Interband excitations in hexagonal two-dimensional conducting systems

Nakon što je uspješno sintetiziran grafen, prvi dvodimenzionalni sustav debljine jednog atoma, eksperimentalno i teorijsko istraživanje dvodimenzionalnih sustava značajno se inteziviralo. Posljedica je to velikih tehnoloških mogućnosti dvodimenzionalnih materijala koji su po svojim elektronskim karakteristikama vodiči (grafen) ili izolatori (heksagonalni borov nitrid i različiti dihalkogenidi s prijelaznim metalima). Stoga je vrlo važno razviti teorijski formalizam pomoću kojeg ćemo moći opisati najvažnija elektrodinamička svojstva ovih sustava. Na primjer, jednočestična pobuđenja koja imaju centralnu ulogu u objašnjenju električnog transporta, ili kolektivna pobuđenja, na primjer, plazmone i ekscitone. Sustavna formulacija teorijskih metoda uključuje nekoliko faza. Prvo će biti potrebno odrediti elektronske disperzije koje čine temelj analize elektrodinamičkih svojstava. U ovom razmatranju ograničit ćemo se na nekoliko vrpci u blizini Fermijevog nivoa. Vrpce određujemo pomoću aproksimacije čvrste veze te ćemo vidjeti da je za opis grafena i hBN-a dovoljno uzeti dvije orbitale u bazi dok je broj orbitala potrebnih za opis 2DMoS2 znatno veći. Zatim ćemo promotriti međudjelovanje elektrona i različitih bozonskih stupnjeva slobode. Prvo ćemo odrediti konstante elektron-fonon vezanja promatrajući promjenu matričnih elementa preskoka elektrona na prve susjede i promjenu energija atomskih orbitala uzrokovanih titranjem rešetke. Izračunat ćemo konstante elektron-fonon vezanja za akustične i optičke fonone i odrediti njihovu formu u dugovalnoj granici. Vezanje elektrona i vanjskih elektromagnetskih polja opisati ćemo pomoću Peierlsove supstitucije u aproksimaciji čvrste veze u režimu linearnog odziva. Izvest ćemo standardne oblike interakcije nabojnih i strujnih gustoća s vanjskim skalarnim i vektorskim potencijalima i objasniti ulogu nabojnih, strujnih i dipolnih vršnih funkcija. Transportna svojstva vodljivih elektrona studirati ćemo u režimu linearnog odziva razmatrajući semiklasične i kvantne transportne jednadžbe. U oba slučaja centralnu ulogu ima neravnotežna funkcija raspodjele koja se u semiklasičnim jednadžbama definira kao neravnotežna funkcija raspodjele za elektron impulsa ħk na položaju r, dok je u kvantnim transportnim jednadžbama povezana s propagatorom elektron-šupljina para čiji su valni vektori k+q i k. Dinamiku elektron šupljina para unutar kvantnih transportnih jednadžbi odrediti ćemo pomoću Heisenbergove jednadžbe, te ćemo pokazati da sva raspršenja elektrona, na drugim elektronima, na nečistoćama, i na fononima, možemo opisati preko tri doprinosa relaksacijskoj funkciji koju ćemo zvati memorijska funkcija. Alternativa formalizmu jednadžbi gibanja za elektron-šupljina propagator su standardne perturbativne metode računa odzivnih funkcija. Na taj način obično defniramo različite korelacijske funkcije, koje nazivamo Kubo formule; na primjer, Kubo formula za naboj-naboj korelacijsku funkciju, za struja-struja korelacijsku funkciju, itd. Pomoću jednadžbe kontinuiteta i zahtjeva baždarne invarijatnosti pokazat ćemo veze između korelacijskih funkcija te pokazati da je tenzor vodljivosti zapravo struja-dipol korelacijska funkcija. Izračunati ćemo istosmjernu i dinamičku vodljivost grafena i dopiranog 2DMoS2, analizirati strukturu momentum distribucijske funkcije i njenu ulogu u računu istosmjerne vodljivosti u nedopiranom grafenu. Zatim razmatramo unutarvpčanu dinamičku vodljivost dopiranog grafena u slučaju kada u memorijskoj funkciji zadržavamo samo raspršenja na nečistoćama i fononima. Rezultati dobiveni na taj način su uspoređeni s rezultatima aproksimacije relaksacijskog vremena. Na kraju promatramo međuvrpčanu vodljivost dopiranog grafena. Ponovno promatramo dvije aproksimacije: aproksimaciju relaksacijskog vremena i aproksimaciju međuvrpčane memorijske funkcije. Konačno, analizu kolektivnih pobuđenja vršimo razmatrajući dinamiku onog istog elektron-šupljina propagatora kojeg smo razmatrali u transportnim jednadžbama, no sada dugodosežnu kulonsku interakciju uključujemo eksplicitno u Heisenbergovu jednadžbu. U jednadžbama gibanja za elektron-šupljina propagator za generalni problem s više vrpci jasno ćemo razlikovati unutarvrpčana od međuvrpčanih kolektivnih pobuđenja. Pritom ćemo pokazati da se ladder i Fockov doprinos u unutarvpčanom kanalu međusobno dokidaju u vodećem članu. Preostaje RPA doprinos koji ponovno ulazi u definiciju makroskopskog električnog polja te Fockovi doprinosi višeg reda koji ulaze u doprinos memorijskoj funkciji koji je porijeklom od direktnih elektron-elektron interakcija, te ćemo odrediti disperziju plazmona. U slučaju izolatora izvest ćemo jednadžbu gibanja za međuvrpčani elektron-šupljina propagator i pronaći energije kolektivnih modova, ekscitona, koje imaju slične karakteristike kao energije dvodimenzionalnog pozitronija. Tako dobivene energije slabo se slažu s eksperimentalnim podacima. Stoga će biti potrebno poopćiti model i uključiti elektron-šupljina korelacijske efekte koji dolaze od međuvrpčanih prijelaza. Te efekte uključujemo u jednadžbe preko zasjenjene kulonske interakcije.CHAPTER 1 Experimental and theoretical research of two-dimensional materials is highly intensifying. This is due to their fascinating physical properties which are of technological importance. We shall give a simple overview of the history and physical properties of the three most important members of the two-dimensional materials, graphene, hexagonal boron nitride and molybdenum disulfide. CHAPTER 2 Every thorough analysis of the electromagnetic properties of crystals begins with the calculation of electronic dispersions, which generally is not an easy task. Some approximations are necessary in order to find simple analytical expressions for electronic dispersions. The most important one is to focus our attention only to a few bands nearest to the Fermi level. We calculate these dispersions by using the tight binding approximation (TBA). We start from atomic orbitals or their linear combinations (hybrids), for which the hopping probability is highest. The choice of atomic orbitals is determined by crystal symmetry, which will be different for all three systems considered in this thesis: graphene, hexagonal boron nitride and molybdenum disulfide. We shall briefly introduce the hexagonal lattice with a basis, display the electronic dispersions of the valence bands in aforementioned systems for typical TBA parameters, and estimate the values of these parameters by comparison with ab initio calculations. CHAPTER 3 In this chapter we study the interaction of conduction electrons with various bosonic modes. First, we determine the electron-phonon coupling constants by looking at how lattice vibrations change the hopping matrix elements between neighbouring sites and atomic energies on those sites. We calculate these constants for acoustic and optical phonons and find their long-wavelength forms. The electron coupling to external electromagnetic fields is described by the Peierls substitution in the TBA, for a linear response regime. We shall derive the standard form of the interaction of the charge and current densities with external scalar and vector potentials and explain the role of charge, current and dipole vertex functions. These coupling constants and vertex functions are naturally defined in the direct-space representation, or in the delocalized orbital representation. Therefore, we shall use the Fourier transformations to determine the corresponding contributions to the total Hamiltonian in the Bloch representation. CHAPTER 4 Transport properties of conducting systems are usually studied in the linear response regime. This can be done using semi-classical transport equations or quantum transport equations. In both cases, the non-equilibrium distribution function has a central role. The semiclassical Boltzmann equations define it as a non-equilibrium distribution function for an electron of momentum ħk at a position r, whereas in the quantum equations and their semiclassical regimes it is related to the propagator of an electron-hole pair with momenta k + q and k. The Landau transport equations are a generalization of Boltzmann equations, where electron-electron interactions are treated in a self-consistent way and the scattering from the static disorder and from phonons is included phenomenologically. We can reach the Landau equations following the usual Landau’s procedure, or from the Heisenberg equations of motion for the electron-hole propagator. The advantage of the latter approach is that all electron scattering, from electrons, from disorder and from phonons, can be described in the same way, by studying the three contributions to the relaxation function, which will be called here the memory function. In this chapter we shall introduce the non-equilibrium distribution function n (k, r, t), derive the semiclassical equations of motion, determine the structure of the memory function for the scattering by phonons and describe the role of its real and imaginary parts in the dynamical conductivity tensor. CHAPTER 5 An alternative to the equations of motion approach from the previous chapter are the usual perturbative methods of calculating response functions. In this way various correlation functions are usually defined, named the Kubo formulae; for example, the Kubo formula for the charge-charge or current-current correlation functions, etc. There are two basic formalisms; the first one at T = 0 and the Matsubara finite-temperature formalism. The conductivity tensor, introduced in the previous chapter, represents the current-dipole correlation function, linking the current (response of the system) induced in the system by a macroscopic field (external perturbation). The external field couples to the dipole moment operator. In a similar manner, other response functions of interest can be defined. In this chapter, we shall use the T =0 formalism where response functions represent the retarded correlation functions (the response follows the external perturbation). We shall define all correlation functions related to the conductivity tensor, derive their general relations, and make sure that these relations are satisfied in the lowest order, usually called the one fermion loop approximation. We shall study the effects of phenomenologically introduced relaxation processes. We shall show that in general considerations, the continuity equation and Kramers-Kronig relations play an important role. CHAPTER 6 In this chapter, we study the DC and dynamic conductivity of graphene and doped molybdenum disulfide, using the general expressions derived in chapters 4 and 5. First, we analyze the structure of the momentum distribution function and explain its role in the calculation of the DC conductivity of undoped graphene. Afterwards, we study the intraband dynamical conductivity of doped graphene, keeping in the memory function only the scattering by phonons by disorder. These results are compared with the relaxation-time approximation results. In the end, we consider the interband conductivity of doped graphene, once again in two approximations: the relaxation-time and interband memory function approach. CHAPTER 7 In this chapter, we analyze elementary excitations in the electron subsystem. Unlike the transport equations studied in the third chapter, where long-range Coulomb interactions were taken into account implicitly through the macroscopic electric field, here we will include various electron-electron interaction contributions step by step. The primary interest here is to study the collective excitations in the electronic subsystem. In the electron-hole equations of motion for a general problem with multiple bands, we shall clearly distinguish the intraband and interband collective modes. We shall show that the "ladder" and Fock contributions cancel out in the intraband channel. What remains is the RPA contribution (being part of the macroscopic field definition) and higher order Fock contributions which lead to the electron-electron contribution to the memory functions. For the insulating case, we shall derive the equation of motion for the interband electron-hole propagator and find the energies of collective modes, excitons, which have energies similar to the 2D positronium. Energies calculated in this way do not agree well with experimental data, so it will be necessary to include electron-hole correlation effects. These effects are included in our equations by using screened Coulomb interaction instead of the bare one