Assignment 5 - Electron gas - Aula de 01/09 (Prazo: 08/09 23h55)
Condições de conclusão
The tight-binding Hamiltonian is given by:
Obs: "H.c." means ``Hermitian conjugate" of the previous operator (we will use it quite a bit to short Hamiltonians). Here, it represents the term \(\hat{c}^\dagger_{i+1} \hat{c}_{i}\).
Single-particle states: Consider the case of \(N=1\) (only one particle in the entire chain of \(N_s\) sites) with equal orbital energies (\(\epsilon_i=\epsilon_1\)).
Aberto: segunda-feira, 10 ago. 2020, 00:00
Vencimento: terça-feira, 8 set. 2020, 23:59
Problem 1 - Density of states for the 2D electron gas
[See pdf]
Problem 2 - Tight-binding model for non-interacting electrons
Consider the so-called tight-binding model for fermions in 1D. This model describes a set of \(N_s\) orbitals located in the sites of a 1D chain. The \(N\)-particle system involves non-interacting fermions which can occupy these orbitals (zero or one fermion per site) but can also "hop" between the sites. The probability for the hopping is encoded in the "overlap integral" t with dimensions of energy and t>0.The tight-binding Hamiltonian is given by:
\( H= \sum_{i=1,N_s} \epsilon_i \hat{n}_{i} - t \sum_{i=1,N_s-1}\left(\hat{c}^\dagger_{i} \hat{c}_{i+1} + \mbox{H.c.} \right) \)
where \(\epsilon_i\) is the energy associated with the orbital at site \(i\), \(\hat{c}^\dagger_{i}\) creates a fermion at a site \(i\) and \(\hat{n}_{i} \equiv \hat{c}^\dagger_{i} \hat{c}_{i}\) counts the number of fermions at site \(i\).Obs: "H.c." means ``Hermitian conjugate" of the previous operator (we will use it quite a bit to short Hamiltonians). Here, it represents the term \(\hat{c}^\dagger_{i+1} \hat{c}_{i}\).
Single-particle states: Consider the case of \(N=1\) (only one particle in the entire chain of \(N_s\) sites) with equal orbital energies (\(\epsilon_i=\epsilon_1\)).
- Write the Hamiltonian in the occupation number basis \(|n_1, n_2,\dots n_{N_s} \rangle\).
Hint: Choose the ordering such that the resulting matrix is tri-diagonal. - Diagonalize the Hamiltonian.
Hint: Use the fact that the eigenvalues \(\lambda_n\) (\(n=1,\dots,N_s\)) of an \(N_s \times N_s\) tridiagonal matrix with diagonal elements \(D\) and off-diagonal elements \(T\) is given by:
\( \lambda_n = D + 2T\cos{\left( \frac{n \pi}{N_s + 1}\right)} \)
- If the chain length is L, and the spacing between the sites is a with \(L \gg a\), write the single-particle energies in terms of \(k_n \equiv n \pi a/L\).
- Show that, in the limit \(k_n . a \rightarrow 0\) (\(N_s \) large) the single-particle energies (up to a constant) show a quadratic dispersion similar to the non-interaction electron gas, namely:
\( \varepsilon_{k_n} - \varepsilon_{0} \propto (k_n)^2 \)
- Finally, calculate the Fermi energy for the special case of \(N=N_s/2\) particles in the system. This situation referred to as "half-filling" and has interesting properties such as electron-hole symmetry.
- 31 agosto 2020, 16:25 PM