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. Recall how we went to bonds
Here ‘grid’ (or basis sets)
are hydrogen 1s orbitals
u1(r) and u2(r) with scalar
coefficients f1 and f2
xn-1 xn
xn+1 f
fn-1 fn
fn+1
Just as a real-space finite-diff grid gives a wave
function that involves coefficients fn at diff points,
this involves coefficients fn for different atomic sites
f ≈ ∑ fnun(r) n

6. Recall how we went to bonds
In this atomic basis set (rather than
grid basis set), we calculate H, which
becomes a matrix. The diagonal elements
= H11 =
diagonal elements
-t = ∫u1(r)Hu2(r)dV
∫u1(r)Hu1(r)dV, while the off-
e t
-t e
H =
in this atomic basis set
Eigenvalues e ± t (ignoring non-orthogonality)
Eigenvector coefficients f1,2 = ±1/√2, ie, (u1 u2)/√2
Lower energy state occupied by electrons

7. Extend now to infinite chain
1-D Solid
-t e -t
-t e -t
H =
e: Onsite energy (2t+U)
-t: Coupling (off-diag. comp.
of kinetic energy)
Note how this looks like the
Finite-element H !

8. 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

9. 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=

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

11. Simplify algebra E k k -t(fn+1 + fn-1 -2fn) = Efn
The results are not sensitive to boundary conditions, as seen above
But periodic bcs make the algebra simple.
Let’s now look at the equation for the nth row.
-t(fn+1 + fn-1 -2fn) = Efn
Periodicity allows us to try a solution fn = f0eikna,
substituting which gives us E = 2t[1-coska]

12. Band properties E = 2t[1-coska] gives us a band of states.
(a) Need to go only upto ka = p (Brillouin Zone)
(b) Band bottom at k=0 (value 0) and top at ka=p (value 4t)
(c) Bandwidth = 4t depends on overlap between atoms x1 x2 x3 x4 f1 f2 f3 f4
f(x)
Equiv states
as far as
coeffs go
2p/a

13. Band properties + + + +
Periodic nuclear potential

14. Band properties Electronic wavefunctions overlap
and their energies form bands

15. Band properties Shallower potentials give bigger overlaps.
Greater overlap creates greater bonding-antibonding splitting of
each multiply degenerate level, creating wider bandwidths
Since shallower potentials allow electrons to escape easier,
they correspond to smaller effective mass
Thus, effective mass ~ 1/bandwidth ~ 1/t (t: overlap)

16. Effective Mass Near bottom of band, can fit a parabola
So Effective Mass m* = ħ2/(∂2E/∂k2) ~ 1/curvature of E-k
In case of our 1-D band E=2t(1-coska), m* = ħ2/2ta2 (~ 1/t, as argued earlier)
Electronic properties at low bias depend on these effective masses
Eg. GaAs m* = me, Si ml*=0.91me, mt* = 0.19 me
But instead of using a free electron mass
to calculate t (chapter 2), we are using a
differently computed t (overlap of basis
sets un) to extract m*. This does not
need to match the free electron mass!!
In fact it incorporates nuclear attraction
terms effectively.

17. Effective Mass Effective Mass m* = ħ2/(∂2E/∂k2)
Larger E-k curvature means smaller m*
(Steeper bowl in k-space gives mobile electrons)
Is that sensible?
Recall that x and k are Fourier transforms
This means that a steeper parabola in k-space
actually corresponds to a shallower bowl-shaped
potential in real space, which an electron can
easily get out of. This corresponds to a more
mobile electron with a smaller m* and a larger
band-width t.

18. Approximations to bandstructure
Properties important near band tops/bottoms

19. Filling these band states
Periodic bcs mean exp(ikL) = exp(ikNa) = 1
 kL = 2pn (n=0,1,2,…,N-1)
N sites give N states, and N electrons

20. Filling these band states
But each state can accommodate two electrons
(up and down spins)
So only half the states are filled
EF near band middle where lot of states exist
 METAL
N sites give N states, and N electrons

21. What do the wavefunctions look like?
≈ ∑ fnun(r)
with
fn = f0eikna x1 x2 x3 x4 f1 f2 f3 f4
Coeffs
for a particular
eigenstate with a
Specific k
u(x) u1 u2
un-1 un f2 f4 x1 x2 x3 x4

22. What do the wavefunctions look like?
≈ ∑ fnun(r)
with
fn = f0eikna
f(x) x1 x2 x3 x4
Bloch solutions
(part ‘atomic’ through u’s and part
plane wave through fns)

23. Els between bound and free

24. need two diff basis sets (e.g. s and px), or + + -
This flips the sign of t
To get both bands,
need two diff basis sets
(e.g. s and px), or
equivalently, two different
types of atoms E k
Valence
Band E k
Conduction
Band

25. Kronig-Penney Model +
Periodic nuclear potential
Simpler abstraction

26. Kronig-Penney Model N domains 2N unknowns (A, B, C, Ds)
Usual procedure
Match y, dy/dx at each of the N-1 interfaces
y(x  ∞) = 0

27. Can’t we exploit periodicity?
Bloch’s Theorem
This means we can work over 1 period alone!

28. Match BCs and Bloch’s Theorem for piece-meal solutions

29. Working thro’ the algebra patiently...

30. Set Determinant of coeffs to zero

31. Let’s plot this
k=p/(a+b)

32. Allowed energies appear in bands !
Like earlier,
but folded into
-p/(a+b) < k < p/(a+b)

33. Solve numerically Un=Ewell/2[sign(sin(n/(N/(2*pi*periods))))+1];
Like Ptcle in a box
but does not vanish
at ends

34. Bloch’s theorem y(x) = eikxu(x) y(x+a+b) = eik(a+b)y(x)
u(x+a+b) = u(x)
y(x+a+b) = eik(a+b)y(x)
Plane wave part eikx
handles overall Xal
Periodicity
‘Atomic’ part u(x)
handles local bumps
and wiggles

35. Energy bands emerge
E/Ewell ~
~1-1.35
~0.35

36. What about edges of Xal? Periodic BCs at edges
Wavefunction must come back full circle
Using Bloch’s theorem,
eik(a+b)N = 1

37. What about edges of Xal? Periodic BCs at edges eik(a+b)N = 1
k = 2pn/(a+b)N,
n=-N/2,...,-1,0,1,...,N/2
N discrete k points for N unit cells
In practice, N large, so continuous k

38. Number of states and Brillouin Zone
Only need points within BZ
(outside, states repeat
themselves on the atomic grid)

39. The overall solution looks like

40. More accurately...

41. What is the real-space velocity?
Superposition of nearby Bloch waves
y(x) ≈ Aei(kx-Et/ħ) + Aei[(k+Dk)x-(E+DE)t/ħ]
≈ Aei(kx-Et/ħ)[1 + ei(Dkx-DEt/ħ)]
Fast varying
components
Slowly varying
envelope (‘beats’)

42. Band velocity Aei(kx-Et/ħ)[1 + ei(Dkx-DEt/ħ)] y(x) ≈
Envelope (wavepacket) moves at speed v = DE/ħDk = 1/ħ(∂E/∂k)
i.e., Slope of E-k gives real-space velocity

43. Band velocity v = 1/ħ(∂E/∂k) Slope of E-k gives real-space velocity
This explains band-gap too!
Two counterpropagating waves give
zero net group velocity at BZ
Since zero velocity means flat-band, the
free electron parabola must distort at BZ
Flat
bands

44. Effective mass v = 1/ħ(∂E/∂k), p = ħk F = dp/dt = d(ħk)/dt
a = dv/dt = (dv/dk).(dk/dt)
= 1/ħ2(∂2E/∂k2).F
1/m* = 1/ħ2(∂2E/∂k2)
Curvature of E-k gives m*