1. Solids and Bandstructure

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

3. QM of solids QM interference creates bandgaps and separates
metals from insulators and semiconductors

4. In the spirit of ‘bottom-up’ theory, we will identify
minimal models to create metallic or semiconducting
bands
A simple 1-D chain of atoms with 1 electron/atom will yield
a metal
A chain of dimers with 1 electron/atom will yield a semi-
conductor
A real 3D solid will involve dimerization of atoms or orbitals

5. Extend now to infinite chain
1-D Solid
-t e -t
-t e -t
H =
Let’s now find the eigenvalues
of H for different matrix sizes N
This is because our basis sets
are localized on atoms and look like grid points

6. Eigenspectra
If we simply find eigenvalues of each NxN [H] and plot them in a sorted
fashion, a band emerges!
Note that it extends over a band-width of 4t (here t=1).
The number of eigenvalues equals the size of [H]
Note also that the energies bunch up near the edges, creating large DOS there N=

7. Eigenspectra
If we simply list the sorted eigenvalues vs their index, we get
the plot below. Can we understand this analytically ?

8. How would we get
multiple bands leading
to a semiconductor?

9. Dimerized Chain -t2 -t1 e H = How would we solve this?
-t1 e -t2
-t2 e -t1
H =
How would we solve this?
Once again, let’s do this numerically for various sized H

10. Eigenspectra
If we keep the t’s different, two bands and a bandgap emerges N=
t1=1, t2=0.5

11. Dimerized Chain -t2 -t1 e H = a = e -t1 -t1 e b = 0 0 t2 0
-t1 e -t2
-t2 e -t1
H =
a = e -t1
-t1 e
b =
t2 0
Take two atoms as one unit

12. Dimerized Chain -t2 e -t1 a -b H = a = e -t1 -t1 e -b† a b = 0 0 t2 0
Take two atoms as one unit

13. Dimerized Chain a b
H =
-b† a -b

14. Solving for the dispersion
-b† a -b f1 f2
fn-1 fn
fn+1 . f1 f2
fn-1 fn
fn+1 .
= E
a = e -t1
-t1 e
Let’s look at the nth row
a[fn] – b[fn+1] – b†[fn-1] = E[fn]
Periodic bcs imposed
Try [fn] = [f0]eikna
This gives a – beika – b+e-ika = EI2x2
b =
t2 0

15. Solving for the dispersion
Substituting expressions for a, b etc gives
E – e t1-t2eika
-t1-t2e-ika E - e
= 0
E± = e ± √t12 + t22 – 2t1t2cos(ka)
Eigenvalues:

16. Solving for the dispersionka E
Conduction
Band
Valence
2(t1-t2)
2t2 E+ E-
E± = e ± √t12 + t22 – 2t1t2cos(ka)

17. Solving for the dispersion
2(t1-t2)
2t2 E+ E- E ka
We now have a material with a bandgap
So we just need a lattice with a basis to get multibands

18. Solving for the dispersionka E
2(t1-t2)
2t2 E+ E-
If parameters chosen properly, EF can lie in the gap
(e.g. 1 electron per dimer atom)
No states around the Fermi energy here
 semiconductor

19. Semiconductor: Lattice + Basis
We need a lattice with a basis
(Note that we could have also made the onsite
energies oscillate, and make that oscillation
periodic but infrequent  Wells and barriers)

20. Atomic levels  Bands
Deeper potential due to nuclear attraction effectively makes
intraatomic ‘box’ width > interatomic separation, so that s-p
separation < bonding-antibonding split

21. Why do we get a gap?
Because a free electron cannot propagate at the Brillouin zone
(“Bragg reflection”)
Consider the dispersion of a free electron
What happens at the Brillouin zones? E
p/a
-p/a k

22. Why do we get a gap? A forward propagating wave has
wavelength l = 2p/k = 2a
Its backward (reflected) wave has
the same wavelength, but is ahead
by DK = 2p/a, meaning it’s ahead by
one lattice constant a E k
p/a
-p/a

23. Why do we get a gap? Because crests sit on troughs,
the forward and reverse waves
cancel, which is why there can be
no propagating free electron states
at the BZ
The parabola must thus deform to
open a gap at the BZ E
p/a
-p/a k

24. Other ways to get a gap
Usual bandstructure theories don’t include strong
Electron-electron interaction effects
Add charging
exactly
(increases gap)
eHOMO = EGN-1 – EGN
eLUMO = EGN+1 - EGN U0
eHOMO
eLUMO
Bare levels in
absence of charging

25. Metal-insulator transitions
Levels broaden into a band with width D = 4t, where
-t = ∫uN*HuN±1dV. As atoms moved apart, t decreases,
until bandwidth D is smaller than U0 and Coulomb Blockade
sets in. We then get a gap in an otherwise metallic band. U0 EF
Conduction
Band
Valence EF
“Mott Transition”

26. Two opposite limits invoked to describe bands
Nearly free-electron model, Au, Ag, Al,...
Parabolic electron bands distort near BZ
to open bandgaps (slide 32)
Tight-binding electrons, Fe, Co, Pd, Pt, ...
Localized atomic states spill over so that their
discrete energies expand into bands
(slides 9, 38)

27. Electron and Hole fluxes
(For every positive J2 or J3 component, there is an equal
negative one!)

28. Electron and Hole fluxes

29. How does m* look?

30. Xal structure in 1D
(K: Fourier transform of real-space)

32. Bandstructure along G-X direction

33. Bandstructure along G-L direction

34. 3D Bandstructures

35. GaAs Bandstructure

36. Constant Energy Surfaces for conduction band
Tensor effective mass

37. 4-Valleys inside BZ of Ge

38. Valence band surfaces
These are warped (derived from ‘p’ orbitals)