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 field and the quantum Hall effect Light is a Wave Hallmark of a wave : Interference constructive destructive Wave Particle Duality A photon 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 Dispersion Relation Relation between • Frequency and Wavelength • Energy and Momentum E For a photon: c fckEcp= ⇒ ω = ⇒ = λ 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 Niels Bohr 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 “Fermions”: Each quantum state 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 Insulator 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 quasiparticles 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 magnetic field 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 mechanics of electrons in a magnetic field Quantization in a magnetic field G G G Cyclotron Motion: =− × 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 eB Cyclotron frequency ω = c m π h Magnetic flux 2 nn= Magnetic Flux φ = π 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 Lev Landau 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.