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==Problem 1== | ==Problem 1== | ||
Consider a | Consider a tight-binding model of a 1D nanowire: | ||
<math> \hat{H} = \sum_m \cos(2 \pi m \ | <math> \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)</math>, | ||
The integer <math> m </math> is indexing sites at which the atoms are located. The distance between two sites defined the lattice spacing <math> a </math>. The ket <math>| m \rangle</math> is quantum state of an electron on atom <math> m </math>, so that <math>\langle x | m \rangle = \psi(x-m)</math> is the corresponding wave function in coordinate representation (or single "orbital" per site) which decays fast away from the position of an atom <math> m</math>. | |||
(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 | |||
(a | (b) Use Bloch theorem to reduce the eigenvalue problem of an infinite matrix <math> \mathbf{H} </math>, obtained by representing the Hamiltonian in the basis of orbitals <math> |m \rangle </math>, to diagonalization of a small matrix [whose size <math> n </math> is equal to the periodicity of the Hamiltonian found in (a)]. | ||
(c) The <math> n \times n </math> matrix in (b) will depended on the Bloch wave vector <math> k </math>. Compute and plot <math> n </math> bands as a function of Bloch wave vector <math> k </math> throughout the first Brillouin zone (this task will have to be carried our numerically). | |||
( | |||
==Problem 2== | ==Problem 2== | ||
Revision as of 22:39, 18 September 2009
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 } . 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).
