ECE 663-1, Fall ‘08

Solids and Bandstructure

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

Energy bands emerge

~0.35

~1-1.35

~1.7-2.7

E/Ewell

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

The overall solution looks like

ECE 663-1, Fall ‘08

More accurately...

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

Nearly Free Electrons

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

Approximations to bandstructure

Properties important near band tops/bottoms

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

Els between bound and free

ECE 663-1, Fall ‘08

Band properties

Electronic wavefunctions overlapand their energies form bands

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

Electron and Hole fluxes

ECE 663-1, Fall ‘08

How does m* look?

ECE 663-1, Fall ‘08

Xal structure in 1D

(K: Fourier transform of real-space)

ECE 663-1, Fall ‘08

ECE 663-1, Fall ‘08

Bandstructure along -X direction

ECE 663-1, Fall ‘08

Bandstructure along -L direction

ECE 663-1, Fall ‘08

3D Bandstructures

ECE 663-1, Fall ‘08

GaAs Bandstructure

ECE 663-1, Fall ‘08

Constant Energy Surfaces for conduction band

Tensor effective mass

ECE 663-1, Fall ‘08

4-Valleys inside BZ of Ge

ECE 663-1, Fall ‘08

Valence band surfaces

These are warped (derived from ‘p’ orbitals)

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