（x a) i cos
3 ( x a) 3x ika e i cos a a

a
eika 1
电子在此状态中的简约波矢为
Reduced Zone Scheme
(1) (2)
(3)
(4)
All the energy bands in the first Brillouin zone; An energy band is a single branch of energy vs k surface; Two wavefunctions at the same k but of different energies will be independent of each other; The wavefunctions will be made up of different combinations of the plane wave components
Example
一维周期场中电子波函数k(x)应当满足布洛赫 定理，若晶格常数为a，电子波函数为 ， 3 x k x i cos
a
试求电子在此状态中的简约波矢。 ik Rn （r Rn ) e (r )
（x a) eika ( x)
n,k=GCn(k+G)exp[i(k+G)· r],
Periodic Zone Scheme
repeat a given Brillouin zone periodically through all of wavevector space The engery of a band is a periodic function in the reciprocal lattice

The alkali metals are the simplest metals, with weak interactions between the conduction electrons and the lattice: only one valence electron per atom, the first Brillouin zone boundaries are distant from the Fermi surface
Fermi surface of Na: closely spherical
of Cs: deformed by 10 percent from a sphere

The divalent metals Be and Mg have weak interactions and nearly spherical Fermi surfaces. The volume enclosed by the Fermi surface is exactly equal to that of a zone.

Example: two hydrogen atoms

Each with an electron in the 1s ground state. Wavefunctions: A ,B As the atoms are brought together, their wavefuntions overlap. A+B A-B ground state excited state
(c) open orbit
CALCULATION OF ENERGY BANDS
trial wave function
calculation
computer

the tight binding method, useful for interpolation The Wigner-Seitz method, useful for the visualization and the understanding of the alkali metals The pseudopotential method, shows the simplicity of many problems
Reduced Zone Scheme
select the wavevector index k of any Bloch function to lie within the first Brillouin zone using Bloch function Bloch theorem The eigenfunctions of the wave equation for a periodic potential are the product of a plane wave exp(ik•r) times a function uk(r) with the periodicity of the crystal lattice. The solutions of the Schrodinger equation for a periodic potential must be of a special form:
ELECTRON ORBITS, HOLE ORBITS, AND OPEN ORBITS
Electrons in a static magnetic field move on a curve of constant energy on a plane normal to B Fermi surface Fig12 closed orbit: (a) and (b) (a) holelike (b) electronlike


CONSTRUCTION OF FERMI SURFACES
construction the Brillouin zones in k space
Fermi surfaces and Brillouin zones in k space free electrons
Nearly Free Electrons

Tight binding approximation (紧束缚近似） LACO (linear combination of atomic orbitals) approximation Good for the inner electrons of atoms, d bands of the transition metals and the valence bands of diamondlike and inert gas crystals Not good for the conduction electrons Solution: an electron at the ground state of an isolated atom: potential: U(r), wavefunction: (r)
k(r)=N-1/2jexp(ik· rj)(r-rj) Prove it: k(r+T)=N-1/2jexp(ik· rj)(r+T-rj) =exp(ik· T)N-1/2jexp[ik· (rj-T)][r-(rj-T)] =exp(ik· T)k(r) T: translation vector
Chapter 9: Fermi Surfaces and Metals
Chapter 9: Fermi Surfaces and Metals
THREE DIFFERENT ZONE SCHEMES: (1)reduced zone scheme (2)periodic zone scheme (3)extended zone scheme related terms： Brillouin zone energy band (band structure: energy vs wavevector) Fermi surface (one of the surfaces of constant energy in k space)

The interaction of the electron with the periodic potential of the crystal causes energy gaps at the zone boundaries. Almost always the Fermi surface will intersect zone boundaries perpendicularly. The crystal potential will round out sharp corners in the Fermi surfaces.

Tight binding method for energy bands
As free atoms are brought together , the coulomb interaction between the atom cores and the electron splits the energy levels, spreading them into bands.



The total volume enclosed by the Fermi surface depends only on the electron concentration and is independent of the details of the lattice interaction.
the first-order energy: calculate the diagonal matrix elements of the hamiltonian of the crystal <kHk>=N-1jmexp[ik· (rj-rm)]<mHj> m=(r-rm), m=rm-rj
k = k+G

The extended zone scheme in which different bands are drawn in different zones in wavevector space. The reduced zone scheme in which all bands are drawn in the first Brillouin zone. The periodic zone scheme in which every band is drawn in every zone.