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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 3 3 3 Every Brillouin zone has the same area 3 • At zone boundary, k points to the plane bi-secting the G G vector, thus satisfying the Laue condition k Gˆ 2 • Bragg reflection at zone boundaries produce energy gaps (Peierls, 1930) Beyond the 1st Brillouin zone BCC crystal FCC crystal Fermi surface for (2D) empty lattice 3 2 • For a monovalent element, the Fermi wave vector kF 2 a • For a divalent element 1 kF 4 a • For a trivalent element kF 6 a • Distortion due to lattice potential 1st BZ 2nd 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) kF = (3π2n)1/3 n = 2/a3 → kF = (3/4π)1/3(2π/a) ΓN=(2π/a)[(1/2)2+(1/2)2]1/2 ∴ kF = 0.877 ΓN Fermi spheres of alkali metals 4π/a Percent deviation of k from the free electron value (1st octant) Fermi surface of noble metals (monovalent, FCC lattice) Band structure (empty lattice) kF = (3π2n)1/3, n = 4/a3 → kF = (3/2π)1/3(2π/a) ΓL= ___ kF = ___ ΓL Fermi surface (a cross-section) Fermi surfaces of noble metals Periodic zone scheme Fermi surface of Al (trivalent, FCC lattice) 1st BZ • Empty lattice approximation 2nd 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 1 n (k ) r ( k ) k k q E r (k ) B c 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) eEa g g F c g g , F May not be valid in small gap or heavily doped semiconductors, but “never close to being violated in a metal” c 1.16 104 B / T eV • 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 g Bloch electron in an uniform electric field (Kittel, p.197) dk eE k (t ) eEt dt • Energy dispersion (periodic zone scheme, 1D) ε(k) k π/a -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 eE / • Current density 1 j ( e) vk V k filled states • Why the oscillation is not observed in ordinary crystals? To complete a cycle (a is the lattice constant), eET/ = 2π/a → T=h/eEa 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 - 0.1mm (Waschke et al, PRL, 1993) important Bloch electron in an uniform magnetic field 1 (k ) k 1 d (k ) k B 0, k vk 0 dt dk v e B, dt c vk 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? e c k r B r 2 Bk r c eB c ˆ r (t ) r (0) B [k (t ) k (0)] eB r-orbit k-orbit ⊙ • r-orbit is rotated by 90 degrees from the k-orbit and scaled by c/eB ≡ λB2 • 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 Resistance in Ga (Shubnikov-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, 1 dr p n h 2 Why (q/c)A is momentum of field? See Kittel App. G. q A, q e c Gauge dependence prob? e e dr k dr r B 2 Not worse than the gauge c c dependence in qV. e e dr A c c 1 hc 1 n n , An n n 2B2 2 e B 2 where p pkin p field k also • the flux through an r -orbit is quantized in units of flux quantum: hc/e≡Φ0=4.14·10-7 gauss.cm2 Mansuripur’s Paradox Kirk T. McDonald 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) Sn 1 2 e 1 2 n B n B4 2 c 2 B2 An B=0 • Energy of the orbit (for a spherical FS) n kn 1 n c Landau levels 2m 2 2 • Degeneracy of the Landau level D2 spin 2 eB / c 2 / L B≠0 2 2 BL2 2 sample hc / e 0 Highly degenerate • Notice that, in 3D, the kz direction is not quantized n,kz 2 2 kz 1 n c 2 2m In the presence of B, the Fermi sphere becomes a stack of cylinders. Remarks: • Fermi energy ~ 1 eV, cyclotron energy ~ 0.1 meV (for B = 1 T) ∴ the number of cylinders usually ~ 10000 need low T and high B to observe the fine structure. • Radius of cylinders B , 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) 1 1 2 e S c B B' equal increment of 1/B reproduces similar orbits Quantitative details (2D) partially filled N=50 filled partially filled filled DB partially filled filled M E B Oscillation of the DOS at Fermi energy (3D) • 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 dHvA experiment of silver, the two different periods of oscillation are due two different extremal orbits. Determination of FS 1 1 2 e Recall that Se c B B' Therefore, from the two periods we can determine the ratio between the sizes of the "neck" and the "belly“. A111(belly)/A111(neck)=27 A111(belly)/A111(neck)=51 B A111(belly)/A111(neck)=29