Experimental methods in fermi surface studiesExperimental methods:magnetoresistanceanomalous skin effect cyclotron resonance magneto-acoustic geometric effectsShubnikow- de Haas effect de Haas-van Alphen effect de Haas-van Alphen effect exhibits very well the characteristic periodicity in 1/B of the properties of a metal in a uniform magnetic field. Quantization of orbits in a magnetic fieldthe kinetic momentum:the potential momentum:(22)(23)Quantization of orbits in a magnetic field(24)The equation of motion of a particle of charge q in a magnetic field is(25a)Thus one of the path integrals in (24) is(25b)The other path integral in (24) isQuantization of orbits in a magnetic field(25c)Thus (26)The total magnetic flux (27)Quantization of orbits in a magnetic field(28)It follows that(29)(30)Quantization of orbits in a magnetic fieldThe areas are equal when(31)Equal increments of 1/B reproduce similar orbits.The population of orbits on or near the Fermi surface oscillates as B is varied, causing a wide variety of effects.De Haas-van Alphen EffectThe de Haas-van Alphen effect is the oscillation of the magnetic moment of a metal as a function of the static magnetic field intensity.The effect can be observed in pure specimens at low temperatures in strong magnetic fields.for a two-dimensional (2D) system(absolute zero):The area between successive orbits:(32)De Haas-van Alphen EffectThe number of free electron orbitals that coalesce in a single magnetic level is(33)Such a magnetic level is called a Landau level.De Haas-van Alphen EffectLandau levelthe total energy of the electrons raisedDe Haas-van Alphen EffectAllowed electron orbitals in two dimensionsThe angular position of the points has no significanceA magnetic fieldno magnetic fieldThe area between successive circles：The area of the adjacent circle:De Haas-van Alphen Effecta system of N electrons at absolute zeropartly filledentirely filledno filledFermi level moves down to the level sLandau levelWhen s+1 is vacated, the Fermi level moves down abruptly to the next lower level s.De Haas-van Alphen EffectThe electron transfer to lower Landau levels can occur because their degeneracy D increases as B is increased.As B is increased there occur values of B at which the quantum number of the uppermost filled level decreases abruptly by unity. At the critical magnetic fields labeled Bs no level is partly occupied at absolute zero, so that(34)De Haas-van Alphen Effectthe energy of the Landau level:cyclotron frequency:The total energy of the electrons in levels that are fully occupied is(35)The total energy of the electrons in the partly occupied level s+1 is(36)De Haas-van Alphen EffectThe thermal and transport properties of the metal also oscillate as successive orbital levels cut through the Fermi level when the field is increased.De Haas-van Alphen EffectThis oscillatory magnetic moment of the Fermi gas at low temperatures is the de Haas-van Alphen effect.The oscillations occur at equalintervals of 1/B such that(37)S is the extremal area of the Fermi surface normal to the direction of B.Extremal orbitsExtremal orbitsThe dominant response of the system comes from extremal orbits.Essentially it is a question of phase cancellation.The contributions of different nonextremal orbits cancel, but near the extrema the phase varies only slowly and there is a net signal from these orbits.Sharp resonances are obtained even from complicated Fermi surfaces because the experiment selects the extermal orbits. Fermi surface of copperThe Fermi surface of copper is distinctly nonspherical: eight necks make contact with the hexagonal faces of the first Brillouin zone of the fcc lattice. (38)Fermi surface of copperThe shortest distance across the Brillouin zone (the distance between hexagonal faces)：The free electron sphere does not touch the zone boundary, but we know that the presence of a zone boundary tends to lower the band energy near the boundary. Thus it is plausible that the Fermi surface should neck out to meet the closest (hexagonal) faces of the zone.The square faces of the zone are more distant, with separation 12.57/a,and the Fermi surface does not neck out to meet these faces.Example: fermi surface of gold Example: fermi surface of gold Anotherextremalorbit:“dogs bone ”Its area in Au is about0.4 of the belly area.Multiply-connected hole surface of magnesium inbands 1 and 2Magnetic breakdown Electrons in sufficiently high magnetic fields will move in free particle orbits. Here the magnetic forces are dominant, and the lattice potential is a slight perturbation.The eventual breakdown of this description as the magnetic field is increased is called magnetic breakdown.The onset of magnetic breakdown will be revealed by physical properties such as magnetoresistance that depend sensitively on the connectivity.Magnetic breakdown The condition for magnetic breakdown:Small gaps may be found in hcp metals where the gap across the hexagonal face of the zone would be zero except for a small splitting introduced by the spin-orbit interaction.