4   Electrons in a periodic potential (41 pts)
Due date: Wednesday, 27th November 2024.
4.1    Concept of bands: ID approximation of a crystal
Consider a one-dimensional crystal with the potential given by:
Vie---    U    [0, a[   U   [a + b, (a + b) + a[   U • • • , U(x) = 0
Vie ---U    [~b,0[   U   [a,a + b[   U   [(a + b) + a, 2 (a + b)[   U---,  U(x) = V>0
periodically repeated over JV» 1 periods of length d = a+b. We use periodic boundary conditions. The x axis is therefore divided in zones, where the potential is either 0 or V. The region [0, a] (U(x) = 0) is labeled region I, the region [a, a + b] is labeled region II (U(x) = V).
U(x) = V U{x) = 0 U(x) = V
— b 0 a        a + b
(i) (1 pt) Give the expression for a wave function for a particle in this potential, using Bloch's theorem. The corresponding wave vector will be denoted k.
(ii) (1 pt) Give the general expressions for the solutions of the Schrodinger equation ipi(x) and ipn(x) for a particle of mass m, in region I and in region II respectively. These involve plane waves with wave vector k± and ki respectively, as well as two unknowns each, which will be denoted A, B for region I, C, D for region II. For simplicity, assume E > V, where E is the energy of the particle. Please pay attention to the specific domain of definition for each of the functions ipi(x) and ^j-j-(x)!
(iii) (4 pts) Use continuity constraints at points x = 0 and x = a on the wave function, in order to obtain a set of four independent linear equations involving the unknowns introduced in the preceding question.
(iv) (6 pts) Show that the dispersion relation is given by:
i) sin(/j2&) = cos(kd)
(13)
k2 + k2
cos(kia) cosfefr)--7t,—r~^ sm(kia) sinfe^) = cos(kd)
2kik2
2      2 _ 2mV kl    k2~ h2
This is not a difficult calculation, but it is long; you need to keep things organized.
(v) (2 pts) Deduce from the above the expression for the dispersion relation in the configuration
h2x2
where E < V. (Use the notation-= V — E, you do not need to repeat the derivation!).
2m
(vi) (5 pts) Use a =   ^   , b = —j=, and V = 1. Plot the the profile of the left hand side of the
dispersion relation, as a function of E, for 0 < E < 7. Provide me with a transcript of your code, explaining how it works. Comment on your results, in relation to the concept of bands in solids.
8
4.2   Band structure in a periodic potential
Consider a two dimensional square lattice, with periodic boundary conditions, and with the periodic potential given by
f2Trx\ (2iry
U (x, y) = —AV cos - cos -
\ a J      \ a
where a is the lattice parameter and V > 0. In this problem, we explore the band structure near the point (^, ^), i.e., the corner of the 1st BZ. In all questions, we assume that V <C (ti2/2m)(ir/a)2. The arguments used when solving the problem can also be used when discussing band structures of simple metals.
(i) (1 pt) Give a description of the reciprocal lattice.
(ii) (2 pts) Give the expression of the Fourier expansion of U(r).
(iii) (1 pt) Explain in plain words what the statement V <C (fi2/2m)(7r/a)2 tells us about the system. In particular, explain why this allows us to look for solutions of the Schrodinger equation in the form
v>w= c^ikr ■ (14)
fc^BVK conditions
(iv) (3 pts) Derive the so-called central equation, that is, the Schrodinger equation in fe-space. This provides you with a relation between all the coefficients c^: for any vector K from the reciprocal lattice,
Ck-k + ^2 Uk'-kcu-k' = 0 (15)
k'
Hint: Substitute the solution in the form given in (iii) to Schrodinger equation.
(v) (2 pts) Show that this allows you to write the solutions of the Schrodinger equation in the form
Mr) = Y,c^(k~Kyr(16)
k
where the sum runs over all reciprocal lattice vectors, and k is any vector in the first Brillouin zone.
(IT 7T\
For k close to   —, —  , only the four following coefficients in the expansion of i/>fc(r), Eq. (16) \a a J
are non-negligible:
«.k.*k* {(}).(;). (;).(»
in units of —.
a
(IT 7T\
(vi) (4 pts) Solve the central equation for k close to (—,— ): calculate the perturbed energy
levels E(k).
a a
4.3   Density of states from the tight-binding model
Consider a tight-binding model on a square lattice, whose band structure is given by (we assume the lattice parameter is a = 1 for simplicity)
e(k) = -2t^2cos (ki), (17)
i
where i runs over all dimensions. We consider lattices in dimensions d = 1, 2, 3, so that the density of states can be written in a general manner
D^ = ^H^-^ (18)
k
9
(i) (3 pts) Compute analytically the expression for D(e) for d = 1.
(ii) (3 pts) For d = 2, d = 3, use numerics. Plot the profiles of D(e) for d = 1 (analytic and numerical results together), d = 2, d = 3. Provide me with a transcript of your code, which should implement the elementary summation in some form (do not use the "histogram" feature of some numerical libraries).
(iii) (3 pts) Discuss and commment your results.
10