Solid State Physics I
PY 741 Homework 2
(1) One-dimensional model of ionic solids ABABABA
n-2 n-2 n-1 n-1 n n n+1 dd
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A composite 1-dimensional crystal of period a = 2d consisting of two types of atoms
The periodic arrangement of the two types of atoms shown in the above figure is the simplest 1-dimensional model of an ionic solid. Each atom has one orbital and one electron, with atom A being the cation and B the anion (✏A > ✏B).
h A|H| Ai = ✏A
h B|H| Bi = ✏B
h A,n|H| B,ni = h B,n|H| A,n+1i = V2
Derive an expression for the energy dispersion of its electronic bands. Plot the corresponding band dispersions.
(2) One-dimensional solid with 2 electrons per primitive cell sss
1-dimensional solid with sp hybrids
Let us consider a 1-dimensional solid, of period a, consisting of a single atom type, but now each
atom has a single s and a single p orbital, with 2 electrons per atom. Setting
h np|H| npi = ✏p,
h ns|H| nsi = ✏s
h np|H| n±1,pi = Vpp > 0
h ns|H| n±1,si = Vss < 0
h ns|H| n+1,pi = h ns|H| n 1,pi = Vsp > 0
(a) Construct the Bloch functions associated with the s and p orbitals.
(b) Determine the set of coupled equations obtained by minimizing the energy expectation value,
assuming that orbitals on di↵erent sites are orthogonal.
(c) Derive an expression for the dispersion of the electronic bands, and plot the dispersion curves for the two sets of parameters:
1. Parameter set I
✏s = 10.725 eV, ✏p = 3.525 eV, Vss = 2.08 eV, Vpp = 3.49 eV, Vsp = 2.24 eV Identify the maximum energy of occupied states.
2. Parameter set II
✏s = 9.725 eV, ✏p = 4.525 eV, Vss = 4.294 eV, Vpp = 5.2 eV, Vsp = 2.24 eV Identify the the valence band maximum and conduction band minimum in set II.
(3) In the example of s-p basis on a square lattice with nearest-neighbor interactions (a = 2 ̊A), extend the interactions to next-nearest neighbor.
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