Temp
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 \frac{1}{2} \left ( |m \rangle \langle m+1 | + |m \rangle \langle m-1| \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 defined 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 atoms the chain starts to repeat itself. This atoms define the unit cell of the wire whose periodic repetition in both direction
(b) Use 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 } . Compute 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 as a function of 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 } throughout the first Brillouin zone (this task will have to be carried our numerically).
Problem 2
A nanowire consists of 1000 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 \frac{1}{2} \left ( |m \rangle \langle m+1 | + |m \rangle \langle m-1| \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 1000 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_500 = 5t } . Compute (DOS) for this case and comment on differences between (a) and (b). What is the highest eigenenergy in (a) vs. (b)?
