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Monte Carlo study of disorder in electronic tight-binding models

Pettersson, Carl LU (2023) FYSK03 20231
Department of Physics
Mathematical Physics
Abstract
Oftentimes solids are described by uniform, periodic lattices. In reality, however, there is often some disorder on some of the sites in the lattice. This disorder may come from, for example, there being a different type of atom or tightly bound electrons resulting in a larger on-site potential, affecting the electronic properties of the lattice. In this thesis the electronic properties of square, periodic lattices with a number of these impurity sites are studied. The main electronic properties that are studied are the electronic energies of the
lattice, and the density of states. Several properties of the impurity sites are also studied. The Hamiltonians that describe the different lattices were created with the tight-binding model. Two... (More)
Oftentimes solids are described by uniform, periodic lattices. In reality, however, there is often some disorder on some of the sites in the lattice. This disorder may come from, for example, there being a different type of atom or tightly bound electrons resulting in a larger on-site potential, affecting the electronic properties of the lattice. In this thesis the electronic properties of square, periodic lattices with a number of these impurity sites are studied. The main electronic properties that are studied are the electronic energies of the
lattice, and the density of states. Several properties of the impurity sites are also studied. The Hamiltonians that describe the different lattices were created with the tight-binding model. Two models for the placements of impurities were considered. The binary mixture model which considers random placements of disorders and the Falicov-Kimball model which considers thermodynamic placements of disorder. The results from these models were discussed and compared. For the Falicov-Kimball model a Monte Carlo algorithm was developed to examine the system and provide accurate results. Some papers that had previously looked at approximate solutions to the Falicov-Kimball model were examined and some of the results were recreated in the hope that the developed program could be used as a benchmark for approximate solutions. Recreations of works with similar algorithms were very accurate and for the more approximate solutions my simulations shared all of the important characteristics. The efficiency of the Monte Carlo algorithm was evaluated by first studying a benchmark smaller system that could be calculated and verified by hand and then using a benchmark of a previous paper that solved the systems in a similar way. It was determined that the Monte Carlo algorithm worked as expected. (Less)
Popular Abstract (Swedish)
Studier av material och deras egenskaper har nog aldrig varit mer relevant. Mycket av den teknologin som vi använder idag existerar just på grund av att vi förstår oss på de uppbyggande materialens egenskaper. Det är därför av stort intresse att fortsätta undersöka vad det är som ger material sina egenskaper och hur vi skulle kunna använda oss av det för att utveckla smartare, billigare, och mer effektiva material.

I denna tes så har två olika modeller som beskriver hur elektronerna i material beter sig studerats. Den ena modellen är lite simplare och tar inte speciellt många fysiska parametrar i åtanke. Den andra modellen, den så kallade Falicov-Kimball modellen är lite mer verklighetstrogen. Den har använts i decennier för att studera... (More)
Studier av material och deras egenskaper har nog aldrig varit mer relevant. Mycket av den teknologin som vi använder idag existerar just på grund av att vi förstår oss på de uppbyggande materialens egenskaper. Det är därför av stort intresse att fortsätta undersöka vad det är som ger material sina egenskaper och hur vi skulle kunna använda oss av det för att utveckla smartare, billigare, och mer effektiva material.

I denna tes så har två olika modeller som beskriver hur elektronerna i material beter sig studerats. Den ena modellen är lite simplare och tar inte speciellt många fysiska parametrar i åtanke. Den andra modellen, den så kallade Falicov-Kimball modellen är lite mer verklighetstrogen. Den har använts i decennier för att studera en mängd olika fenomen som övergångar mellan metall och insulator, kristallisering och mycket annat.

Materialen som studerades bestod av fyrkantiga rutnät av atomer i två dimensioner med sidolängder av 16 atomer. Beräkningarna har skett under den approximativa modellen ``Tight-binding model". Den beskriver att elektronerna sitter fast på sina respektive atomer men att de också har en viss sannolikhet att hoppa till en grannatom. På ungefär hälften av alla atomer så finns det så kallade ``impurities" vilket fysiskt kan motsvaras av en mängd olika saker. Ibland kan det motsvara hårt bundna elektroner på atomerna eller andra sorters atomer jämfört med resten av rutnätet.

För studeringen av Falicov-Kimball modellen så utvecklades en Monte Carlo, eller mer specifikt, en Metropolis-Hastings algoritm från grunden. Monte Carlo algoritmer används i en mängd olika fält, bl.a. finans, ingenjörskap, och självklart fysik.
Den algoritmen som utvecklades för just den här tesen fungerar på det sättet att slumpmässiga drag föreslås och om draget leder till en situation som är bättre än den nuvarande så sparas den som utgångspunkt för nästa drag. Genom att repetera denna process massvis med gånger så kan man på ett väldigt exakt sätt undersöka ett system.

Hoppet med denna tes är att den ska kunna användas som något slags utgångsläge som man kan jämföra mer approximativa metoder med för att se om ens resultat ser rimliga ut. På det sättet skulle man också kunna hitta saker som skulle förbättra de approximativa metoderna. Det övergripande målet är att forskningen inom det här området är utvecklingen av framtidens material. (Less)
Please use this url to cite or link to this publication:
author
Pettersson, Carl LU
supervisor
organization
course
FYSK03 20231
year
type
M2 - Bachelor Degree
subject
keywords
Monte Carlo, Tight-binding model, Falicov-Kimball model
language
English
id
9117275
date added to LUP
2023-05-31 08:58:36
date last changed
2023-05-31 08:58:36
@misc{9117275,
  abstract     = {{Oftentimes solids are described by uniform, periodic lattices. In reality, however, there is often some disorder on some of the sites in the lattice. This disorder may come from, for example, there being a different type of atom or tightly bound electrons resulting in a larger on-site potential, affecting the electronic properties of the lattice. In this thesis the electronic properties of square, periodic lattices with a number of these impurity sites are studied. The main electronic properties that are studied are the electronic energies of the
lattice, and the density of states. Several properties of the impurity sites are also studied. The Hamiltonians that describe the different lattices were created with the tight-binding model. Two models for the placements of impurities were considered. The binary mixture model which considers random placements of disorders and the Falicov-Kimball model which considers thermodynamic placements of disorder. The results from these models were discussed and compared. For the Falicov-Kimball model a Monte Carlo algorithm was developed to examine the system and provide accurate results. Some papers that had previously looked at approximate solutions to the Falicov-Kimball model were examined and some of the results were recreated in the hope that the developed program could be used as a benchmark for approximate solutions. Recreations of works with similar algorithms were very accurate and for the more approximate solutions my simulations shared all of the important characteristics. The efficiency of the Monte Carlo algorithm was evaluated by first studying a benchmark smaller system that could be calculated and verified by hand and then using a benchmark of a previous paper that solved the systems in a similar way. It was determined that the Monte Carlo algorithm worked as expected.}},
  author       = {{Pettersson, Carl}},
  language     = {{eng}},
  note         = {{Student Paper}},
  title        = {{Monte Carlo study of disorder in electronic tight-binding models}},
  year         = {{2023}},
}