1. 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

2. 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

3. 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)?

4. 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)

5. |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)

6. 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.

7. 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?)

9. 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.

10. 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.

11. 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?

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

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

14. 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

15. 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

16. 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

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

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