Solids and Bandstructure

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

Recall numerical trick y 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

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 =

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

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=2 4 6 8 10 20 50 500

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?

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

Eigenspectra If we keep the t’s different, two bands and a bandgap emerges Bandgap N=2 4 6 8 10 20 50 500 t1=1, t2=0.5

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

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

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); axis([0 500 -0.1 3]) end

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

Energy bands emerge E/Ewell ~1.7-2.7 ~1-1.35 ~0.35

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

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

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

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

The overall solution looks like

More accurately...

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

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

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

Nearly Free Electrons

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

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

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

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*

Approximations to bandstructure Properties important near band tops/bottoms

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!

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

Els between bound and free

Band properties Electronic wavefunctions overlap and their energies form bands

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)

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)

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

Electron and Hole fluxes

How does m* look?

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

Bandstructure along G-X direction

Bandstructure along G-L direction

3D Bandstructures

GaAs Bandstructure

Constant Energy Surfaces for conduction band Tensor effective mass

4-Valleys inside BZ of Ge

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

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