Modelling atomic chains

In Periodic problems and plane-wave discretisations we already summarised the net effect of Bloch's theorem. In this notebook, we will explore some basic facts about periodic systems, starting from the very simplest model, a tight-binding monoatomic chain. The solutions to the hands-on exercises are given at the bottom of the page.

Monoatomic chain

In this model, each site of an infinite 1D chain is a degree of freedom, and the Hilbert space is $\ell^2(\mathbb Z)$, the space of square-summable biinfinite sequences $(\psi_n)_{n \in \mathbb Z}$.

Each site interacts by a "hopping term" with its neighbors, and the Hamiltonian is

\[H = \left(\begin{array}{ccccc} \dots&\dots&\dots&\dots&\dots \\ \dots& 0 & 1 & 0 & \dots\\ \dots&1 & 0 &1&\dots \\ \dots&0 & 1 & 0& \dots \\ \dots&\dots&\dots&\dots&… \end{array}\right)\]

Exercise 1

Find the eigenstates and eigenvalues of this Hamiltonian by solving the second-order recurrence relation.

Exercise 2

Do the same when the system is truncated to a finite number of $N$ sites with periodic boundary conditions.

We are now going to code this:

function build_monoatomic_hamiltonian(N::Integer, t)
    H = zeros(N, N)
    for n = 1:N-1
        H[n, n+1] = H[n+1, n] = t
    end
    H[1, N] = H[N, 1] = t  # Periodic boundary conditions
    H
end

build_monoatomic_hamiltonian (generic function with 1 method)

Exercise 3

Compute the eigenvalues and eigenvectors of this Hamiltonian. Plot them, and check whether they agree with theory.

Diatomic chain

Now we are going to consider a diatomic chain A B A B ..., where the coupling A<->B ($t_1$) is different from the coupling B<->A ($t_2$). We will use a new index $\alpha$ to denote the A and B sites, so that wavefunctions are now sequences $(\psi_{\alpha n})_{\alpha \in \{1, 2\}, n \in \mathbb Z}$.

Exercise 4

Show that eigenstates of this system can be looked for in the form

\[ \psi_{\alpha n} = u_{\alpha} e^{ikn}\]

Exercise 5

Show that, if $\psi$ is of the form above

\[ (H \psi)_{\alpha n} = (H_k u)_\alpha e^{ikn},\]

where

H_k = \left(\begin{array}{cc}
0                & t_1 + t_2 e^{-ik}\\
t_1 + t_2 e^{ik} & 0
\end{array}\right)

Let's now check all this numerically:

function build_diatomic_hamiltonian(N::Integer, t1, t2)
    # Build diatomic Hamiltonian with the two couplings
    # ... <-t2->   A <-t1-> B <-t2->   A <-t1-> B <-t2->   ...
    # We introduce unit cells as such:
    # ... <-t2-> | A <-t1-> B <-t2-> | A <-t1-> B <-t2-> | ...
    # Thus within a cell the A<->B coupling is t1 and across cell boundaries t2

    H = zeros(2, N, 2, N)
    A, B = 1, 2
    for n = 1:N
        H[A, n, B, n] = H[B, n, A, n] = t1  # Coupling within cell
    end
    for n = 1:N-1
        H[B, n, A, n+1] = H[A, n+1, B, n] = t2  # Coupling across cells
    end
    H[A, 1, B, N] = H[B, N, A, 1] = t2  # Periodic BCs (A in cell1 with B in cell N)
    reshape(H, 2N, 2N)
end

function build_diatomic_Hk(k::Integer, t1, t2)
    # Returns Hk such that H (u e^ikn) = (Hk u) e^ikn
    #
    # intra-cell AB hopping of t1, plus inter-cell hopping t2 between
    # site B (no phase shift) and site A (phase shift e^ik)
    [0                 t1 + t2*exp(-im*k);
     t1 + t2*exp(im*k) 0                 ]
end

using Plots
function plot_wavefunction(ψ)
    p = plot(real(ψ[1:2:end]), label="Re A")
    plot!(p, real(ψ[2:2:end]), label="Re B")
end

plot_wavefunction (generic function with 1 method)

Exercise 6

Check the above assertions. Use a $k$ of the form $2 π \frac{l}{N}$ in order to have a $\psi$ that has the periodicity of the supercell ($N$).

Exercise 7

Plot the band structure, i.e. the eigenvalues of $H_k$ as a function of $k$ Use the function build_diatomic_Hk to build the Hamiltonians. Compare with the eigenvalues of the ("supercell") Hamiltonian from build_diatomic_hamiltonian. In the case $t_1 = t_2$, how do the bands follow from the previous study of the monoatomic chain?

Exercise 8

Repeat the above analysis in the case of a finite-difference discretization of a continuous Hamiltonian $H = - \frac 1 2 \Delta + V(x)$ where $V$ is periodic Hint: It is advisable to work through Comparing discretization techniques before tackling this question.