Trạng thái ngưng tụ exciton trong các hệ có chuyển pha bán kim loại bán dẫn tt tiếng anh

Electrons and holes have semi-integer spin, so the excitonsact as bosons and if the temperature is sufficiently low, these excitons can condense in a newmacroscopic phase-coherent quantu

MINISTRY OF EDUCATION VIETNAM ACADEMY OF SCIENCE

GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY

———————————-DO THI HONG HAI

EXCITONIC CONDENSATION

IN SEMIMETAL – SEMICONDUCTOR TRANSITION SYSTEMS

Major: Theoretical Physics and Maths Physics

Code: 9.44.01.03

SUMMARY OF PHYCICS DOCTORAL THESIS

Hanoi – 2020

The thesis has been completed at Graduate University of Science and Technology, VietnamAcademy of Science and Technology.

Supervisor 1: Assoc.Prof.Dr Phan Van Nham

Supervisor 2: Assoc.Prof.Dr Tran Minh Tien

Hardcopy of the thesis can be found at:

- Library of Graduate University of Science and Technology

- National Library of Vietnam

1 Motivation

The condensate state of the electron-hole pairs (or excitons) has recently become one

of the attractive research objects Electrons and holes have semi-integer spin, so the excitonsact as bosons and if the temperature is sufficiently low, these excitons can condense in a newmacroscopic phase-coherent quantum state called an excitonic insulator – EI

Although first theoretical of the excitonic condensation state in the semimetal (SM) andsemiconductor (SC) systems was proposed over a half of century ago but the experimentalrealization has proven to be quite challenging In recent years, materials promising to observe

EI state have been investigated, such as mixed-valent rare-earth chalcogenide TmSe0.45Te0.55,transition-metal dichalcogenide 1T-TiSe2, semiconductor Ta2NiSe5, layer double graphene, which have increased the studies of the excitonic condensation state both the theoretical sideand the experimental side

On the theoretical side, the excitonic condensation state is often studied through gating the extended Falicov-Kimball model by many different methods such as the mean-field(MF) theory andT −matrices, an SO(2)-invariant slave-boson approach, the approximate vari-ational cluster method, projector-based renormalization (PR) method, The authors haveshown the existence of the excitonic condensation state near the SM – SC transition However,

investi-in the above studies, investi-investigatinvesti-ing the EI state was mainvesti-inly based on purely electronic teristics with the attractive Coulomb interaction between electrons and holes Therefore, thecoupling of electrons or excitons to the phonon was completely neglected

charac-Besides, when studying the EI state of the semimetallic1T-TiSe2 by applying BCS perconductivity theory to the electron – hole pairs, C Monney and co-workers have con-firmed that the condensation of excitons affects the lattice through an electron – phonon in-teraction at low temperature Recently, when studying the condensation state of excitons intransition metal Ta2NiSe5 by using the band structure calculation and MF analysis for thethree-chain Hubbard model phonon degrees of freedom, T Kaneko has confirmed the origin

su-of the orthorhombic-to-monoclinic phase transition Without any doubt, lattice distortion orphonon effects are significantly important in this kind of material, particularly, in establishingthe excitonic condensation state Based on this, B Zenker and co-workers studied the EI state

in a two-band model by using the Kadanoff-Baym approach and mean-field Green function,

or in the EFK model concluding one valence and three conduction bands by using the MFapproximation and the frozen-phonon approximation when considering both the Coulomb in-teraction between the electron – hole and the electron – phonon interaction The authors haveaffirmed that that both the Coulomb interaction and the electron – phonon coupling act to-gether in binding the electron – hole pairs and establishing the excitonic condensation state

However, B Zenker has studied only for the ground state, i.e., at zero temperature.

Recently, in Vietnam, investigating EI state in EFK model was also studied by Phan VanNham and co-workers in a completely quantum viewpoint By PR method, lattice distortioncausing EI state is also intensively studied on the theoretical side, however, only for the groundstate In general, as a kind of superfluidity, the EI state possibly occurs at finite temperature,and at high temperature, it might be deformed by thermal fluctuations Clearly, the study ofthe excitonic condensation in Vietnam need to be further promoted In order to contribute tothe development of new research in Vietnam on the excitonic condensation, in the present

thesis, we focus on the problem of “Excitonic condensation in semimetal – semiconductor

transition systems” to investigate the nature of the excitonic condensation state in these

mod-els by using MF theory Electronic correlation in the systems is described by the two-bandmodel including electron – phonon interaction and the extended Falicov-Kimball model in-volving electron – phonon interaction Under the influence of Coulomb interaction betweenelectron – hole, the electron – phonon interaction as well as the influence of the temperature

or the extenal pressure, the nature of the excitonic condensation state especially the BCS –BEC crossover or competition with the CDW state in the system is clarified

• Studying the properties of electronic systems in EI state through investigating the abovemodels Then, we compare the nature of each condensation state on both sides of theBCS – BEC crossover or the competition with the CDW state

3 Main contents

The content of the thesis includes: Introduction of exciton and excitonic condensationstates; Mean-field theory and application; The results of the study about excitonic condensa-tion state in the two-band models when considering effects of phonon, the Coulomb interac-tion, the extenal pressure and the temperature by mean-field theory The main results of thethesis are presented in chapters 3 and 4

CHAPTER 1 EXCITON AND EXCITONIC CONDENSATION STATES

1.1 The concept of excitons

1.1.2 The exciton creation and annihilation operators

Considering a two-band model with fp† and c†p are hole creation operators in valenceband and electron creation operators in conduction band with momentump, respectively Wecan write exciton creation operators relating with electron and hole creation operators

a†k,n= √1

VX

p,p 0

δk,p+p0 ϕn(q)c†pfp†0 (1.17)

From the anticommuting properties of creation, annihilation operators of electrons andholes, the excitons atc as bosons with the creation and annihilation operators satisfying thecommutation relations

1.2 BEC and excitonic condensation states

Bose-Einstein condensed (BEC) is the condensation state of bosons at low ture with a large number of particles in the same quantum state Because the excitons arepseudo-bosons, they condensate in the BEC state in the low density limit as the independentatoms and the Fermi surface does not play a role in the formation of electron – hole pairs

tempera-In contrast, the excitons condensate in the BCS state in the high density limit similar to thesuperconducting state described by the BCS theory Studying the BCS – BEC crossover ofexcitons is considered an interesting problem when examining excitonic condensation state

As the temperature increases, condensased states are broken by temperature fluctuations Thesystem transfers to a free exciton gas state from the BEC-type, while the BCS-type transfers

to an plasma of electrons and holes

1.3 Achievements of excitonic condensation research

1.3.1 Theoretical research

By applying from the MF approximation to the more complex methods for the EFKmodel, the existence of EI state in both BCS-type and BEC-type near the SM – SC transitionhas been confirmed Then the BCS – BEC crossover of EI phase is also considered

When studying the EI state of the SM structure 1T-TiSe2, C Monney and co-workersconfirmed the existence of the EI state at low temperature and the electron-hole pairing maylead to the Ti ionic displacement In other words, the exciton causes a lattice displacement

through electron – phonon interaction at low temperature B Zenker et al studied the EI

state in the EFK model by using the MF theory and the frozen-phonon approximation whenconsidering the influence of electron – phonon interaction The authors have confirmed thatboth the Coulomb interaction and the electron – phonon coupling act together in binding theelectron – hole pairs and establishing the excitonic condensation state However, B Zenkerhas studied only at zero temperature

1.3.2 Experimental observation

In strongly correlated electronic systems, it is difficult to observe excitonic condensationstate However, increasing of experimental observations on some materials has confirmedthe existence of the EI state which is theoretically predicted For example, in semiconductor

Ta2NiSe5or in transition metal dichalcogenide1T-TiSe2, ARPES shows the flattening of thevalence peak at low temperature, this only is explained by the formation of an EI state In anarrow band SC TmSe0.45Te0.55, studying of P Wachter and co-workers have proposed that

an excitonic bound state of a 4f hole at the Γ-point and a 5d electron at the X-point can becreated These excitons condense into an EI superfluid state at sufficiently low temperatures

CHAPTER 2 MEAN FIELD THEORY 2.1 The basic concepts

2.1.1 Mathematical representation of mean-field theory

Considering a system with two kind of particles, described by operatorsaν and bµ, spectively Let us assume that only interactions between different kind of particles are impor-tant By relapcing the pairing operatorsa†νa ν 0 with their average values and a small correction.Neglecting the constant contribution, Hamiltonian is written

2.1.2 The art of mean field theory

In the MF approach, Hamiltonian of the system is often separated into separate parts

containing single-particle operators, so it is easy to calculate the expected values based onHamiltonian Thus, the MF approximation gives a physical significance result to the study ofthe interaction systems, in which the correlations are less important The choice of the mean-field is important, depending on the particular problem.

2.2 Hartree-Fock approximation

Hartree-Fock approximation (HFA) is one of the methods of MF theory For differentparticles system, we applied the approximation to the interaction term so-called the Hartreeapproximation However, for the like particles, Hamiltonian not only contains the Hartree termbut also the Fock term when taking into account the contribution of the exchange interaction.The mean-field Hamiltonian in HFA is written in the form

HHF = H0+VintFock+VintHartree, (2.21)

op-2.3 Broken symmetry

2.3.1 The concept of phase transition and broken symmetry

At the critical temperature, the thermodynamical state of the system develops non-zeroexpectation value of some macroscopic quantities which have a symetry lower than the orig-inal Hamiltonian, it is called spontaneous breaking of symmetry Those quantities are calledorder parameters that indicate the phase transition For the mean field theory, we select thefinite mean field through order parameters, then we derive a set of self-consistent equationsdetermining the order parameters

2.3.2 The Heisenberg model of ionic ferromagnets

Applying the MF theory to the Hamiltonian of Heisenberg ferromagnetic model, weobtain MF Hamiltonian which is diagonalized in the site index

2.3.3 The Stoner model of metallic ferromagnets

Applying HFA to the metallic ferromagnet model, based on the Hubbard model, the MFHamiltonian becomes

kσ

εM Fkσ c†kσckσ− U V

2X

σ

nσn−σ+U V

2X

2.3.5 The excitonic insulator – EI

Applying MF approximation to the electronic system in the two-band model with Coulombinteraction between them1 Similar to the superconducting state survey in BCS theory, exci-tonic condensation state is characterized by quantityhc†kfki 6= 0 In HFA, neglecteing constants

1 Note that, the electronic representation is completely equivalent to the hole representation by electronic transformation – hole Then the annihilation operator of electron is replaced with the creation operator of hole and vice versa.

we can rewrite Hamiltonian

This equation is similar to the gap equation of superconducting in BCS theory ∆k 6= 0 dicates the hybridization between electrons in the valence band and the conduction band.Therefore, the system turn into the excitonic insulator state

in-CHAPTER 3 EXCITONS CONDENSATE IN THE TWO-BAND MODEL

INVOLVING ELECTRON – PHONON INTERACTION 3.1 The two-band electronic model involving electron – phonon interaction

The Hamiltonian for the two-band electronic model involving electron – phonon action can be written

where εc,f are the on-site energies; tc,f are the nearest-neighbor particle transfer amplitudes.

In a 2D square lattice, γk = 2 (cos kx+ cos ky)andµis the chemical potential At sufficientlylow temperature, the bound pairs with finite momentumQ = (π, π)might condense, indicated

by a non-zero value ofdk = hc†k+Qfkiand

NX

k

These quantities express the hybridization between c and f electrons so they are called theorder parameters of the excitonic condensate The order parameter is nonzero representing thesystem stabilize in excitonic condensation state

3.2 Applying mean field theory

Applying MF theory with mean fields

NX

k

act as the order parameters which specify to spontaneous broken symmetry, Hamiltonian in(3.1) is reduced to Hamiltonian Hartree-Fock involving two parts, the electronic part (He) andthe phononic part (Hph) are as follows

withWk = (εck+Q− εfk) 2 + 4|∆| 2 The quadratic form of Eq (3.17) allows us to compute allexpectation values, resulting in

nck+Q = hc†k+Qck+Qi = ξk2nF(Ek1) + ηk2nF(Ek2),

dk = hc†k+Qfki = −[nF(Ek1) − nF(Ek2)]sgn(εfk− εck+Q) ∆

Wk,here nF(Ek1,2)are the Fermi-Dirac distribution functions; ξk and ηk are the prefactors of theBogoliubov transformation which satisfyξk2+ η2k = 1 The lattice displacement in the EI state

ω 0

r2

ω 0

3.3 Numerical results and discussion

For the two-dimensional system consisting ofN = 150 × 150lattice sites, the numericalresults are obtained by solving self-consistently Eqs (3.9), (3.10), (3.22) and (3.24) startingfrom some guessed values for hb†Qi and dk with a relative error 10−6 In what follows, allenergies are given in units oftc and we fixtf = 0.3 to consider the half-filled band case, i.e

nc+ nf = 1 The chemical potentialµhas to be adjusted such that this equation is satisfied

3.3.1 The ground state

0.0 0.5 1.0 1.5 2.0 2.5 3.0 -0.2

0.0 0.2 0.4 0.6 0.8 1.0

Fig 3.2 shows the dependence of the excitonic condensate order parameterd at T = 0

on the phonon frequencyω 0 for different values of g atεc− εf = 1 For a given value of thecoupling constant, the order parameter decreases when increasing phonon frequency This is

also shown in Fig 3.5 the dependence of the order parameter d and the lattice displacement

xQ on εc − ε f for some values of ω0 at g = 0.5, T = 0 The diagram shows that d and xQare intimately related When increasingω0, bothdandxQ decrease significantly, indicating aweakened condensation state.dandxQare non-zero, the systems thus stabilize in the excitoniccondensation state with the charge density wave state (EI/CDW)

g=0.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0



g=0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0

0.5 1.0 1.5 2.0 2.5 3.0

ofg The excitonic condensation phase is indicated in orange.

Fig 3.6 shows the phase diagram of the model in the(εc−ε f , ω0)plane in the ground statefor different g Ifg is large enough, we always find the excitonic condensate regime EI/CDW(orange) when the phonon frequency is less than the critical value ω0c This critical valueincreases when increasingg The excitonic condensation regime is narrowed if decreasing thetwo energy bands overlap and the electron – phonon interaction constant

3.3.2 The effect of thermal fluctuations

Fig 3.7 describes the dependence of the order parameterd on the phonon frequencyω0when varying the temperature atεc− εf = 1andg = 0.5 For a given value of temperature, thevalue of the order parameter decreases rapidly when increasing the phonon frequency Thedependence of the order parameter d the lattice displacement xQ on the electron – phononinteraction when the temperature changes forεc− ε f = 1 andω0 = 0.5are shown in Fig 3.8

d andxQ are always closely related, they are non-zero i.e the system exists in EI/CDW state

0.0 0.1 0.2 0.3 0.4 0.5 -0.5

0.0 0.5 1.0 1.5

0

T=0.2

0.5 1.0 1.5 2.0 2.5 3.0 0.0

temperature The excitonic condensation phase is indicated in orange

when the electron – phonon coupling is larger than a critical valueg c

Fig 3.9 shows the phase diagram in the(ω0, g)plane whenεc− ε f = 1for some values oftemperature The larger phonon frequency, the greater critical value gc for phase transition ofthe excitonic condansation state The higher temperature, the narrower condensation region