Questions tagged [tight-binding]
The tight-binding tag has no usage guidance.
195
questions
0
votes
0
answers
25
views
What's the difference in chemical potentials between nearly free electron model and tight-binding model?
In nearly free electron model, we know that when temperature is zero, the chemical potential $\mu$ is same as the Fermi energy $E_F$, $\mu=E_F$. For a good metal, $E_F$ is roughly $10\mathrm{eV}$, so ...
-1
votes
0
answers
18
views
Tight binding hamiltonian graphene [duplicate]
I'm stuck on solving a question regarding the tight-binding hamiltonian for graphene.
I have been given a hamiltonian that looks like this (where the spin has been omitted since hopping is independent ...
-3
votes
0
answers
92
views
Tight-Binding Hamiltonian Graphene
I'm stuck on solving a question regarding the tight-binding hamiltonian for graphene.
I have been given a hamiltonian that looks like this (where the spin has been omitted since hopping is independent ...
0
votes
0
answers
35
views
How to build up the creation/anihilation operators in Numerical Renormalization group?
Let's define a simple 1D linear tight-binding model (no matter that NRG is trivial here or even fails, this is just an simple example to illustrate my point)
$$H = \sum t_j \hat{c}^\dagger_{j}\hat{c}_{...
- 1,234
2
votes
1
answer
143
views
Tight binding on a square lattice with three orbitals symmetries
I came across a tricky problem while studying tight binding within the second quantization frame:
Consider a square lattice with one atom per unit cell, where each atom has three active hydrogen atom ...
- 198
0
votes
1
answer
73
views
Triangular Tight Binding Model
I have the following Hamiltonian for a triangular tight binding model:
$$
H = e^{i\phi} \Big(|2\rangle \langle1| + |3\rangle \langle2| + |1\rangle \langle3| \Big) + e^{-i\phi} \Big(|1\rangle \langle2| ...
- 1
2
votes
1
answer
51
views
How can I identify momentum eigenstates in a tight binding model with degenerate energy eigenstates?
Summary:
I numerically diagonalize a tight binding Hamiltonian to get energy eigenvectors, some of which are degenerate. However, the numerically diagonalized degenerate eigenvectors are not ...
- 21
0
votes
0
answers
28
views
Orthonormality of Wannier functions [duplicate]
In "Modern Condensed Matter Physics" by Girvin and Yang, a question invites the reader to prove the orthonormality of the Wannier functions. My proof will be complete if
$$
\langle n',0\vert ...
- 225
0
votes
1
answer
120
views
Dispersion Relation for one dimensional monoatomic lattice in Kronig-Penny model and in Tight-Binding Approximation model
In the tight binding approximation model, we have the dispersion relation for a one-dimensional atomic lattice given as:
$$E(k) = E_0 - \alpha - 2\beta \cos(ka)$$
Here, $\alpha$ and $\beta$ are ...
- 321
2
votes
0
answers
51
views
Confusion about linear combination of atomic orbitals (LCAO)
In an YouTube video, that I used some information from it to work out my homework,
Link: https://youtu.be/5vOvjeTRcqM?si=NzIcx_9QTNQ9Nhff
the linear combination of atomic orbitals (LCAO) was discussed ...
- 69
0
votes
1
answer
62
views
Orthogonality in the Tight-binding model
In the linear combination of atomic orbitals and the Tight-binding model we write the wave functions for some k as a linear combination of Bloch states like this
$$
\Psi_n(\textbf{r}, \textbf{k}) = \...
- 69
1
vote
0
answers
19
views
Impact of labeling in a bloch-crystal with orbital basis
If I'm diagonalizing a hamiltonian of electrons in a crystal that is written in the orbital basis, does it matter whether I calculate the matrix element between one atom and another atom (or the image ...
- 1,240
1
vote
0
answers
61
views
Current Operators on Lattice
Peierls substitution method by taking the functional derivative of Hamiltonian can be used to determine the form of current-operator in continuum model (See Bruus-Flensberg) as well as lattice model. ...
- 189
0
votes
0
answers
42
views
Signal in hopping term in diatomic chain using the tight binding method
I am currently studying the tight binding method, and while solving solving a problem I came across something I don't understand. There are two atoms A and B, A has only type s orbitals and B has only ...
- 31
0
votes
0
answers
58
views
How the $-2t\cos(k)$ term appears in the dispersion of the $1D$ tight binding model?
I am trying to derive the tight binding dispersion relation with periodic boundary condition with $N$ lattice sites of the simplest Hamiltonian:
\begin{equation}
\label{ham}
\tag{1}
H = -t\sum\...
- 9
1
vote
1
answer
72
views
Complex values for the dispersion relation obtained through an $s$-band only tight binding model for diamond cubic crystal
Any given atom in a diamond cubic lattice (Like Si or Ge) has four nearest neighbours at at a distance $\sqrt{3}a/4$, being $a$ the lattice constant. The translation vectors to these neighbours can be ...
- 207
0
votes
1
answer
48
views
How to calculate a momentum space of a semi-finite lattice?
If we have a 2D square lattice of lattice constant a whose $x$ axis has only $N_x$ cells each with one atom and no with spin degeneracy, and periodic boundary conditions on $y$ with $N_y$ cells along ...
- 13
0
votes
1
answer
57
views
Position space representation of tight-binding model with composite lattice
Consider a composite lattice such as silicon or graphene (with sublattice A and B). We may call the sublattice tight-binding Bloch sums $\left| k, \alpha \right>$ with $\alpha$ being the sublattice ...
- 109
0
votes
2
answers
56
views
Can we use hybridised orbitals in Tight-Binding Method for calculation energy dispersion?
I am working on a model of a Fe square planar complex with nitrogen and oxygen given below is a monomer of that polymer. While constructing the hamiltonian matrix of this monomer I'm confused as to if ...
1
vote
0
answers
15
views
Term simplification in single atomic orbital (LCAO) model
I was following the text of one of our lectures where we derived for a single-orbial $\phi(\vec{r})$ LCAO model for the band-structure of a crystal:
$$E_{\vec{k}} = E_a + \frac{\sum_{\vec{R}}e^{i\vec{...
- 303
0
votes
0
answers
26
views
Why is the effective mass of top band just $-1$ times that of the bottom band? [duplicate]
In band theory, the formula for effective mass is $$m^* = \frac{\hbar^2}{\frac{d^2E}{dk^2}}$$ and $m_{top}^* = -1 \times m_{bottom}^*$ the effective mass of top band is just -1 times the bottom band. ...
1
vote
0
answers
32
views
Why perturbational approach fails to describe the bands?
I'm trying to find the band structure of Graphene using tight-binding for a unit cell with 6 carbon atoms(this is a toy model for my own research). The hamiltonian is as below:
$$H=
\left[\begin{...
- 83
2
votes
3
answers
86
views
Time evolution of non-interacting field operators [closed]
I learned that for a non-interacting tight-binding system $H=\sum_{n}\varepsilon_na^\dagger_n a_n$, the time evolution of $a_n$ is simply $a_n(t)=e^{-i\varepsilon_nt}a_n$. I tried to prove this:
\...
- 133
0
votes
1
answer
85
views
Can I calculate partial density of states (PDOS) using tight binding approximation? [closed]
I am using tight binding approximation for a 2D material by 2*2 Hamiltonian, and I have ploted the density of states correctly. Can I also calculate the partial density of states using tight binding? ...
0
votes
1
answer
80
views
Berry connection in SSH model
In the SSH model for the 1D case, we get the eigenvectors as $$|(\pm)k>= \begin{pmatrix} \pm e^{-i\phi} \\ 1 \end{pmatrix}$$ where $\phi = tan^{-1}(\frac{wsin(k)}{v+wcos(k)})$. We can calculate the ...
- 41
1
vote
1
answer
141
views
Dispersion Relation and Eigenvectors of SSH Model in Tight Binding
Consider a one-dimensional chain of atoms as shown in the figure. Let the spacing between the atoms be $a$. Assume that the onsite energy is the same at each point and is equal to
$0$ (without any ...
- 41
1
vote
0
answers
35
views
Empirical tight-binding sp$^3$s* band structure of semiconductors
I'm simulating on code the tight-binding sp3s* bandstructure of certain semiconductors, such as GaAs, AlP, InP, ZnSe, etc. with spin-orbit coupling at a temperature of $T = 0$ K but I'm having trouble ...
- 211
1
vote
0
answers
32
views
Is there any method for folding a Hamiltonian matrix to lower dimension?
I want to solve a tight-binding Hamiltonian which is $6\times6$. I'm only interested in two of the six bands which lie near zero energy at $\vec{k}=(0,\frac{4\pi}{3\sqrt{3}a})$. Is there any way to ...
- 83
0
votes
0
answers
37
views
How should I plot a tight-binding dispersion when I use multiple gridpoints in real space?
Consider a simple tightbinding calculation with spacing $a$. I can write down a dispersion relation $$E(k) = 2t - te^{ika} - te^{-ika}$$
Say I am solving a simple tightbinding problem numerically, on ...
- 285
2
votes
1
answer
75
views
Variational method in the tight binding approach
I'm trying to read Professor David Tong's notes to understand the principles behind the tight-binding model - section 2.3.5 'Deriving the Tight-Binding Model'.
He first considers the Hamiltonian of ...
0
votes
0
answers
23
views
Interface between two different Su-Schrieffer-Heeger (SSH) chains
When I was exploring the interface (edge) states between two different SSH chains, I noticed some strange solutions.
I was considering the following system. Two different SSH chains are connected as a ...
- 43
0
votes
0
answers
91
views
Band structure of Graphene and Tight Binding
Consider the $sp^2$ hybridization of Carbon in Graphene depicted in the following picture:
When considering the LCAO TB method, one would expect conduction properties to be determined by the ...
- 101
0
votes
0
answers
71
views
Tight binding model, get dispersion relation in crystal
I know the theory of tight binding model but don't know how to apply them in the real lattice
1
vote
0
answers
64
views
What does band inversion mean for occupation?
I am thinking about Topological Insulators at the moment and am not clear about how to understand the occupation of the inversed band.
I understand that due to energetic and symmetry reasons the ...
- 122
1
vote
0
answers
177
views
Chemical potential $\mu$ controls filling?
If total magnetization of a spin 1/2 system is zero, does it mean that system is at half filling? or chemical potential $\mu=0$? I was trying to show that chemical potential controls filling by taking ...
- 427
0
votes
1
answer
163
views
Green's functions of disordered tight binding models
A research problem has led me to calculate a Green's function of a tight binding model with both onsite disorder and hopping amplitudes which vary in space. Since so much is known about tight binding ...
- 707
2
votes
1
answer
224
views
Is one-body tight-binding Hamiltonian always diagonal in Fourier space?
A generic tight-binding Hamiltonian for one-body operators is (assume one state per site)
$$
H=\sum_{ij}C_i^\dagger t_{ij} C_j \quad; \quad t_{ij}=\int d\mathbf{r} \phi_i^*(\mathbf{r}) t(\mathbf{r})\...
- 801
2
votes
1
answer
208
views
Second quantization: Hamiltonian in field operators vs Tight binding form
Hamiltonian written in terms of field operators: The kinetic energy (KE) part of Hamiltonian is
$$
H=-\frac{\hbar^2}{2m}\int d\mathbf{r} \Psi^\dagger(\mathbf{r}) \nabla^2\Psi(\mathbf{r}) \tag{1}
$$
...
- 801
1
vote
1
answer
139
views
Confusion tight binding model graphene
I think I have a conceptual doubt on the calculation of the matrix elements in momentum space for the tight binding Hamiltonian of graphene. I will break down my question into sections to make it as ...
- 731
1
vote
0
answers
55
views
Introduce a model with logarithmic dispersion relation
Having a model Hamiltonian on a lattice, one can compute the band structure of a system by employing the Bloch theorem. Here for simplicity, let's focus on noninteracting models. This procedure for a ...
- 384
1
vote
1
answer
172
views
Tight-binding: aren't the electronic bands just the eigenvalues of the Hamiltonian?
In the tight-binding model we choose a set of atomic orbitals $\{\phi_1(\textbf{r}), ..., \phi_N(\textbf{r})\}$ and estimate a transfer matrix and hopping matrix
$$
\begin{cases}
S_{ij}(\textbf{k}) = \...
- 715
2
votes
0
answers
117
views
New representation of annihilation operator in tight binding model for graphene
In this article about the electronic properties of graphene, in section b the author considers the usual tight binding hamiltonian for graphene
$$ H=-t\sum_{<i,j>,\sigma}(a_{\sigma,i}^{\dagger} ...
- 731
1
vote
0
answers
38
views
Parity operator for a general $N\times N$ Hamiltonian
Assume I've a $N\times N$ tight-binding Hamiltonian that depends on spatial coordinates $k=(k_x, k_y)$; it holds
$$ H(k)|\psi_k\rangle = E_k |\psi_k\rangle \quad .$$
Now I want to find the parity ...
- 135
2
votes
2
answers
522
views
Does particle-hole symmetry always imply half-filling and real correlations $\langle c^\dagger_n c_{n+1} \rangle$?
Suppose we had a lattice Hamiltonian $H$ which was symmetric under the particle-hole transformation
$$ c_n \mapsto U^\dagger c_nU=(-1)^nc^\dagger _n$$
such that $[H,U] = 0$, where $c_n$ are Fermionic ...
- 3,776
1
vote
1
answer
107
views
Un-equal time correlation via non-interacting tight-binding Hamiltonian
Let's assume we have a model, which is initially defined by the tight-binding Hamiltonian with a random on-site energy $f_n$, as follows:
$$H^i=-J\sum_n^{L-1}\left(a_n^\dagger a_{n+1}+h.c\right)+\...
- 13
8
votes
1
answer
153
views
Alternating Tight Binding Hamiltonian
The alternating Hamiltonian may be written as:
$$H = t \sum_{n} (-1)^{n} \left[c^{\dagger}_{n+1}c_{n} + c^{\dagger}_{n}c_{n+1} \right] \; \; .$$
I wanted to know the energy dispersion for this system, ...
- 1,151
1
vote
1
answer
154
views
Tight-Binding method and orthogonality of Bloch functions
Tight binding summary
When computing the electronic bands of a crystal in the tight-Binding approximation, the standard way to do it is to construct Bloch solution as
$$
\Psi_n(\textbf{r}, \textbf{k}) ...
- 715
1
vote
0
answers
86
views
Number density equation in BCS Theory
In this paper, The author has given two-equation
$$\frac{1}{U}=\sum_k \frac{1}{E(k)}\tanh\left(\frac{\beta E(k)}{2}\right)$$
and $$2\rho-1=-\sum_k \frac{\epsilon(k)}{E(k)}\tanh\left(\frac{\beta E(k)}{...
- 11.5k
1
vote
1
answer
122
views
Questionable Taylor expansion for Peierls substitution
In this paper, on page 3, the authors go from the tight binding model w the Peierls substitution
$$ H = \sum_{i,j} \sum_{a,b} t_{a,b} \exp\left(i \int_{\textbf{R}_{j,b}}^{\textbf{R}_{i,a}} dr'_{\mu} ...
- 182
6
votes
1
answer
922
views
What's a charge density wave?
I'm reading chapter 2 of Condensed Matter Field Theory by Alexander Altland, Ben D. Simons, Section on Interacting fermions in one dimension.
From what I understood, they considered the system of ...
- 11.5k