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Solids and Bandstructure

Solids and Bandstructure. QM of solids. QM interference creates bandgaps and separates metals from insulators and semiconductors. Recall numerical trick. -t. Periodic BCs H(1,N)=H(N,1)=-t. -t. y. y n-1. y n. y n+1. x n-1. x n. x n+1. -t U n-1 +2t -t. t = ħ 2 /2ma 2. H =.

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Solids and Bandstructure

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  1. Solids and Bandstructure

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

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

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

  5. -t e e -t e -t -t e -t H = -t e -t 1-D Solid Extend now to infinite chain Let’s now find the eigenvalues of H for different matrix sizes N

  6. N=2 4 6 8 10 20 50 500 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

  7. Eigenspectra 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. -t2 -t1 -t1 e e e e -t2e -t1 -t1e -t2 -t2e -t1 -t1e -t2 Dimerized Chain H = Once again, let’s do this numerically for various sized H

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

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

  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([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

  13. Bloch’s theorem y(x) = eikxu(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 ~1.7-2.7 ~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. Like earlier, but folded into -p/(a+b) < k < p/(a+b) Allowed energies appear in bands ! 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. At the interface (BZ), we have two counter-propagating waves eikx, with k = p/a, that Bragg reflect and form standing waves y -p/a p/a Why do we get a gap? Let us start with a free electron in a periodic crystal, but ignore the atomic potentials for now E Its periodically extended partner k

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

  23. y+ |U0| y- -p/a p/a 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 k

  24. Nearly Free Electrons

  25. k+Dk time k 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’)

  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. Flat bands Flat bands 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

  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. http://fermi.la.asu.edu/ccli/applets/kp/kp.html Band properties 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)

  40. Bandstructure along G-X direction

  41. Bandstructure along G-L direction

  42. 3D Bandstructures

  43. GaAs Bandstructure

  44. Constant Energy Surfaces for conduction band Tensor effective mass

  45. 4-Valleys inside BZ of Ge

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

  47. 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

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