9. Fermi Surfaces and Metals Construction of Fermi Surfaces Electron Orbits, Hole Orbits, and Open Orbits Calculation of Energy Bands Experimental Methods in Fermi Surface Studies

Fermi Surface : surface of ε =εF in k-space Separates filled & unfilled states at T = 0. Close to a sphere in extended zone scheme. Looks horrible in reduced zone scheme. 2nd zone nearly half-filled

Reduced Zone Scheme Reduced Zone Scheme: k  1st BZ. k is outside 1st BZ. k = k + G is inside. Both & are lattice-periodic. So is → is a Bloch function

1-D Free Electrons / Empty Lattice Reduced zone scheme: εk is multi-valued function of k. Each branch of εk forms an energy band Bloch functions need band index: PWE

Periodic Zone Scheme εk single-valued εk multi-valued εnk single-valued εnk = εnk+G periodic E.g., s.c. lattice, TBA

Construction of Fermi Surfaces Zone boundary:

3rd zone: periodic zone scheme

Harrison construction of free electron Fermi surfaces Points lying within at least n spheres are in the nth zone.

Nearly free electrons: Energy gaps near zone boundaries → Fermi surface edges “rounded”. Fermi surfaces & zone boundaries are always orthogonal.

Electron Orbits, Hole Orbits, and Open Orbits Electrons in static B field move on intersect of plane  B & Fermi surface.

Nearly filled corners: P.Z.S. Simple cubic TBM P.Z.S.

Calculation of Energy Bands Tight Binding Method for Energy Bands Wigner-Seitz Method Cohesive Energy Pseudopotential Methods

Tight Binding Method for Energy Bands 2 neutral H atoms Ground state of H2 Excited state of H2 1s band of 20 H atoms ring.

TBM / LCAO approximation Good for valence bands, less so for conduction bands. α = s, p, d, … j runs over the basis atoms Bravais lattice , s-orbital only: ψk is a Bloch function since 1st order energy: Keep only on site & nearest neighbor terms:

For 2 H atoms ρ apart: Simple cubic lattice: 6 n.n. at  Band width = 12  2 N orbitals in B.Z.  =  surface. 1 e per unit cell. periodic zone scheme

Fcc lattice: 12 n.n. at surface Band width = 24 

Band Structure

Wigner-Seitz Method Bloch function: Schrodinger eq.: For k = 0, we have u0 is periodic in R l . is a Bloch function; can serve as an approximate solution of the Schrodinger eq. for k  0.

d  /d r = 0 at cell boundaries.   Prob 8 Wigner-Seitz B.C.: d  /d r = 0 at cell boundaries. Wigner-Seitz result for 3s electrons in Na. Table 3.9, p.70  ionic r = 1.91A r0 of primitive cell = 2.08A  n.n. r = 1.86A   is constant over 7/8 vol of cell.

Cohesive Energy linear chain Na Table 6.1, p.139: F ~ 3.1 eV. K.E. ~ 0.6 F ~ 1.9 eV.  5.15 eV for free atom. 0 ~ 8.2 eV for u0 .  +2.7 eV for k at zone boundary.  ~  8.2+1.9 ~ 6.3 eV Cohesive energy ~  5.15 +6.3 ~ 1.1 eV exp: 1.13 eV

Pseudopotential Methods Conduction electron ψ plane wave like except near core region. Reason: ψ must be orthogonal to core electron atomic-like wave functions. Pseudopotential: replace core with effective potential that gives true ψ outside core. Empty core model for Na (see Chap 10) Rc = 1.66 a0 . U ~ –50.4 ~ 200 Ups at r = 0.15 With Thomas-Fermi screening.

Typical reciprocal space Ups (see Chap 14) Empirical Pseudopotential Method Cohen

Experimental Methods in Fermi Surface Studies Quantization of Orbits in a Magnetic Field De Haas-van Alphen Effect Extremal Orbits Fermi Surface of Copper Example: Fermi Surface of Gold Magnetic Breakdown

Experimental methods for determining Fermi surfaces: Magnetoresistance Anomalous skin effect Cyclotron resonance Magneto-acoustic geometric effects Shubnikov-de Haas effect de Haas-van Alphen effect Experimental methods for determining momentum distributions: Positron annihilation Compton scattering Kohn effect Metal in uniform B field → 1/B periodicity

Quantization of Orbits in a Magnetic Field q = –e for electrons Bohr-Sommerfeld quantization rule: Phase corrector γ = ½ for free electrons B = const → Flux quantization Dirac flux quantum

→ For Δr  B : Let A = Area in r-space, S = Area in k-space. → Hence Area of orbit in k-space is quantized If then Properties that depend on S are periodic functions of 1/B.

De Haas-van Alphen Effect dHvA effect: M of a pure metal at low T in strong B is a periodic function of 1/B. 2-D e-gas: PW in  (B) dir. # of states in each Landau level = (spin neglected) See Landau & Lifshitz, “QM: Non-Rel Theory”, §112. Allowed levels B = 0 B  0

Number of e = 48 D = 16 D = 19 D = 24 For the sake of clarity, n of the occupied states in the circle diagrams is 1 less than that in the level diagrams.

Critical field (No partially filled level at T = 0): s = highest completely filled level Black lines are plots of n = s ρ B, n = N = 50 at B = Bs . Red lines are plots of n = s N / ( N / ρ B ), n = N = 50 at N / ρ B = s .

Total energy in fully occupied levels: → for Total energy in partially occupied level s + 1: for for where

S = extremal area of Fermi surface  B for where S = extremal area of Fermi surface  B Section AA is extremal. Its contribution dominates due to phase cancellation effect.

Fermi Surface of Copper Cu / Au Monovalent fcc metal: n = 4 / a3 Shortest distance across BZ = distance between hexagonal faces Band gap at zone boundaries → band energy there lowered → necks Distance between square faces  12.57/a : necking not expected

Example: Fermi Surface of Gold dHvA in Au with B // [110]: Dogbone μ has period 210–9 gauss–1 for most directions → Table 6.1: → Period along [111] is 610–8 gauss–1 → → neck Dogbone area ~ 0.4 of belly area

Multiply-connected hole surface of Mg in bands 1 & 2

Magnetic Breakdown → breakdown Change of connectivity ~ free electron-like Affected quantities ~ sensitive to connectivity : magnetoresistance E.g., hcp metals with zero (small if spin-orbit effect included) gap at hexagonal zone boundary Mg: Eg ~ 10–3 eV, εF ~ 10 eV, breakdown :