Temp HW 2

Problem 1

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

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m } is indexing sites at which the atoms are located. The distance between two sites defines the lattice spacing Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a } , while the nearest neighbor hopping Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t } sets the unit of energy. The ket Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle | m \rangle} is quantum state of an electron on atom Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m } , so that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{H} } , obtained by representing the Hamiltonian in the basis of orbitals Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |m \rangle } , to diagonalization of a small matrix [whose size Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n } is equal to the periodicity of the Hamiltonian found in (a)].

(c) The Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n \times n } matrix in (b) will depended on the Bloch wave vector Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k } . For each value of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k } , diagonalize this matrix and plot Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n } bands Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E_n(k) } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varepsilon_m =0 } . In numerical calculations use Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 code dos_negf.m written in the lab on the Computing 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:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varepsilon_m \in \left [-\frac{W}{2},\frac{W}{2}\right] } ,

plot the wave functions whose eigenenergy is close to Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E=0.5t } for different disorder strengths Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle W=0, 0.5, 1, 1.5, 2, 2.5, 3 } . In MATLAB, you can generate values of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 10 \times 10 } .

(b) If you increase your quantum dot array size to Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 100 \times 100 } , does your Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E \ \mathrm{vs.} \ \alpha } plot resemble Hofstadter butterfly on an infinite lattice plotted in Fig. 6 of the same reference as in (a).