Solids and Bandstructure

Solids: From Bonds to Bands Atom Band Bond E Levels Molecule 1-D Solid

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

In the spirit of ‘bottom-up’ theory, we will identify minimal models to create metallic or semiconducting bands A simple 1-D chain of atoms with 1 electron/atom will yield a metal A chain of dimers with 1 electron/atom will yield a semi- conductor A real 3D solid will involve dimerization of atoms or orbitals

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 This is because our basis sets are localized on atoms and look like grid points

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 If we simply list the sorted eigenvalues vs their index, we get the plot below. Can we understand this analytically ?

How would we get multiple bands leading to a semiconductor?

Dimerized Chain -t2 -t1 e H = How would we solve this? -t1 e -t2 -t2 e -t1 H = How would we solve this? 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 N=2 4 6 8 10 20 50 500 t1=1, t2=0.5

Dimerized Chain -t2 -t1 e H = a = e -t1 -t1 e b = 0 0 t2 0 -t1 e -t2 -t2 e -t1 H = a = e -t1 -t1 e b = 0 0 t2 0 Take two atoms as one unit

Dimerized Chain -t2 e -t1 a -b H = a = e -t1 -t1 e -b† a b = 0 0 t2 0 Take two atoms as one unit

Dimerized Chain a b H = -b† a -b

Solving for the dispersion -b† a -b f1 f2 fn-1 fn fn+1 . f1 f2 fn-1 fn fn+1 . = E a = e -t1 -t1 e Let’s look at the nth row a[fn] – b[fn+1] – b†[fn-1] = E[fn] Periodic bcs imposed Try [fn] = [f0]eikna This gives a – beika – b+e-ika = EI2x2 b = 0 0 t2 0

Solving for the dispersion Substituting expressions for a, b etc gives E – e -t1-t2eika -t1-t2e-ika E - e = 0 E± = e ± √t12 + t22 – 2t1t2cos(ka) Eigenvalues:

Solving for the dispersion ka E Conduction Band Valence 2(t1-t2) 2t2 E+ E- E± = e ± √t12 + t22 – 2t1t2cos(ka)

Solving for the dispersion 2(t1-t2) 2t2 E+ E- E ka We now have a material with a bandgap So we just need a lattice with a basis to get multibands

Solving for the dispersion ka E 2(t1-t2) 2t2 E+ E- If parameters chosen properly, EF can lie in the gap (e.g. 1 electron per dimer atom) No states around the Fermi energy here  semiconductor

Semiconductor: Lattice + Basis We need a lattice with a basis (Note that we could have also made the onsite energies oscillate, and make that oscillation periodic but infrequent  Wells and barriers)

Atomic levels  Bands Deeper potential due to nuclear attraction effectively makes intraatomic ‘box’ width > interatomic separation, so that s-p separation < bonding-antibonding split

Why do we get a gap? Because a free electron cannot propagate at the Brillouin zone (“Bragg reflection”) Consider the dispersion of a free electron What happens at the Brillouin zones? E p/a -p/a k

Why do we get a gap? A forward propagating wave has wavelength l = 2p/k = 2a Its backward (reflected) wave has the same wavelength, but is ahead by DK = 2p/a, meaning it’s ahead by one lattice constant a E k p/a -p/a

Why do we get a gap? Because crests sit on troughs, the forward and reverse waves cancel, which is why there can be no propagating free electron states at the BZ The parabola must thus deform to open a gap at the BZ E p/a -p/a k

Other ways to get a gap Usual bandstructure theories don’t include strong Electron-electron interaction effects Add charging exactly (increases gap) eHOMO = EGN-1 – EGN eLUMO = EGN+1 - EGN U0 eHOMO eLUMO Bare levels in absence of charging

Metal-insulator transitions Levels broaden into a band with width D = 4t, where -t = ∫uN*HuN±1dV. As atoms moved apart, t decreases, until bandwidth D is smaller than U0 and Coulomb Blockade sets in. We then get a gap in an otherwise metallic band. U0 EF Conduction Band Valence EF “Mott Transition”

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)