Fermi surfaces and metals construction of Fermi surface

Fermi surfaces and metals • construction of Fermi surface • semiclassical electron dynamics • de Haas-van Alphen effect • experimental determination of Fermi surface (the Sec on “Calculation of energy bands” will be skipped) Dept of Phys M. C. Chang

• Higher Brillouin zones (for square lattice) • Reduced zone scheme 3 2 1 2 2 2 3 3 3 Every Brillouin zone has the same area 3 • At zone boundary, k points to the plane bi-secting the G vector, thus satisfying the Laue condition • Bragg reflection at zone boundaries produce energy gaps (Peierls, 1930)

Beyond the 1 st Brillouin zone BCC crystal FCC crystal

Fermi surface for (2 D) empty lattice 3 • For a monovalent element, the Fermi wave vector 2 • For a divalent element 1 • For a trivalent element • Distortion due to lattice potential 1 st BZ 2 nd BZ

A larger Fermi "sphere " (empty lattice) • Extended zone scheme • Reduced zone scheme • Periodic zone scheme Again if we turn on the lattice potential, then the corners will be smoothed.

Fermi surface of alkali metals (monovalent, BCC lattice) k. F = (3π2 n)1/3 n = 2/a 3 → k. F = (3/4π)1/3(2π/a) ΓN=(2π/a)[(1/2)2+(1/2)2]1/2 ∴ k. F = 0. 877 ΓN

Fermi spheres of alkali metals 4π/a Percent deviation of k from the free electron value (1 st octant)

Fermi surface of noble metals (monovalent, FCC lattice) Band structure (empty lattice) k. F = (3π2 n)1/3, n = 4/a 3 → k. F = (3/2π)1/3(2π/a) ΓL= ___ k. F = ___ ΓL Fermi surface (a cross-section)

Fermi surfaces of noble metals Periodic zone scheme

Fermi surface of Al (trivalent, FCC lattice) 1 st BZ • Empty lattice approximation 2 nd BZ • Actual Fermi surface

Fermi surfaces and metals • construction of Fermi surface • semiclassical electron dynamics • de Haas-van Alphen effect • experimental determination of Fermi surface

important Semiclassical electron dynamics (Kittel, Chap 8, p. 192) Consider a wave packet with average location r and wave vector k, then Derivation neglected here • Notice that E is the external field, which does not include the lattice field. The effect of lattice is hidden in εn(k) ! Range of validity • This looks like the usual Lorentz force eq. But It is valid only when Interband transitions can be neglected. (One band approximation) May not be valid in small gap or heavily doped semiconductors, but “never close to being violated in a metal” • E and B can be non-uniform in space, but they have to be much smoother than the lattice potential. • E and B can be oscillating in time, but with the condition

Bloch electron in an uniform electric field (Kittel, p. 197) • Energy dispersion (periodic zone scheme, 1 D) ε(k) k -π/a v(k) k • In a DC electric field, the electrons decelerate and reverse its motion at the BZ boundary. • A DC bias produces an AC current ! (called Bloch oscillation)

• Partially filled band without scattering E • Partially filled band with scattering time • Current density

• Why the oscillation is not observed in ordinary crystals? To complete a cycle (a is the lattice constant), e. ET/ = 2π/a → T=h/e. Ea For E=104 V/cm, and a=1 A, T=10 -10 sec. But electron collisions take only about 10 -14 sec. ∴ a Bloch electron cannot get to the zone boundary without de-phasing. To observe it, one needs • a stronger E field → but only up to about 106 V/cm (for semicond) • a larger a → use superlattice (eg. a = 100 A) • reduce collision time → use crystals with high quality (Mendez et al, PRL, 1988) • Bloch oscillators generate THz microwave: frequency ～ 1012~13, wave length λ ～ 0. 01 mm - 0. 1 mm (Waschke et al, PRL, 1993)

important Bloch electron in an uniform magnetic field Therefore, 1. Change of k is perpendicular to the B field, k|| does not change and 2. ε(k) is a constant of motion This determines uniquely an electron orbit on the FS: B • For a spherical FS, it just gives the usual cyclotron orbit. • For a connected FS, there might be open orbits.

Cyclotron orbit in real space The above analysis gives us the orbit in k-space. What about the orbit in r-space? r-orbit k-orbit ⊙ • r-orbit is rotated by 90 degrees from the k-orbit and scaled by c/e. B ≡ λB 2 • magnetic length λB = 256 A at B = 1 Tesla

Fermi surfaces and metals • construction of Fermi surface • semiclassical electron dynamics • de Haas-van Alphen effect • experimental determination of Fermi surface

De Haas-van Alphen effect (1930) Silver In a high magnetic field, the magnetization of a crystal oscillates as the magnetic field increases. Similar oscillations are observed in other physical quantities, such as • magnetoresistivity (Shubnikov- Resistance in Ga de Haas effect, 1930) • specific heat • sound attenuation … etc Basically, they are all due to the quantization of electron energy levels in a magnetic field (Landau levels, 1930)

Quantization of the cyclotron orbits • In the discussion earlier, the radius of the cyclotron orbit can be varied continuously, but due to their wave nature, the orbits are in fact quantized. • Bohr-Sommerfeld quantization rule (Onsager, 1952) for a closed cyclotron orbit, Why (q/c)A is momentum of field? See Kittel App. G. Gauge dependence prob? Not worse than the gauge dependence in q. V. also • the flux through an r -orbit is quantized in units of flux quantum: hc/e≡Φ 0=4. 14· 10 -7 gauss．cm 2

Mansuripur’s Paradox Kirk T. Mc. Donald

important • Since a k-orbit (circling an area S) is closely related to a r-orbit (circling an area A), the orbits in k-space are also quantized (Onsager, 1952) B=0 • Energy of the orbit (for a spherical FS) Landau levels • Degeneracy of the Landau level spin B≠ 0 Highly degenerate • Notice that, in 3 D, the kz direction is not quantized In the presence of B, the Fermi sphere becomes a stack of cylinders.

Remarks: • Fermi energy ～ 1 e. V, cyclotron energy ～ 0. 1 me. V (for B = 1 T) ∴ the number of cylinders usually ～ 10000 need low T and high B to observe the fine structure. • Radius of cylinders , so they expand as we increase B. The orbits are pushed out of the FS one by one. Cyclotron orbits FS E Landau levels larger level separation, and larger degeneracy (both B) EF B • Successive B’s that produce orbits with the same area: Sn = (n+1/2) 2πe/ c B Sn-1'= (n-1/2) 2πe/ c B' (B' > B) equal increment of 1/B reproduces similar orbits

Quantitative details (2 D) partially filled N=50 filled partially filled

Oscillation of the DOS at Fermi energy (3 D) • Number of states in dε are proportional to areas of cylinders in an energy shell. Two extremal orbits • The number of states at EF are highly enhanced when there are extremal orbits on the FS. • There are extremal orbits at regular interval of 1/B. • This oscillation in 1/B can be detected in any physical quantity that depends on the DOS .

In the d. Hv. A experiment of silver, the two different periods of oscillation are due two different extremal orbits. Determination of FS Therefore, from the two periods we can determine the ratio between the sizes of the "neck" and the "belly“. A 111(belly)/A 111(neck)=27 A 111(belly)/A 111(neck)=51 B A 111(belly)/A 111(neck)=29