Post on 25-Dec-2015
ECE 663-1, Fall ‘08
QM of solids
QM interference creates bandgaps and separatesmetals from insulators and semiconductors
ECE 663-1, Fall ‘08
Recall numerical trick
xn-1 xn xn+1
n-1 n n+1
-t Un-1+2t -t
H = -t Un+2t -t
-t Un+1+2t -t
t = ħ2/2ma2
-t
-t
Periodic BCsH(1,N)=H(N,1)=-t
ECE 663-1, Fall ‘08
Extend now to infinite chain
1-D Solid
-t -t
-t -t
-t -tH =
-t
Onsite energy (2t+U)-t: Coupling (off-diag. comp. of kinetic energy)
ECE 663-1, Fall ‘08
Extend now to infinite chain
1-D Solid
-t -t
-t -t
-t -tH =
-t
Let’s now find the eigenvaluesof H for different matrix sizes N
ECE 663-1, Fall ‘08
Eigenspectra
N=2 4 6 8 10 20 50 500
If we simply find eigenvalues of each NxN [H] and plot them in a sortedfashion, 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
ECE 663-1, Fall ‘08
Eigenspectra
If we simply list the sorted eigenvalues vs their index, we getthe plot below showing a continuous band of energies.
How do we get a gap?
ECE 663-1, Fall ‘08
Dimerized Chain
H =
-t1 -t2
-t2 -t1
-t1 -t2
-t2 -t1
-t2
-t1
-t1
Once again, let’s do this numerically for various sized H
ECE 663-1, Fall ‘08
Eigenspectra
t1=1, t2=0.5
N=2 4 6 8 10 20 50 500
If we keep the t’s different, two bands and a bandgap emerges
Bandgap
ECE 663-1, Fall ‘08
One way to create oscillations
+ + + +
Periodic nuclear potential(Kronig-Penney Model)
Simpler abstraction
ECE 663-1, Fall ‘08
Solve numerically
Un=Ewell/2[sign(sin(n/(N/(2*pi*periods))))+1];
Like Ptcle in a boxbut does not vanishat ends
ECE 663-1, Fall ‘08
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([1 80 0 3])• figure(2)• plot(n,Un);• %axis([0 500 -0.1 2])• • hold on• • for k=1:N• plot(n,real(v(:,k))+d(k)/Ewell,'k','linewidth',3);• hold on• axis([0 500 -0.1 3])• end
ECE 663-1, Fall ‘08
Bloch’s theorem
(x) = eikxu(x)
u(x+a+b) = u(x)
Plane wave part
eikx
handles overall X-alPeriodicity
‘Atomic’ part u(x)handles local
bumpsand wiggles
(x+a+b) = eik(a+b)(x)
ECE 663-1, Fall ‘08
Can do this analytically, if we can survive the algebra
N domains2N unknowns (A, B, C, Ds)
Usual procedureMatch , d/dx at each of the N-1 interfaces(x ∞) = 0
ECE 663-1, Fall ‘08
Can’t we exploit periodicity?
Bloch’s Theorem
This means we can work over 1 period alone!
Need periodic BCs at edgesSolve transcendental equations graphically
ECE 663-1, Fall ‘08
Allowed energies appear in bands !
Like earlier, but folded into -/(a+b) < k < /(a+b)
The graphical equation:Solutions subtended between black curve and red lines
ECE 663-1, Fall ‘08
Number of states and Brillouin Zone
Only need points within BZ(outside, states repeatthemselves on the atomic grid)
ECE 663-1, Fall ‘08
Why do we get a gap?
E
k/a-/a
At the interface (BZ), we have two counter-propagating waves eikx,
with k = /a, that Bragg reflect and form standing waves
Its periodicallyextended partner
Let us start with a free electron in a periodic crystal,but ignore the atomic potentials for now
ECE 663-1, Fall ‘08
Why do we get a gap?
E
k/a-/a
-+
Its periodicallyextended partner
+ ~ cos(x/a) peaks at atomic sites
- ~ sin(x/a) peaks in between
ECE 663-1, Fall ‘08
Let’s now turn on the atomic potential
The + solution sees the atomic potential and increases its energy
The - solution does not see this potential (as it lies between atoms)
Thus their energies separate and a gap appears at the BZ
k/a-/a
+
-
|U0|
This happens only at the BZ where we have standing waves
ECE 663-1, Fall ‘08
What is the real-space velocity?
Superposition of nearby Bloch waves
(x) ≈ Aei(kx-Et/ħ) + Aei[(k+k)x-(E+E)t/ħ]
≈ Aei(kx-Et/ħ)[1 + ei(kx-Et/ħ)]Fast varyingcomponents
Slowly varyingenvelope (‘beats’)
k
k+k
time
ECE 663-1, Fall ‘08
Band velocity
(x) ≈ Aei(kx-Et/ħ)[1 + ei(kx-Et/ħ)]
Envelope (wavepacket) moves at speed v = E/ħk = 1/ħ(∂E/∂k)
i.e., Slope of E-k gives real-space velocity
ECE 663-1, Fall ‘08
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
Flat bands
ECE 663-1, Fall ‘08
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*
ECE 663-1, Fall ‘08
What does Effective mass mean?
1/m* = 1/ħ2(∂2E/∂k2)
Recall this is not a free particle butone moving in a periodic potential.
But it looks like a free particle near the band-edges, albeit with an effective massthat parametrizes the difficulty faced bythe electron in running thro’ the potential
m* can be positive, negative, 0 or infinity!
ECE 663-1, Fall ‘08
Band properties
Electronic wavefunctions overlapand their energies form bands
http://fermi.la.asu.edu/ccli/applets/kp/kp.html
ECE 663-1, Fall ‘08
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)
ECE 663-1, Fall ‘08
• Nearly free-electron model, Au, Ag, Al,...
Parabolic electron bands distort near BZto open bandgaps (slide 32)
• Tight-binding electrons, Fe, Co, Pd, Pt, ...
Localized atomic states spill over so that theirdiscrete energies expand into bands
(slides 9, 38)
Two opposite limits invoked to describe bands
ECE 663-1, Fall ‘08
(For every positive J2 or J3 component, there is an equal
negative one!)
Electron and Hole fluxes
ECE 663-1, Fall ‘08
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