1. Solids and Bandstructure

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

3. Recall numerical tricky
yn-1 yn
yn+1
xn-1 xn
xn+1 -t
-t Un-1+2t -t
t = ħ2/2ma2
H =
-t Un+2t -t
-t Un+1+2t -t
Periodic BCs
H(1,N)=H(N,1)=-t

4. Extend now to infinite chain
1-D Solid
e: Onsite energy (2t+U)
-t: Coupling (off-diag. comp.
of kinetic energy)
-t e -t
-t e -t
H =

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

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 How do we get a gap?
If we simply list the sorted eigenvalues vs their index, we get
the plot below showing a continuous band of energies.
How do we get a gap?

8. Dimerized Chain -t2 -t1 e H =
-t1 e -t2
-t2 e -t1
H =
Once again, let’s do this numerically for various sized H

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

10. One way to create oscillations+
Periodic nuclear potential
(Kronig-Penney Model)
Simpler abstraction

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

12. Matlab code hbar=1.054e-34;m=9.1e-31;q=1.6e-19;ang=1e-10; Ewell=10;
alpha0=sqrt(2*m*Ewell*q/hbar^2)*ang;
period=2*pi/alpha0;
periods=25;span=periods*period;
N=505;a=span/(N+0.3);
t0=hbar^2/(2*m*q*(a*ang)^2);
n=linspace(1,N,N);
Un=Ewell/2*(sign(sin(n/(N/(2*pi*periods))))+1);
H=diag(Un)+2*t0*eye(N)-t0*diag(ones(1,N-1),1)-t0*diag(ones(1,N-1),-1);
H(1,N)=-t0;H(N,1)=-t0;
[v,d]=eig(H);
[d,ind]=sort(real(diag(d)));v=v(:,ind);
% figure(1)
% plot(d/Ewell,'d','linewidth',3)
% grid on
% axis([ ])
figure(2)
plot(n,Un);
%axis([ ])
hold on
for k=1:N
plot(n,real(v(:,k))+d(k)/Ewell,'k','linewidth',3);
axis([ ])
end

13. 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 X-al
Periodicity
‘Atomic’ part u(x)
handles local bumps
and wiggles

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

15. Can do this analytically, if we can survive the algebra
N domains
2N unknowns (A, B, C, Ds)
Usual procedure
Match y, dy/dx at each of the N-1 interfaces
y(x  ∞) = 0

16. Can’t we exploit periodicity?
Bloch’s Theorem
This means we can work over 1 period alone!
Need periodic BCs at edges
Solve transcendental equations graphically

17. Allowed energies appear in bands !
Like earlier, but folded
into -p/(a+b) < k < p/(a+b)
The graphical equation:
Solutions subtended between
black curve and red lines

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

19. The overall solution looks like

20. More accurately...

21. Why do we get a gap?
Let us start with a free electron in a periodic crystal,
but ignore the atomic potentials for now
At the interface (BZ), we have two counter-propagating waves eikx,
with k = p/a, that Bragg reflect and form standing waves y E
p/a
-p/a
Its periodically
extended partner k

22. Why do we get a gap? y+ y- y+ ~ cos(px/a) peaks at atomic sites
y- ~ sin(px/a) peaks in between E
p/a
-p/a
Its periodically
extended partner k

23. Let’s now turn on the atomic potential
The y+ solution sees the atomic potential and increases its energy
The y- solution does not see this potential (as it lies between atoms)
Thus their energies separate and a gap appears at the BZ
This happens only at the BZ where we have standing waves
p/a
-p/a y+ y-
|U0| k

24. Nearly Free Electrons

25. 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’) k
k+Dk
time

26. 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

27. 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

28. 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*

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

30. What does Effective mass mean?
1/m* = 1/ħ2(∂2E/∂k2)
Recall this is not a free particle but
one moving in a periodic potential.
But it looks like a free particle near the
band-edges, albeit with an effective mass
that parametrizes the difficulty faced by
the electron in running thro’ the potential
m* can be positive, negative, 0 or infinity!

31. Band properties http://fermi.la.asu.edu/ccli/applets/kp/kp.html
Electronic wavefunctions overlap
and their energies form bands

32. Els between bound and free

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

34. 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)

35. 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)

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

37. Electron and Hole fluxes

38. How does m* look?

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

41. Bandstructure along G-X direction

42. Bandstructure along G-L direction

43. 3D Bandstructures

44. GaAs Bandstructure

45. Constant Energy Surfaces for conduction band
Tensor effective mass

46. 4-Valleys inside BZ of Ge

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

48. In summary Solution of Schrodinger equation tractable for
electrons in 1-D periodic potentials
Electrons can only sit in specific energy bands. Effective mass and bandgap parametrize these states.
Only a few bands (conduction and valence) contribute to conduction.
Higher-d bands harder to visualize. Const energy ellipsoids help visualize where electrons sit