Quantum Transport in Solids

• Waves and particles in quantum mech.

• Quantization in atoms

• Insulators : Energy gap

• Emergent particles in a solid

• Landau quantization in a magnetic fieldand the quantum Hall effect

Light is a WaveHallmark of a wave : Interference

constructive

destructive

Wave Particle Duality

A photon is a particle too

Electrons are Waves

Energy and Momentum

Max Planck 1858-1947

Louis de Broglie 1892-1987

Planck : Energy ~ Frequency

E hf ω= =

hp kλ

= =

de Broglie: Momentum ~ (Wavelength)-1

34/ 2 1.05 10h Jsπ −= = ×

2 fω π=

2 /k π λ=

Dispersion RelationRelation between

• Frequency and Wavelength• Energy and Momentum

For a photon:

cf ck E cpωλ

= ⇒ = ⇒ =

2 2 221 v

2 2 2p kE mm m

ω= = ⇒ =

For an electron in vacuum:

p

E

p

E

QuantizationWaves in a confined geometry have discrete modes

1v

2f

L=

2v2

2f

L=

3v3

2f

L=

Quantization of circular orbits

Circular orbits have quantizedangular momentum.

2n rλ π=2 np

rπλ

= =

L rp n= =

Bohr Model

Niels Bohr 1885-1962

vL m r n= =2 2

2

ve mF kr r

= =

2/nE Ry n= −2

0nr n a=

2 4

2 13.6 eV2

mk eRy = =

2

0 2 .5amke

= = Å

• Two equations :

• Bohr radius :

• Rydberg energy:

• Solve for rn and En(rn,vn)

Explains line spectra of atoms

Flatland ... (1980’s)Semiconductor Heterostructures : “Top down technology”→ Two dimensional electron gas (2DEG)

Ga As

AlxGa1-x As

2DEGz

V(z)

∆E ~ 20 mV ~ 250°K

2D subband

conduction bandenergy

Fabricated with atomic precision using MBE. 1980’s - 2000’s : advances in ultra high mobility samples

z

Pauli Exclusion Principle

Wolfgang Pauli 1900-1958

Enrico Fermi 1901-1954

Electrons are “Fermions”:

Each quantum state canaccommodate at most one electron

n=1

n=2

n=3

n=4

n=1

n=2

n=3

n=4

Ground state Excited State

Insulator

EnergyGap

An insulator is inert because a finite energy is required to unbind an electron.

A semiconductor is an insulator with a smallenergy gap.

Add electrons to a semiconductor

EnergyGap

Accomplished by either

• Chemical doping • Electrostatic doping (as in MOSFET)

Added electrons are mobile.

Add electrons to a semiconductor

EnergyGap

Accomplished by either

• Chemical doping • Electrostatic doping (as in MOSFET)

Added electrons are mobile.

Add electrons to a semiconductor

EnergyGap

Accomplished by either

• Chemical doping • Electrostatic doping (as in MOSFET)

Added electrons are mobile.

Added electrons form a band

p

E2

2 *pEm

=

m* = “Effective mass”

Electrons in the “conduction band” behave just likeordinary electrons with charge e, but with a renormalized mass.

“Emergent quasiparticles at low energy”

“Holes” in the valence band

EnergyGap

Added holes are mobile

“Holes” in the valence band

EnergyGap

Added holes are mobile

“Holes” in the valence band

EnergyGap

Added holes are mobile

Holes are particles too

p

E2

2 *h

pEm

=

mh* = Hole Effective mass

Holes in the “valence band” behave just likeordinary particles with charge + e, with a mass mh* .

The sign of the charge of the carriers can be measured with the Hall effect.

Band Structure of a Semiconductor

p

E

Valence Band: Occupied states

Conduction Band: Empty states

electrons

holes

Gap

The Quantized Hall Effect

Von Klitzing,1981 (Nobel 1985)

Hall effect in 2DEG MOSFET at large magnetic field

• Quantization:

• Quantum Resistance:

• Explained by quantum mechanics of electrons in a magnetic field

/xy QR Nρ =2/ 25.812 807 kQR h e= = Ω

N = integer accurate to 10-9 !

Quantization in a magnetic fieldCyclotron Motion:1D Quantization argument:

vL m r n= =2vv mF e B

r= =

ceBm

ω =

nnreB

=2 2n ceB nE nm

ω= =

Cyclotron frequency

Magnetic flux enclosed by orbit

Solve for rn and En(rn,vn)

BF

v

e

vF e B= − ×

202n

n nr Beππ φ= = Magnetic Flux

quantum 0

15 24.1 10

he

Tm

φ

−

=

= ×

Landau Levels

Lev Landau 1908-19681( )2n cE nω= +

Landau solved the 2D SchrodingerEquation for free particles in a magnetic field

Closely related to the harmonic oscillator

/ cE ω

Each Landau level has one stateper flux quantum

total flux Area# states = flux quantum ( / )

Bh e×=

Quantized Hall Effect

yxy

x

E BJ ne

ρ = =

/ cE ω N filled Landau levels:# particles/area:

Gap

0

# flux quantaarea

n N

B eBN Nhφ

= ×

= =

From last time:

2

1xy

hN e

ρ =