Homework Set 2

Problem 1

Consider a tight-binding model of a 1D nanowire:

${\hat {H}}=\sum _{m}\cos \left(2\pi m{\frac {5}{3}}\right)|m\rangle \langle m|+t\sum _{m}\left(|m\rangle \langle m+1|+|m+1\rangle \langle m|\right)$,

The integer $m$ is indexing sites at which the atoms are located. The distance between two sites defines the lattice spacing $a$, while the nearest neighbor hopping $t$ sets the unit of energy. The ket $|m\rangle$ is quantum state of an electron on atom $m$, so that $\langle x|m\rangle =\psi (x-m)$ is the corresponding wave function in coordinate representation (or single "orbital" per site) which decays fast away from the position of an atom $m$.

(a) What is the periodicity of the Hamiltonian? (That is, after how many sites the chain starts to repeat itself? The atoms on those sites define the unit cell of the wire whose periodic repetition in both direction generates the whole wire.)

(b) Use the Bloch theorem to reduce the eigenvalue problem of an infinite matrix $\mathbf {H}$, obtained by representing the Hamiltonian in the basis of orbitals $|m\rangle$, to diagonalization of a small matrix [whose size $n$ is equal to the periodicity of the Hamiltonian found in (a)].

(c) The $n\times n$ matrix in (b) will depended on the Bloch wave vector $k$. For each value of $k$, diagonalize this matrix and plot $n$ bands $E_{n}(k)$ where $k$ vector belongs to the first Brillouin zone (this task will have to be carried our numerically).

Problem 2

A nanowire consists of 500 atoms described by a 1D tight-binding Hamiltonian:

${\hat {H}}=\sum _{m}\varepsilon _{m}|m\rangle \langle m|+t\sum _{m}\left(|m\rangle \langle m+1|+|m+1\rangle \langle m|\right)$.

(a) Compute numerically the density (DOS) of states for this wire assuming periodic boundary conditions and $\varepsilon _{m}=0$. In numerical calculations use $t=1$ as the unit of energy. How does DOS change if you increase the number of atoms from 500 to 5000?

(b) Replacement of the original atom in the middle of the chain by an impurity atom can be modeled by using $\varepsilon _{250}=5t$. Compute (DOS) for this case and comment on differences between (a) and (b). What is the highest eigenenergy in (a) vs. (b)? What is the physical meaning of this energy in the case (b)?

(c) Use the Green function method (see Matlab script dos_negf.m on the Computer Lab page) to compare its result for DOS to (a) and (b) obtained using eigenvalue method for 500 atom wire.

Problem 3

In 2009, different physics communities are celebrating 50 years of the theoretical discovery of Anderson localization (see special focus of Physics Today, August 2009 issue). Anderson localization, where quantum wave function is reduced to be non-zero only in a small region of space due to disorder, plays an important role in low-dimensional structures at low temperatures since all quantum states are localized in 1D wires and two-dimensional electron gases for arbitrary small concentration of impurities (if the concentration is really small, the system has to be large enough for electrons to realized that they are localized). Thus, the resistance of such systems decays exponentially with the system size.

For the same wire described by the Hamiltonian as in problem 2, but with disorder potential introduced via the so-called Anderson model, where presence of impurity is simulated via on-site energy as uniform random variable:

$\varepsilon _{m}\in \left[-{\frac {W}{2}},{\frac {W}{2}}\right]$,

plot the wave functions whose eigenenergy is close to $E=0.5t$ for different disorder strengths $W=0,0.5,1,1.5,2,2.5,3$. In MATLAB, you can generate values of $\varepsilon _{m}$ as the random variable with uniform distribution using rand function.

Problem 4

The Hofstadter butterfly is the energy spectrum of an infinite square lattice plotted as a function of the magnetic field. In this problem we will examine similar spectra for finite lattices modeling arrays of quantum dots in a magnetic field using methods that consider the appropriate molecular orbitals and compare their spectra to the Hofstadter butterfly.

(a) Reproduce panels (a)-(f) in Fig. 5 of American Journal of Physics 72, 5 (2004) for small arrays up to $10\times 10$.

(b) If you increase your quantum dot array size to $100\times 100$, does your $E\ \mathrm {vs.} \ \alpha$ plot resemble Hofstadter butterfly on an infinite lattice plotted in Fig. 6 of the same reference as in (a).

Problem 5

Consider semi-infinite 1D lead modeling reservoir, which can be described as 1D tight-binding chain (with hopping $t$ between neighboring atomic orbitals and energy $\varepsilon _{0}$ on each site) terminated at site $n=1$. By solving the matrix equation for the Green function:

$(E\mathbf {I} -\mathbf {H} )\cdot \mathbf {g} =\mathbf {I}$

Find the expression for the surface Green function $g_{11}^{r}$ (i.e., the matrix element 11 of $\mathbf {g}$). This surface Green function defines the self-energy of the lead via $\Sigma =t^{2}g_{11}^{r}$ employed in the Computer Lab exercise.