Questions tagged [tight-binding]
The tight-binding tag has no usage guidance.
145
questions
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I was solving some numerical problems on Soild State Physics, related to Effective mass, but i could not get the approach to solve this [closed]
The energy band relation in a linear chain with inter-atomic distance 'a' is given by $$E(k) = E_o - T \cos(ka)$$ When width of this bond is increased by 20%, calculate the % change in the effective ...
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Assumptions of tight binding model
I am referring to chapter 10 of Ashcroft & Mermin's Solid State Physics text (from the 2d edition):
In developing the tight-binding approximation. we assume that in the vicinity of
each lattice ...
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1answer
59 views
What is the difference between lattice models and tight-binding simulations?
In condensed-matter physics, people use different methods to solve the many-particle Schrödinger equation. I was wondering about two of those methods, the lattice model and tight-binding simulation. ...
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29 views
Nuclear binding and internal energy [duplicate]
A section of some article:
When you cool a body at rest its internal energy decreases. Since energy is related to mass, here $E=m_0c^2$, where $m_0$ is the rest mass, the rest mass of the body ...
2
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1answer
72 views
Why aren't the eigenvectors of a tight-binding Hamiltonian periodic?
I try to calculate the Berry connection for a simple graphene model and stumbled across the following question. Suppose I have a tight binding Hamiltonian (further details here or here):
$$H = \begin{...
2
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1answer
35 views
Tight-binding model for decorated square lattice
How do I go about determining the tight-binding Hamiltonian for the crystal structure below? I have identified the primitive lattice vectors $\mathbf{a}_1=(a,0)$ and $\mathbf{a}_2=(0,a)$ for lattice ...
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37 views
Pair correlation function in a 1D tight binding chain
I want to calculate the pair correlation function in a non-interacting Fermi system which is defined in "Quantum Theory of the Electron Liquid" by Gabriele Giuliani and Giovanni Vignale as:
$...
2
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1answer
93 views
Complex energies of non-reciprocally coupled chain with hoppings of equal absolute value
In the Hatano-Nelson chain (i.e. the simplest 1D tight-binding model with nonreciprocal hopping) for positive hoppings $t_{1,2}>0$ you get a purely real spectrum.
However, as soon as you change the ...
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1answer
30 views
Fermionic occupation for an inhomogeneous tight-binding model
The model
Consider the simple one-dimensional fermionic tight-binding chain of $N$ sites with inhomogenous hopping couplings $t_n$:
$$ H = - \sum_n t_n c^\dagger_{n+1} c_n + \text{h.c.} \equiv \sum_{n,...
1
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1answer
36 views
Peierls substitution, periodic boundary condition, gauge invariance
I am considering a tight-binding model in a magnetic field, and studying the Peierls substitution
$$t_{jk} \to t_{jk} e^{i\frac{q}{\hbar}∫_j^k \vec{A} \cdot d \vec{r}}$$
In some papers, such as this ...
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34 views
Wave-function (in real space) of electron in Graphene nano-ribbons
I was trying to solve the tight binding model of Graphene nano-strip in the zig-zag configuration. It looks something like this:
It has a very beautiful band structure. In order to calculate the band ...
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2answers
211 views
Numerically transforming Hamiltonian into $k$-space
A rather computational question:
Suppose you have a very simple tight-binding Hamiltonian in matrix form, i.e. for example something like this for a 1D chain with open ends:
$$H =\begin{bmatrix}
0 ...
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17 views
Greens function of tight-binding chain (literature request)
The retarted one-particle Greens function $G = \frac{1}{E-H+i\epsilon}$ of a tight-binding chain:
$$H = -J \sum_n c_{n}^\dagger c_{n+1} + c_{n+1}^\dagger c_{n}$$
can easily be evaluated using contour ...
1
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0answers
74 views
Why can $\hbar q_x$ and $\hbar q_y$ be replaced by $\hat{p}_x=-i\hbar\frac{\partial}{\partial x}$ and $\hat{p}_y=-i\hbar\frac{\partial}{\partial y}$?
It is written in the book The Physics of Graphene (Page 10 and 17) that when the intervalley scattering is neglected, we can make the following substitution in the Hamiltonian of the graphene when ...
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29 views
Pseudo-Hamiltonian of atom
Consider the pseudo-Hamiltonian of atom i,
It is said that "the more the states
are localized on the atom and the less they overlap with the neighbouring orbitals, the
more this formalism is ...
2
votes
1answer
54 views
Group velocity (open vs periodic boundary conditions) [duplicate]
I'm trying to understand the meaning of the group velocity for Bloch electrons given by
$$
\mathbf{v}=\frac{1}{\hbar}\frac{\partial E(\mathbf{k})}{\partial \mathbf{k}}
$$
where $E(\mathbf{k})$ is the ...
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45 views
Particle current operator in tight binding
For general non-interacting Hamiltonian $H = \frac{-\hbar^2}{2m}\int dr \Psi_r^\dagger\nabla_r^2\Psi_r$, the particle current operator $J$ can be derived using continuety equation $\nabla_r\cdot J = -\...
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29 views
Position of the Wannier center in tight-binding model
With the Fourier transform of the annihilation and creation operators,
$$c_{m,n} = \frac{1}{\sqrt{N}} \sum_{k_x} \sum_{k_y} c_{kx,ky} e^{-i\mathbf{k}\cdot \mathbf{r}_{m,n}} \quad\text{and}\quad c^{\...
2
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1answer
58 views
Hamiltonian density of a spin Hamiltonian
A many-body Hamiltonian in second quantization is written as
$$
H = \int d\vec r \Psi^\dagger_{\vec r} H_1 (r) \Psi_{\vec r}
$$
where $H_1(r)$ is one-body Hamiltonian. For example, for a non-...
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20 views
Constructing an operator from k.p hamiltonian
I have a question regarding to how to construct an operator from k.p hamiltonian. May be there are some problems in my understanding, I hope you can point me out and correct my description if I made ...
2
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0answers
40 views
Line integral in Peierls substitution
I'm trying to understand the reasoning behind Peierls substitution. The final result seems to be simply replacing the hopping elements
$$t_{ij} \to t_{ij} e^{i \frac{q}{\hbar} \int_i^j \vec{A} \cdot d ...
2
votes
1answer
79 views
Expanding the Graphene Hamiltonian near Dirac points upto second order term
I was trying to solve the Graphene Hamiltonian near the Dirac points upto the second order term for the nearest neighborhood points.
So expanding the function near the Dirac Point, we get$$g(K+q)=\...
0
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1answer
69 views
How to calculate spin texture in $k$-space?
I have a triangular lattice model. In $k$-space, it is written as:
$$
H = \sum_k\sum_\sigma \Psi_{k\sigma}^+ h_k\Psi_{k\sigma}
$$
where $\Psi_{k\sigma} = [a_{k\uparrow}, b_{k\uparrow}, c_{k\uparrow},...
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0answers
17 views
Given a crystal with mirror symmetry along a lattice plane, how to find the correspond plane in first Brillouin zone
Given a crystal with mirror symmetry along a lattice plane, how to find the correspond invariant plane in first Brillouin zone?
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1answer
42 views
Hermiticity of Slater-Koster hamiltonian
From the paper written by Slater and Koster, some of the tight binding Hamiltonian seemed to change sign under a sign flip. For example
$$E_{s,x}=l(sp\sigma)$$
Suppose we have two atoms (A and B), ...
1
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1answer
45 views
Quantum description of the tight-bonding model
In an introduction to the tight-bonding model in Condensed Matter Physics, I came across this statement:
When two atoms go towards one another, the electrons in the atoms start to build existence of ...
1
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1answer
59 views
Bands versus reciprocal lattice vectors in the Bloch basis
In Bloch theory, using the reduced zone scheme, we can index the states of a crystal by either $k,G$ where $k$ is in the first Brillouin zone and $G$ is a reciprocal lattice vector, or $n,k$ where $k$ ...
4
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1answer
155 views
Thermal averages with opposite sign in the exponential
I've recently seen an example where a thermal average was carried using a plus sign instead of the usual minus sign inside the exponential.
$$\langle \mu \rangle = \frac{1}{Z} tr(e^{\beta H }\mu) \...
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16 views
Can I construct the inversion operator in this way given a crystal and a tight binding matrix?
I have some problem about the concept of band parity defined at the TRIM in K-space. Given a crystal and an orbital basis set, I can easily write down the matrix representation of an Inversion ...
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19 views
Graphene tight-binding vs exact band diagram
In undergraduate courses, we study the tight-binding hamiltonian for graphene, assuming only 2pz orbitals overlap. Based on this model, we get the dirac cones, where at the tip graphene behaves like ...
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1answer
170 views
Continuum limit of tight-binding models
Suppose I had a simple $1D$ tight-binding Hamiltonian
$$ H = -t \sum_i a^\dagger_n a_{n +1} + \text{h.c.}$$
with $N$ sites and lattice spacing $a$. This Hamiltonian can be diagonalised with a discrete ...
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33 views
Derivation of Lyapunov exponent in 1D disordered system
What I am considering is a tight-binding model of 1D disordered system. According to the literature (page 1500, equation (60)), Lyapunov exponent $\gamma$ is calculated as follows which I am not ...
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55 views
Meaning of the discrete Fourier transform in condensed matter
I'm in a CMP course now and I think I'm taking for granted what it means for a creation/annihilation operator to be Fourier transformed. I understand what a Fourier transform is for some function, but ...
2
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1answer
72 views
A potential well with 3-fold reflection symmetry
When we are talking about Bloch's theorem and also the tight-binding approximation, we can use them to help finding eigenstates of a system. However, I am so confused how to apply it in this case (...
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0answers
215 views
Dirac equation for the Kagome lattice
Background
To model graphene we often use a nearest-neighbour tight-binding Hamiltonian
$$H = - t \sum_{<ij>} c^{\dagger}_i c_j$$
embedded on a hexagonal lattice. By performing a Fourier ...
4
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1answer
307 views
Tight-binding vs nearly free electron (NFE) model predictions
I came across the following problem:
Suppose that a certain material consists of $N$ atoms that are ordered in a 2D lattice with a lattice constant $a$, and that each atom donates two conduction ...
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1answer
109 views
How to compute band structure from real space Hamiltonian
I will often see a paper where someone has diagonalized a tight binding model in real space, see for example Eq 1 of https://iopscience.iop.org/article/10.1088/2516-1075/ab9f94/pdf. Immediately ...
2
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1answer
93 views
Parallels between Tight-binding wavefunctions and Bloch states
I was reading Robert Knox's Theory of Excitons and I came across a certain statement that I have trouble reconciling, both mathematically and conceptually. The context concerns the electronic ...
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61 views
Ordering of operators in 2nd Quantization of the Tight Binding Model
I am looking at the position space representation of the Tight Binding Hamiltonian (without the spin component and ignoring any constants) as given by:
$$\hat H = \sum_{i}\sum_{\delta}(\hat c_{i}^{\...
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90 views
Best way to calculate local density of states of tight binding model
I am currently trying to analyze a system using a tight binding model. I have a quite complicated unit cell with more than nearest neighbor hopping, that is repeating in one dimension. I have a matrix ...
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1answer
92 views
Clarification regarding the calculation of Effective Mass from a Tight Binding Energy
Suppose we have the following:
$E(\vec{k}) = E_0 - A(\cos(k_x a) + \cos(k_y a) + \cos(k_z a))$, a tight binding energy.
I know that the effective mass is given by $m^{*} = \hbar^2 [\frac{\partial^2 E}{...
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0answers
134 views
Dispersion law for a tight binding Hamiltonian and particle states at $t$$\rightarrow$$\infty$
A spinless fermion (possessing an electric charge) can move across the sites of
the discrete (translationally-invariant) lattice. The structure
features three kinds of sites: $α_n$, $β_n$, $γ_n$ with ...
0
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1answer
174 views
Expanding a tight-binding hamiltonian around a Dirac point (1.D. Graphene) [closed]
My off diagonal terms for this Hamiltonian are the following:
$H = e^{\mp ik_xa}(1+ 2e^{\pm i3k_xa/2}\cos(\sqrt{3}k_ya/2)) $
Next, I want to see the low energy bands, and for this I've been told to ...
1
vote
1answer
326 views
In tight binding 2d square lattice with two atoms basis
How can we get the equations inside the box, which represent hopping between p and d orbital? I mean why we consider only two neighbouring?
0
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1answer
69 views
Disorders in tight binding model and periodicity
Does introducing disorders in tight binding models disrupt the periodicity of the lattice? If it does, doesn't this contradict with the assumptions and purposes of the tight binding model as a ...
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0answers
178 views
Deriving Continuum BdG equation from Tight-Binding model for Graphene-Superconductor interface
Background
My question concerns Beenakker's paper on "Specular Andreev Reflection in Graphene": cond-mat/0604594 (arxiv)/Phys.Rev.Lett. 97, 067007 (The same topic is also discussed in Rev. ...
3
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1answer
270 views
Exact Diagonalization of a tight-binding Hamiltonian with periodically alternating potential
My question is, can we diagonalize a general Hamiltonian ,
$$H=-t\sum_i^N (c_i^{\dagger}c_{i+1}+h.c.)+\sum_i \mu_i c_i^{\dagger}c_i$$ where,
$$\mu_i=\begin{cases}
\mu_0, &\text{if mod}(i,p)=0 \\
0,...
2
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0answers
48 views
Is dynamical localisation an interference effect?
The question refers to the well-known phenomenon of dynamical localisation due to an oscillating electric field, as explained in Dunlap & Kenkre, PRB 34,6 (1986). Here, a particle initially ...
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29 views
How can I easily generate $\rm GaAs$ band structure?
Is there a program where I can easily generate $\rm GaAs$ band structure that shows lowest conduction band valleys?
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154 views
Instrinsic spin orbit coupling in tight-binding Hamiltonian
I'm looking to write down a second quantized Hamiltonian to include the intrinsic spin-orbit coupling term in addition to the hopping spin-orbit coupling Rashba effect.
How would I construct the term ...
