1、固體物理固體物理Chapter 9Fermi surfaces Chapter 9Fermi surfaces and metalsand metalsThe Fermi surface is the surface of constant energy F in k space. The Fermi surface separates the unfilled orbitals from the filled orbitals at absolute zero T = 0K.The electrical properties of the metal are determined by th

2、e shape of the Fermi surface, because the current is due to charges in the occupancy of states near the Fermi surface.Few people would define a metal as a solid with a Fermi surface. This may nevertheless be the most meaning definition of a metal one can give today; it represents a profound advance

3、in the understanding of why metals behaves as they do.Reduced zone schemeIt is always possible to select the wavevector index k of any Bloch function to lie within the first Brillouin zone.)()()(rueeruerkrGirk ikrk ikDefine )()(rueruGkrGik)()( )()()( )(ruruerueeTrueTrukGkrGiGkrGiTGiGkTrGikThen)()()(

4、rerurkrk ikkProve: supposeis a Bloch function, and .)(rkGkkWe can always mapping the band in the reduced zone scheme. aaa2a20k1st BZ2nd BZ2nd BZAACCC”In the reduced zone scheme, one can find different energies at same value of wavevector. Each different energy characterizes a different band.Two wave

5、functions at same k but different energies will be independent of each other: the wavefunctions will be made up of different combination of the plane waves.)(exprGkiPeriodic zone schemeWe can repeat a given Brillouin zone periodically through all wavevector space. We can translate not only a band fr

6、om other zones into the first zone, but also a band in first zone into every other zones in the periodic zone scheme.kGkHere k+G is understood to refer to the same energy band as k.Extended zone schemeExtended zone schemeReduced zone schemePeriodic zone scheme0kConstruction of Fermi surfaceThe equat

7、ion of the zone boundaries is 2kG + G2 = 0 and is satisfied if k terminates on the plane normal to G at the midpoint of G.1st Brillouin zone2nd Brillouin zone3rd Brillouin zone4th Brillouin zoneThe free electron Fermi gas in a square lattice1st zone2nd zone3rd zone2a2b2c2d3a3b1st zone2nd zone3rd zon

8、e2a2b2c2d3a3bMapping of the 1st, 2nd and 3rd Brillouin zone in the reduced zone scheme.The free electron Fermi surface in the reduced zone scheme.The free electron Fermi surface in the extended zone scheme.The free electron Fermi surface in the third zone as drawn in the periodic zone scheme.The par

9、ts of the free electron Fermi surface in the third zone appear disconnected as drawn in the reduced zone scheme. Nearly free electronsWe can make approximate constructions freehand by the use of four facts: The interaction of the electron with the periodic potential of the crystal causes energy gaps

10、 at zone boundaries. Almost always the Fermi surface will intersect zone boundaries perpendicularly. The crystal potential will round out sharp corners in the Fermi surfaces. The total volume enclosed by the Fermi surface depends only on the electron concentration and is independent of the details o

11、f the lattice interaction.Constant energy surface in the Brillouin zone of a simple square lattice with nearly free electron approximation.Constant energy surface is discontinuous at Brillouin zone boundary.The free electron Fermi surface at 2nd and 3rd zones.The near free electron Fermi surface at

12、2nd and 3rd zones.HolelikeElectronlikeHarrison construction of free electron Fermi surface1) Plot the reciprocal lattice2) Plot the free-electron Fermi spheres around each lattice point with the radius appropriate to the electron concentration.Any point in k space that lies within at least one spher

13、e corresponds to an occupied state in 1st zone. Points within at least two spheres correspond to occupied states in 2nd zone, and similarly for points in three spheres in 3rd zone, and so on.Alkali metals have weak interactions between the conduction electrons and the lattice. Because the alkali met

14、als have only one valence electron per atom, the first Brillouin zone boundaries are distant from the approximately spherical Fermi surface that filled one-half of the volume of the zone.The divalent metals, e.g. Be and Mg, also have weak lattice interactions and nearly spherical Fermi surface. But

15、because they have two valence electrons each, the Fermi surface encloses twice the volume in k space as for the alkalis, i.e. the volume enclosed by the Fermi surface is exactly equal to that of the a zone. But the Fermi surface is nearly spherical which extends out of the first zone and into the se

16、cond zone.In a static magnetic field, electrons move on a curve of constant energy on a plane normal to B. an electron on the Fermi surface will move in a curve on the Fermi surface. Three types of orbits:Electron orbits, hole orbits , and open orbitsElectron orbitsHole orbitsOpen orbitsBceBvcedtkdk

17、g2Orbits that enclose filled states are electron orbits.Orbits that enclose empty states are hole orbits.Orbits that move from zone to zone without closing are open orbits.CopperAluminumZone 1Zone 2Zone 3Calculation of energy bandsTight binding method for energy bandsThe tight binding approximation

18、or LCAO (linear combination of atomic orbitals) approximation starts out from the wavefunctions of the free atoms.The tight binding approximation is quite good for the inner electrons of atoms, but it is not a good description of the conduction electrons. Suppose that the ground state of an electron

19、 moving in the potential U(r) of an isolated atom is (r). If the influence of one atom on another atom is small, we obtain an approximate wavefunction for one electron in the whole crystal by Njjj kkrrCr)()(This function is of the Bloch form if jrk ij keNC2/1Then Njjjkrrrk iNr)()exp()(2/1Prove:)()ex

20、p( )()(exp)exp( )()(exp)(2/12/1rTk iTrrTrk iNTk irTrrk iNTrkNjjjNjjjkThe first-order energyjmjmmjkHrrk iNkHk)(exp1where .)(mmrrSet , jmmrrmmmrHrdVk ikHk)()( )exp(*Neglect all integrals except those on the same atom and those between nearest neighbors. We write:.)()( ;)()( *rHrdVrHrdVmThen mmkk ikHk)

21、exp(For a simple cubic structure, the nearest-neighbor atoms are. ) , 0 , 0( );0 , , 0( );0 , 0 ,(aaamThen )coscos(cos2akakakzyxkThe energies are confined to a band of width 12. 22*2/amIn the limit ka 1, k 6 + k2a2. The effective mass isFor the bcc structure with eight nearest neighbors,2cos2cos2cos

22、8akakakzyxkFor the fcc structure with twelve nearest neighbors,2cos2cos2cos2cos2cos2cos4akakakakakakyxxzzyk is the overlap integral. The weaker the overlap, the narrower is the energy band, and the higher is the effective mass.Wigner-Seize MethodOver most of a band the energy may depend on the wavev

23、ector nearly as for a free electron. However the Bloch wavefunction, unlike a plane wave, will pile up charge on the positive ion core as in the atomic wavefunction.A Bloch function satisfies the wave equation)()()(22rueruerUmpkrk ikkrk iWith , we haveip)()()2()()()();()()(222ruperupkeruekruepruperu

24、ekruepkrk ikrk ikrk ikrk ikrk ikrk ikrk iThus the wave equation can be written as)()()()(212rururUkpmkkkAt k = 0, we have the wavefunction .)(00ru)()()(210002rururUpmIf k 0, we can use u0(r) to construct the approximate solution)()exp(0rurk ikTreated the term as a perturbation, the perturbation theo

25、ry can develop the effective mass m* at the band edge.pk*2202mkkCohesive energy of metalsThe stability of the simple metals with respect to free atoms is caused by the lowering of the energy of the Bloch orbital with k = 0 in the crystal compared to the ground valence orbital of the free atom.k = 0

26、state8.2 eV6.3 eV5.15 eVAverage energyFermi levelMetalFree atomGround stateCohesive energyPseudopotential methodsConduction electron wavefunctions are usually smoothly varying in the region between the ion cores, but have a complicated nodal structure in the region of the cores.Outside the core the

27、potential energy that acts on the conduction electron is relatively weak. In this outer region, the conduction electron wavefunctions are as smoothly varying as plane waves. If the conduction orbitals in this outer region are approximately plane waves, the energy must depend on the wavevecror approx

28、imately as 222kmkWhat goes on in the core is largely irrelevant to the dependence of on k.We might replace the actual potential energy (and filled shells) in the core region by an effective potential that gives the same wavefunctions outside the core as are given by the actual ion cores. The effecti

29、ve potential or pseudopotential that satisfies this requirement is nearly zero. this result is referred to as the cannellation theorem.The pseudopotential for a problem is not unique nor exact, but it may be very good.The pseudopotential is much weaker than the true potential, but the pseudopotentia

30、l was adjusted so that the wavefunction in the outer region is nearly identical to that for the true potential.Calculation of band structure depends only on the Fourier components of the pseudopotential at the reciprocal lattice vectors. Usually only a few values of the coefficients U(G) are needed

31、to get a good band structure.The Empty Core Model (ECM):The unscreened pseudopotential iseeRrreRrrUfor , /for , 0)(2It is often possible to calculate band structures, cohesive energy, lattice constants, and bulk moduli form first principles. In such ab initio pseudopotential calculations the basic i

32、nputs are the crystal structure type and the atomic number, along with well-test theoretical approximations to exchange energy terms.Experimental methods in Fermi surface studiesThe experimental methods for determination of Fermi surfaces include magnetoresistence, anomalous skin effect, cyclotron r

33、esonance, magneto-acoustic geometric effects, the Shubnikow-de Haas effect, and the de Hass-van Alphen effect. Further information on the momentum distribution is given by positron annihilation, Compton scattering, and the Kohn effect.Quantization of orbits in a magnetic fieldThe momentum of a parti

34、cle in a magnetic field iscAqkpppfieldkin/where is the vector potential related to the magnetic field,AABThe orbits in a magnetic field are quantized by the Bohr-Sommerfeld relation2)( nrdpThenrdAcqrdkrdpFrom the equation of motionBdtrdcqdtkdWe haveBrcqkcqrdrBcqrdBrcqrdk2where is the magnetic flux c

35、ontained within the orbit in real space.cqdBcqdAcqrdAcqThen 2)( ncqrdpThe orbit of an electron is quantized in such a way that the magnetic flux through it is )/2)(ecnndkeBcdr dSeBcdkeBcdkeBcdrdA22222)()(ecnSBecSeBcBABnnnn2)(122Whence the area of an orbit in k space will satisfyBcenSn2)( In Fermi su

36、rface experiments we may be interested in the increment B for which two successive orbits, n and n+1, have the same area in k space on the Fermi surface.SBSBSnnnn)()(11wheni.e.ceBBSnn2111We have the important result that equal increments of 1/B reproduce similar orbits. This periodicity in 1/B is a

37、striking feature of the magneto-oscillatory effects in metals at low temperatures: resistivity, susceptibility, heat capacity.The population of orbits on or near the Fermi surface oscillates as B is varied. From the period of the oscillation we can reconstruct the Fermi surface.de Hass-van Alphen ef

38、fectThe 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.In a magnetic field B parallel to z axis, the area of an orbit in kx

39、, ky space is quantized. The area between successive orbits isBceS2The area in k space occupied by a single orbital is (2/L)2.The number of free electron orbitals that coalesce in a single magnetic level isBLBceD222Such a magnetic level is called a Landau level.For a system of N electrons at 0K, the

40、 Landau levels are entirely filled up to a magnetic quantum number s. Orbitals at the next higher level s+1 will be partly filled to the extent needed to accommodate the electrons. The Fermi level will lie in the Landau level s+1. As the magnetic field is increased the electrons move to lower levels

41、 because the degeneracy D increases. When s+1 is vacated, the Fermi level moves down abruptly to the next lower level s.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 field Bs no level is partl

42、y occupied at 0K so that NBssThe energy of the Landau level of magnetic quantum number n iscnnE)2/1( Where is the cyclotron frequency.cmeBc*/The total energy of the electrons in the levels that are fully occupied is2121)2/1(sDnDcsncThe total energy of the electrons in the partly occupied level s+1 is