1. A return to density of states & how to calculate energy bands
Discuss Drude and Sommerfield and said we have to consider periodicity. So, we looked at several crystal types, but actually we can learn a lot from just knowing there is periodicity, regardless of its exact form.
Making HW and test corrections due March 19

2. Learning Objectives for Today
After today’s class you should be able to:
Relate DOS to energy bands
Compare two main methods for calculating energy bands
Be able to use tight binding model to calculate energy bands
Understand basics of Dirac notation
Gaps may happen in the next lecture

3. (electrons in metals act free)
The nearly-free-electron model
1 electron per atom:
When EF is well away from a gap, dispersion is similar to free-electron case, but with slight change in curvature
(electrons in metals act free)
As we increase the number of electrons per atom (or per unit cell), EF moves up the dispersion relation. EF
dispersion relation

4. The nearly-free-electron model
When EF is close to or within a gap, major changes in the material properties occur...
Conduction Band
(LUMO) EF
Valence Band (HOMO)
HOMO: Highest Occupied Molecular Orbital
LUMO: Lowest Unoccupied Molecular Orbital

5. How DOS g(E) relates to Dispersion
The density-of-states curve
counts levels.
DOS curves plot the
distribution of electrons in energy
There are more states in a given energy interval at the top and bottom of this band.
In general, DOS(E) is proportional to the inverse of the slope of E(k) vs. k
The flatter the band, the greater the density of states at that energy.

6. To find the energy density of states g(E), we need:
DOS g(E) is the number of electron states per unit volume per unit energy at energy E
To find the energy density of states g(E), we need:
the density of states in k-space g(k) and the energy bands.
PbSe
Not quite so simple

7. Van Hove points are discontinuities of the first derivative of g(E)
Silicon
critical points of the Brillouin zone
Van Hove points are discontinuities of the first derivative of g(E)
Van Hove points are discontinuities of the first derivative of D(E), also know as critical points of the B.Z. (no relation to critical points in phase diagrams)
Often where the interesting physics occurs (such as optical absorption)

8. How would you experimentally determine the density of states?

9. X-ray Photoemission Spectroscopy
Like a fancy photoelectric effect

10. How would you theoretically determine the energy bands?
Lots of ways!

11. Methods to calculate bandstructures
Solve the Schrodinger Equation and apply the Bloch theorem
Simplify the complicated crystal potential to something solvable. E.g. Kronig-Penney model.
Treat the complicated crystal potential as a sum of a simpler potential (solvable Schrodinger Eqn) and potential perturbation. E.g. near free-electron model (plane waves + perturbation), k.p perturbation theory and tight-binding model (atomic orbitals + perturbation).
Numerical methods, e.g. density function theory and quantum Monte Carlo
For more information, see the following websites for instance,

12. Two main models for electron distribution:
The main property of solids that determines their electrical properties is the distribution of their electrons.
Two main models for electron distribution:
1) Nearly Free Electron Approximation
Valence electrons are assumed trapped in a box (the sample) with a periodic potential
2) The Tight Binding Approximation
Valence electrons are assumed to occupy molecular orbitals delocalized throughout the solid

13. Summary of Nearly Free Electron Model (What We’ve Already Done!)
Electrons nearly free due to very large overlap (opposite from TB)
Wave functions approximated by plane waves (free electrons)
Assume energy is unchanged and solve for 1st order correction
Works for upper states even if tightly bound electrons in lower states

14. My Summary of the Two Main Approaches
Nearly free e-’s + pseudopotential
Electrons nearly free due (opposite from TB)
Wavefunctions ~ plane waves
Assume energy is unchanged and correct to first order
Works for upper states even if TB electrons in lower states
Tight-binding or LCMO
Assume some electrons independent of each other (often true, tight core)
Linear combination of atomic wave functions (Wannier)
Each Wannier function is equal to the unperturbed atomic orbital (LCAO approx)

15. Comparing Chemists and Physicists
Calculation theories fall into 2 general categories, which have their roots in 2 qualitatively very different physical pictures for e- in solids (earlier):
“Physicist’s View” - Start from an “almost free” e- & add the periodic potential
Nearly free electrons, Pseudopotential methods
“Chemist’s View” - Start with atomic energy levels & build up the periodic solid by decreasing distance between atoms
Now, we’ll focus on the 2nd method.

16. Method #2 (Qualitative Physical Picture #2) “The Chemists’ Viewpoint”
Start with the atomic/molecular picture of a solid.
The atomic energy levels merge to form molecular levels, & merge to form bands as periodic interatomic interaction V turns on.
TIGHTBINDING or
Linear Combination of Atomic Orbitals (LCAO) method.
This method gives good bands, especially valence bands!
The valence bands are almost the same as those from the pseudopotential method! Conduction bands are not so good because electrons act free!

17. The Tightbinding Method
Some believe the Tightbinding / LCAO method gives a clearer physical picture (than pseudopotential method does) of the causes of the bands & the gaps.
In this method, the periodic potential V is discussed as in terms of an Overlap Interaction of the electrons on neighboring atoms.
As we’ll see, we can define these interactions in terms of a small number of parameters.

18. Tightbinding/LCAO
Assume the atomic orbitals ~ unchanged
bare atoms
solid
Atomic energy levels merge to form molecular levels & merge to form bands as periodic interatomic interaction V turns on.

19. Covalent Bonding Revisited
When atoms are covalently bonded electrons are shared by atoms
Example: the ground state of the hydrogen atoms forming a molecule
If atoms far apart, little overlap
If atoms are brought together the wavefunctions overlap and form the compound wavefunction, ψ1(r)+ψ2(r), increasing the probability for electrons to exist between atoms
These two possible combinations represent 2 possible states of two atoms system with different energies

20. LCAO: Electron in Hydrogen Atom (in Ground State)
Second hydrogen atom
Approximation: Only Nearest-Neighbor interactions

21. Chain of 5 H atoms H2 Molecule c0 c1 c2 c3 c4 E Antibonding 4 c1 c2 3
Nonbonding 2 1
Bonding
# of Nodes

22. Group: For 0, 2 and 4 nodes, determine wavefunctions
k=0 c0 c1 c2 c3 c4
k=p/a
k=p/2a a
If there are N atoms in the chain there will be N energy levels and N electronic states (molecular orbits). The wavefunction for each electronic state is:
Yk = S eiknacn
Where:
a is the lattice constant,
n identifies the individual atoms within the chain,
cn represents the atomic orbitals
k is a quantum # that identifies the wavefunction and tells us the phase of the orbitals.
The larger the absolute value of k, the more nodes one has

23. Infinite 1D Chain of H atoms
k = p/a
Yp/a = c0+(exp{ip})c1 +(exp{i2p})c2 +(exp{i3p})c3+(exp{i4p})c4+…
Yp/a = c0 - c1 + c2 - c3 + c4 +…
k=0 c0 c1 c2 c3 c4
k=p/a
k=p/2a a
k = p/2a
Yp/2a = c0+(exp{ip/2})c1 +(exp{ip})c2 +(exp{i3p/2})c3+(exp{i2p})c4+…
Yp/2a = c c c4 +…
k = 0
Y0 = c0+c1 +c2 +c3 +c4 +…
k=0  orbital phase does not change when we translate by a
k=p/a  orbital phase reverses when we translate by a

24. Infinite 1D Chain of H atoms
What would happen if consider k> p/a?
If not obvious, try in groups k=2 p/a. What is the wavefunction?
Yk = S eiknacn

25. LCAO +Bloch notation Where:
 & potential U(r)=U(r+Na) obeys the Born-von Karman condition
Where:
Combining the Bloch theorem and the above gives that
Let’s simplify by considering just two states. (Dirac notation)
This c constant is the same as what we got from doing the one dimensional ring.
The |ket> notation is like a vector.
The collection of all functions of x constitutes a vector space.
But to present a possible physical state, the wave functions must be normalized:

26. Brief Summary of Dirac Notation
Dirac Notation Wave Mechanics (Position Space)
Dirac ket: |(t) System State (x,t) wavefunction
Linear operator A Measurement A differential or
multiplicative operator
Eigenvalue Equation
Expectation Value (average of many identical measurements)
eigenvalue
 gives possible results of a measurement
eigenstate or eigenket eigenfunction
 gives probability of measurement result an

27. Dirac Notation with LCAO Approximation
Dirac Notation for 2 atoms:
If we assume little overlap, can expand to
and
Or expanding 
This is a good lead into LCAO theory.
Eigenvalue problem with two solutions. E1 and E2 are the unperturbed atom energies. Cross term is the overlap.

28. Lower energy result is the bonding state
Solution:
Lower energy result is the bonding state
V12 is overlap integral

29. Example similar to homework
Find the energies at the H point of BCC. H

30. How do we plot the Empty Lattice Bands?
The limit of a vanishing potential is called the “empty lattice”, and the empty-lattice bands are often plotted for comparison with the energy bands of real solids.
Here plotted in the reduced zone scheme (translations back into the 1st BZ).

31. Example: 1D Empty Lattice
V  0:
We assume a periodicity of a. Define the reciprocal lattice constant G = 2p / a. We can therefore restrict k within the range of [-G/2, G/2].
Sorry G=K
From Bloch’s treatment
Bloch’s theorem
implies

32. Free Electrons in 1D
V  0:
Where k1 is a wavevector lying in the 1st BZ.
The symmetry of the reciprocal lattice requires:
The  sign is redundant.

33. Empty Lattice Bands for bcc Lattice
For the bcc lattice, let’s plot the empty lattice bands along the [100] direction in reciprocal space.
General reciprocal lattice translation vector:
Let’s use a simple cubic lattice, for which the reciprocal lattice is also simple cubic:
Could also use BCC primitive lattice, but would be a little more complex to understand.
And thus the general reciprocal lattice translation vector is:

34. Energy Bands in BCC
We write the reciprocal lattice vectors that lie in the 1st BZ as:
The maximum value(s) of x, y, and z depend on the reciprocal lattice type and the direction within the 1st BZ. For example:
[100]
0 < x < 1
[110]
0 < x < ½, 0 < y < ½ ky kx  H N
Remember that the reciprocal lattice for a bcc direct lattice is fcc!
Here is a top view, from the + kz direction:

35. Group: Plot the Empty Lattice Bands for bcc Lattice
Thus the empty lattice energy bands are given by:
Along [100], we can enumerate the lowest few bands for the y = z = 0 case, using only G vectors that have nonzero structure factors (h + k + l = even, otherwise S=0):
{G} = {000}
{G} = {110}
{G} = {200}

36. Empty Lattice Bands for bcc Lattice: Results
Thus the lowest energy empty lattice energy bands along the [100] direction for the bcc lattice are:

37. Summary Band Structures
What is being plotted? Energy vs. k, where k is the wavevector that gives the phase as well as the wavelength of the electron wavefunction (crystal momentum).
How many lines are there in a band structure diagram? As many as there are orbitals in the unit cell.
How do we determine whether a band runs uphill or downhill? By comparing the orbital overlap at k=0 and k=p/a.
How do we distinguish metals from semiconductors and insulators? The Fermi level cuts a band in a metal, whereas there is a gap between the filled and empty states in a semiconductor.
Why are some bands flat and others steep? This depends on the degree of orbital overlap between building units.
Wide bands  Large intermolecular overlap  delocalized e-
Narrow bands  Weak intermolecular overlap  localized e-

38. How the energies split as we increase N
Energy levels get closer as N increases.
Degenerate pairs, except for ground state.

39. Energy Bands and Fermi Surfaces in 2-D Square Lattice
Reminder: the Brillouin zones of the reciprocal lattice can be identified with a simple construction:
The 1st BZ is defined as the set of points reached from the origin without crossing any Bragg planes.
2 /a 3 2 1
A truly free electron system would have a Fermi circle to define the locus of states at the Fermi energy.

40. Group Problem
Show for a square lattice (2D) that the kinetic energy at the corner of the 1st BZ is larger than that of an electron at the midpoint of a side face. By how much?
What is the corresponding factor in 3D?
Draw the energy bands from the zone center to these to points on the boundary.
What bearing might this have on the conductivity of divalent metals?
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