Energy bands (<<) Nearly-free electron limit M.C. Chang Dept of Phys Energy bands (<<) Nearly-free electron limit Bragg reflection and energy gap Bloch theorem The central equation Empty-lattice approximation (>>) Tight-biding limit  For history on band theory, see 半導體的故事, by 李雅明, chap 4

Nearly-free electron (NFE) model NFE Model is good for Na, K, Al… etc, in which the lattice potential is only a small perturbation to the electron sea. Free electron plane wave Consider 1-dim case, when we turn on a lattice potential with period a, the electron wave will be Bragg reflected when k=±π/a, which forms two different types of standing wave. (Peierls, 1930) Density distribution

Q: where are the energy gaps when U(x)=U1 cos(2πx/a)+U2 cos(4πx/a)? These 2 standing waves have different electrostatic energies. The difference is the energy gap. For U(x)=U cos(2πx/a) Energy dispersion Electron’s group velocity is zero near the boundary of the 1st Brillouin zone Q: where are the energy gaps when U(x)=U1 cos(2πx/a)+U2 cos(4πx/a)?

The Kronig-Penny model (1930) Let b0, but keeps bV0 a constant (called a Dirac comb), then Calculated electron energy dispersion Energy bands (Kittel, p.170)

|Ψ(x)|2 is the same in each unit cell. General theory for an electron in a weak periodic potential |Ψ(x)|2 is the same in each unit cell. Bloch theorem (1928) The electron states in a periodic potential is of the form where uk(r)= uk(r+R) is a cell-periodic function The cell-periodic part uk(x) depends on the form of the periodic potential. A simple proof for 1-dim: Ψ(x+a)=CΨ(x) Consider periodic BC, Ψ(x+Na)=Ψ(x)=CNΨ(x) C=exp(i2πs/N), s=0,1,2… N-1 Write Ψ(x) as uk(x) exp(i2πsx/Na) then uk(x) = uk(x+a)

Allowed values of k are determined by the B.C. Periodic B.C. (3-dim case) This is similar to the discussion in chap 4 about the allowed values of k for a phonon. Therefore, there are N k-points in the 1st BZ, where N is the total number of primitive cells in a crystal.

Difference between conductor/insulator (Wilson, 1931) There are N k-points in an energy band, each k-point can be occupied by two electrons (spin up and down).  each energy band has 2N “seats” for electrons. If a solid has one valence electron per primitive cell, then the energy band is half-filled (conductor) If two electrons per primitive cell, then there are 2 possibilities (a) no energy overlap or (b) energy overlap insulator conductor For example, all alkali metals are conductors, while alkali earth elements can be conductor/insulator. (Boron?)

How do we determine uk(x) from the potential U(x)? (1-dim case) Keypoint: go to k-space to simplify the calculation Fourier transform 1. the lattice potential G=2n/a (U0=0) 2. the wave function k=2n/L Schrodinger equation Schrod. eq. in k-space aka. the central eq. Solve one k at a time, but notice each C(k) is related to other C(k-G) (G) For a given k, there are many eigen-energies n, with corresponding eigen-vectors Cn.

Focus on one k in the first BZ (and a specific n) The Bloch state nk(x) is a superposition of … exp[i(k-g)x], exp[ikx], exp[i(k+g)x] … The energy of nk(x) is n(k) (g=2/a) g -g k  exp[i(k+g)x] exp[ikx] exp[i(k-g)x] If k is close to 0, then the most significant component of 1k(x) is exp[ikx] (little superposition from other plane waves) If k is close to g/2, then the most significant components of 1k(x) and 2k(x) are exp[i(k-g)x] and exp[ikx], others can be neglected.

For example, U(x) = 2U cos2x/a = U exp(2ix/a)+U exp(-2ix/a) There are only 2 components Ug=U-g=U (g=2/a) Matrix form of the central eq. cutoff for kg/2! The eigenvalues εn(k) determines the energy band The eigenvectors {Cn(k-G), G} determines the actual form of the Bloch states check What are the eigen-energies and eigen-states when U=0?

Energy levels near zone boundary k=g/2 Cut-off form of the central eq. Energy eigenvalues

Kittel, p.225 Sometimes it is convenient to extend the domain of k with the following requirement

NFE model in more than 1 dim (a rough discussion) Band structure εn(kx,ky) in 2-dim Origin of energy gaps: Bragg reflection k Laue condition G/2  Bragg reflection whenever k hits the BZ boundary Kittel, p.227

Folding in 2-dim, the “empty lattice” approximation How to fold a parabolic “surface” back to the first BZ? Take 2D square lattice as an example k-space  X M 2/a

Folded parabola along ΓX (reduced zone scheme) In reality, there are energy gaps at BZ boundaries because of the Bragg reflection The folded parabola along ΓM is different  X M Usually we only plot the major directions, for 2D square lattice, they are ΓX, XM, MΓ 2/a

Folding in 3-dim The energy bands for an “empty” fcc lattice Actual Band structure for copper (fcc crystal, 3d104s1)

Tight binding model (details in chap 9) Covalent solid, d-electrons in transition metals Alkali metal, noble metal