic conductorMicrowave surface impedance an for organic conductor

52 GHz

Shubnikov-de Haas effect

Magnetoresistance for an for organic superconductorMagnetoresistance for an for organic superconductor

Shubnikov-de Haas effect

Guest Lecture Stephen HillGuest Lecture Stephen HillUniversity of Florida – Department of University of Florida – Department of

PhysicsPhysics

•Reminder about HO and cyclotron motion

•Schrodinger equation

•Wave functions and quantized energies

•Landau quantization

•Some consequences of Landau quantization in metals

Reading: Reading:

My web page: My web page: http://www.phys.ufl.edu/~hill/http://www.phys.ufl.edu/~hill/

Cyclotron motion and the Cyclotron motion and the Quantum Harmonic OscillatorQuantum Harmonic Oscillator

Documents

Guest Lecture Stephen Hill University of Florida – Department of Physics