Questions tagged [tight-binding]

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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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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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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 ...
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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{...
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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: $...
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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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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,...
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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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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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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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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 ...
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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 ...
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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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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^{\...
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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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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 ...
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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 ...
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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)=\...
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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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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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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), ...
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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 ...
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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 $k$ ...
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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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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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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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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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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 ...
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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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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 ...
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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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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 ...
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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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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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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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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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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 ...
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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 ...
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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?
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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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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. ...
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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,...
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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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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 ...