1. Solids: From Bonds to Bands
Atom
Band
Bond E
Levels
Molecule
1-D Solid

2. Real Materials more complex
Many orbitals per atom
Multiple dimensions (3-D)
Let us first recap the 1-D bandstructure, so we can
see how to generalize it in 3-D.
Being systematic helps !!

3. Summary of 1D bandstructure
STEP 1: Find period
in real space
STEP 2: Find k-space periodicity (connecting equivalents points in k-space)
K.a = 2p
STEP 3: Find BZ by bisecting nearest neighbor connectors.
This gives the smallest zone in k-space for a non-repeating
band. In this case, it’s between –p/a and p/a.
R = a
R = a
On occasion, this may need you to
choose a multiatom or multiorbital basis K x
2p/a
-2p/a

4. Summary of 1D bandstructure
STEP 4: Choose N allowed k-points by imposing periodic boundary conditions
after N unit cells. For complex solids, we may need to choose specific directions.
STEP 5: Identify nearest neighbors and find Fourier transform of H terms
over this range for each allowed k.
Each [Hnn] has size bxb (b: # basis sets)
STEP 6: Find eigenvalues E(k). This gives
b bands for each k within the BZ.
k = (n/N)K, where n=0,1,2,…,N-1, and K=2p/a n
n+1
n-1
Hnn
Hn,n+1
Hn,n-1
Hk = [Hnn] + [Hn,n+1]eika + [Hn,n-1]e-ika
b bands x
p/a
-p/a .

5. Summary of 1D bandstructure
STEP 6: Use this bandstructure E-k to calculate DOS D(E), fit parabolas
to extract effective mass m*, etc. These are then used for calculating
electronic properties like transmission, I-V, etc.

6. General prescription in 3-D1 2 4
[H]nm 3
Identify real and k-space lattice vectors
Identify Brillouin zone
Choose grid points along suitable directions in k-space
Find H(k) by summing over nearest neighbor H terms with Fourier phases
Find eigenvalues to get E-k, which we then use as needed

7. Finding real-space periodicity
(Lattice vectors)

8. Lattice Vectors a = (1,0,0)a b = (0,1,0)a c = (0,0,1)a
Simple cubic lattice
Three primitive vectors are ‘coordinates’ in terms of
which all lattice coordinates R can be expressed
R = ma + nb + pc (m,n,p: integers)

9. Face-centered cube a = a(0, ½, ½) b = a(½, 0, ½) c = a(½, ½, 0)
6 face center atoms shared by 2 cubes each, 8 corners
shared by 8 cubes each, giving a total of
8 x 1/8 + 6 x 1/2 = 4 atoms/cell

10. Body-centered cube a = a(½, ½, ½ ) b = a(-½,-½, ½ ) c = a(½,-½,-½ )
8x1/8 corner atom + 1 center atom gives
2 atoms per cell

11. Finding k-space periodicity
(Reciprocal Lattice vectors)

12. First let’s Fourier transform the lattice
R = ma + nb + pc (m,n,p: integers)
Coordinates of periodic lattice (ie atoms)
Not to be confused with coordinate r of electron which
is spread out everywhere
Need to find reciprocal basis sets K1, K2, K3 such
that reciprocal lattice coordinates can be written as
K = MK1 + NK2 + PK3 (M,N,P: integers)

13. What are lattice vectors in reciprocal space?
R = ma + nb + pc (m,n,p: integers)
K = MK1 + NK2 + PK3 (M,N,P: integers)
To create analogy with 1-D, want
K1 to be orthogonal to b and c,
and have a 2p overlap with a so
that exp(iK.R) = 1
K1 = 2p(b x c)/[a. (b x c)]
Pts in k-space spaced by integer # of K’s
represent same electronic state

14. States spaced by K are identical

15. Choosing k values M, N, P unit cells along Reciprocal lattice vectors
k = (m/M)K1 + (n/N)K2 + (p/P)K3
m=0,1,2,…(M-1)
n=0,1,2,…(N-1)
p=0,1,2,…(P-1)

16. A simple 2-D Example K2 b K1 a In 3-D
Reciprocal lattice of BCC  FCC !!

17. Now to find smallest unit cell
volume in k-space (Brillouin zone)
First let’s do the same in real
space (Wigner-Seitz cell)

18. How to pick a “Primitive” Unit cell?
Two atoms in
cell. Want to
take a smaller
‘primitive’ cell
which only
contains 1 atom
Many ways
to do this

19. How to pick a “Primitive” Unit cell?

20. How to pick a “Primitive” Unit cell?

21. How to pick a “Primitive” Unit cell?
The special ‘Wigner-
Seitz Cell’ is the one
that preserves the
translational/rotational
symmetry of the lattice
Join nearest neighbors
Bisect these lines
Join the bisectors

22. 3-D Wigner-Seitz Cell
Complicated, but it’s the smallest volume in real-space
with 1 atom in it
Join nearest neighbors with center atom, draw bisecting planes, and identify volume enclosed by them
To get smallest range of unique k-values for E-k,
just need to Fourier transform this!
ie,Wigner-Seitz in Fourier space  Brillouin Zone

23. Brillouin Zone of BCC  WS of FCC

24. Choose specific directions within
the BZ to plot E-k along

25. Real Materials more complex
Many orbitals per atom (many bands)
Multiple dimensions (3-D)
Let us look at bands for silicon E
K along X direction
(100, 010, 001 etc)

26. Real Materials more complex
Many orbitals per atom (many bands)
Multiple dimensions (3-D)
Let us look at bands for silicon E
L valley along (111) L G X

27. Real Materials more complex
Many orbitals per atom (many bands)
Multiple dimensions (3-D)
Let us look at bands for silicon E E L G X
Combine these into
1 figure

28. Now that we’ve identified k
directions,let’s choose a grid of
k points, and find the E-k

29. General prescriptions1
b basis sets per atom
Equation for mth atom 2 4
m[H]nm{fm} = E{fm} S
(bx1)
[H]nm 3
Try {fm} = {f0}eik.dm (Bloch’s Theorem)
Remember these are coeffs of f in atomic {um} basis
m[H]nmeik.(dm-dn) S
Solution E{f0} = [h(k)]{f0} where
h(k) =
n: any unit cell atom, m: all its nearest neighbors
Eigenvalues of h(k)  E(k) (bandstructure)

30. Some important materials
Cu: Metal

31. Some important materials
GaAs: Direct Bandgap

32. Some important materials
DOS E
Si: Indirect Bandgap

33. More on silicon

34. 2 FCC lattices interpenetrating,
displaced by ¼ the body diag 1 2 3
Where are the dimers??
Along body diagonal (111 dirn), atoms unequally placed
But along x-axis (100) atoms equally placed

35. With s orbitals alone, dimerization takes
care of gap in (111) direction, not (100)
To get gap in (100), need s and p orbitals,
since projection of p orbitals unequal
To get indirect bandgap, need s, p, s* orbitals

36. Approximations to bandstructure
Properties important near band tops/bottoms
Described through Constant Energy ellipsoids

37. Approximations to bandstructure
In contrast, valence bands are more warped and are hard to write
as parabolas. One uses 6 band k.p for instance.. hh lh