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Quantum Transport in

• Waves and particles in quantum mech. • in atoms • Insulators : Energy gap • Emergent particles in a in a magnetic and the quantum Light is a Wave

Hallmark of a wave : Interference constructive

destructive Wave Particle Duality

A is a particle too Electrons are Waves Energy and Momentum Planck : Energy ~ Frequency

Ehf===ω ωπ= 2 f

= ==×hJs/ 2π 1.05 10−34

Max Planck 1858-1947 de Broglie: Momentum ~ (Wavelength)-1 h pk=== k = 2/π λ λ

Louis de Broglie 1892-1987 Relation

Relation between • Frequency and Wavelength • Energy and Momentum E For a photon: = cλ ω = = fckEcp ⇒ ⇒ p

For an electron in vacuum: E

222 1 2 pk= Em==v ⇒ =ω = p 22mm 2 Quantization Waves in a confined geometry have discrete modes

v f = 1 2L v f = 2 2 2L v f = 3 3 2L Quantization of circular orbits

Circular orbits have quantized angular momentum.

nrλπ= 2 2π ==n p == λ r Lrpn=== Bohr Model • Two equations : L ==mrv n= em22v Fk== rr2

• Solve for rn and En(rn,vn) =− 2 En Ry/ n 1885-1962 = 2 rnan 0 • Rydberg energy: mk24 e Ry ==13.6 eV 2=2 • Bohr radius : 2 === a0 2 .5Å Explains line spectra of atoms mke Flatland ... (1980’s) Semiconductor Heterostructures : “Top down technology” → Two dimensional electron gas (2DEG)

V(z) conduction band energy z

AlxGa1-x As

∆E ~ 20 mV ~ 250°K Ga As 2D subband z 2DEG

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

Electrons are “”:

Each can accommodate at most one electron

Wolfgang Pauli 1900-1958 n=4 n=4

n=3 n=3

n=2 n=2

n=1 n=1 Ground state Excited State

Enrico Fermi 1901-1954

Energy Gap

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

A semiconductor is an insulator with a small energy gap. Add electrons to a semiconductor

Energy Gap

Accomplished by either • Chemical doping • Electrostatic doping (as in MOSFET) Added electrons are mobile. Add electrons to a semiconductor

Energy Gap

Accomplished by either • Chemical doping • Electrostatic doping (as in MOSFET) Added electrons are mobile. Add electrons to a semiconductor

Energy Gap

Accomplished by either • Chemical doping • Electrostatic doping (as in MOSFET) Added electrons are mobile. Added electrons form a band E p2 E = 2*m

p m* = “Effective mass”

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

“Emergent at low energy” “Holes” in the valence band

Energy Gap

Added holes are mobile “Holes” in the valence band

Energy Gap

Added holes are mobile “Holes” in the valence band

Energy Gap

Added holes are mobile Holes are particles too E p2 E = 2*mh p mh* = Hole Effective mass

Holes in the “valence band” behave just like ordinary 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

E Conduction Band: Empty states

electrons Gap

holes

Valence Band: Occupied states p The Quantized Hall Effect Hall effect in 2DEG MOSFET at large

Von Klitzing,1981 (Nobel 1985)

ρ = -9 • Quantization: xyRN Q / N = integer accurate to 10 ! ==2 Ω • Quantum Resistance: RheQ / 25.812 807 k • Explained by quantum of electrons in a magnetic field Quantization in a magnetic field G G G Cyclotron : =− × F eBv B v 1D Quantization argument: F mv2 e FeB==v L ==mrv n= r Solve for rn and En(rn,vn) eB n n= En====ω r = nc22m n eB ω = eB Cyclotron frequency c π m π h Magnetic flux 2 nn= Magnetic Flux φ = rB==φ 0 enclosed by orbit n e 2 0 quantum e =×4.1 10−15Tm 2 Landau Levels

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

Closely related to the harmonic oscillator 1908-1968 1 E / =ω En=+=ω () c nc2 Each Landau level has one state per flux quantum total fluxB× Area # states = = flux quantum (he / ) Quantized Hall Effect ω E / = c N filled Landau levels: # particles/area: # flux quanta Gap nN=× area BeB ==NN φ 0 h E ρ ==y B From last time: xy Jnex

ρ = 1 h xy Ne2