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

MINISTRY OF EDUCATION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY 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, Vietnam Academy of Science and Technology Supervisor 1: Assoc.Prof.Dr Phan Van Nham Supervisor 2: Assoc.Prof.Dr Tran Minh Tien Reviewer 1: Reviewer 2: Reviewer 3: The thesis will be defended to the thesis committee for the Doctoral Degree, at Graduate University of Science and Technology – Vietnam Academy of Science and Technology, on Date Month Year 2020 Hardcopy of the thesis can be found at: - Library of Graduate University of Science and Technology - National Library of Vietnam INTRODUCTION 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 excitons act as bosons and if the temperature is sufficiently low, these excitons can condense in a new macroscopic phase-coherent quantum state called an excitonic insulator – EI Although first theoretical of the excitonic condensation state in the semimetal (SM) and semiconductor (SC) systems was proposed over a half of century ago but the experimental realization 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 TmSe0.45 Te0.55 , transition-metal dichalcogenide 1T -TiSe2 , semiconductor Ta2 NiSe5 , layer double graphene, which have increased the studies of the excitonic condensation state both the theoretical side and the experimental side On the theoretical side, the excitonic condensation state is often studied through investigating the extended Falicov-Kimball model by many different methods such as the mean-field (MF) theory and T − matrices, an SO(2)-invariant slave-boson approach, the approximate variational cluster method, projector-based renormalization (PR) method, The authors have shown the existence of the excitonic condensation state near the SM – SC transition However, in the above studies, investigating the EI state was mainly based on purely electronic characteristics with the attractive Coulomb interaction between electrons and holes Therefore, the coupling of electrons or excitons to the phonon was completely neglected Besides, when studying the EI state of the semimetallic 1T -TiSe2 by applying BCS superconductivity theory to the electron – hole pairs, C Monney and co-workers have confirmed that the condensation of excitons affects the lattice through an electron – phonon interaction at low temperature Recently, when studying the condensation state of excitons in transition metal Ta2 NiSe5 by using the band structure calculation and MF analysis for the three-chain Hubbard model phonon degrees of freedom, T Kaneko has confirmed the origin of the orthorhombic-to-monoclinic phase transition Without any doubt, lattice distortion or phonon effects are significantly important in this kind of material, particularly, in establishing the 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 MF approximation and the frozen-phonon approximation when considering both the Coulomb interaction between the electron – hole and the electron – phonon interaction The authors have affirmed that that both the Coulomb interaction and the electron – phonon coupling act together 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 Van Nham and co-workers in a completely quantum viewpoint By PR method, lattice distortion causing EI state is also intensively studied on the theoretical side, however, only for the ground state 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 of the excitonic condensation in Vietnam need to be further promoted In order to contribute to the 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 models by using MF theory Electronic correlation in the systems is described by the two-band model including electron – phonon interaction and the extended Falicov-Kimball model involving electron – phonon interaction Under the influence of Coulomb interaction between electron – 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 Purpose Investigating the excitonic condensation phase transition in SM – SC transition systems In detail: • Developing mean-field theory for a 2D two-band model including electron – phonon interaction and the extended Falicov-Kimball model involving electron/hole – phonon interaction • Studying the properties of electronic systems in EI state through investigating the above models Then, we compare the nature of each condensation state on both sides of the BCS – BEC crossover or the competition with the CDW state Main contents The content of the thesis includes: Introduction of exciton and excitonic condensation states; Mean-field theory and application; The results of the study about excitonic condensation state in the two-band models when considering effects of phonon, the Coulomb interaction, the extenal pressure and the temperature by mean-field theory The main results of the thesis are presented in chapters and CHAPTER EXCITON AND EXCITONIC CONDENSATION STATES 1.1 The concept of excitons 1.1.1 What is an exciton? An exciton is a bound state of an electron in conduction band and a hole in valence band which are attracted to each other by the Coulomb interaction Depending on the role of Coulomb attraction in different systems that the size of the excitons can vary from a few angstroms to a few hundred angstroms 1.1.2 The exciton creation and annihilation operators Considering a two-band model with fp† and c†p are hole creation operators in valence band and electron creation operators in conduction band with momentum p, respectively We can write exciton creation operators relating with electron and hole creation operators a†k,n = √ V δk,p+p ϕn (q)c†p fp† (1.17) p,p From the anticommuting properties of creation, annihilation operators of electrons and holes, the excitons atc as bosons with the creation and annihilation operators satisfying the commutation relations 1.2 BEC and excitonic condensation states Bose-Einstein condensed (BEC) is the condensation state of bosons at low temperature with a large number of particles in the same quantum state Because the excitons are pseudo-bosons, they condensate in the BEC state in the low density limit as the independent atoms and the Fermi surface does not play a role in the formation of electron – hole pairs In contrast, the excitons condensate in the BCS state in the high density limit similar to the superconducting state described by the BCS theory Studying the BCS – BEC crossover of excitons is considered an interesting problem when examining excitonic condensation state As the temperature increases, condensased states are broken by temperature fluctuations The system 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 EFK model, the existence of EI state in both BCS-type and BEC-type near the SM – SC transition has 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-workers confirmed the existence of the EI state at low temperature and the electron-hole pairing may lead 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 when considering the influence of electron – phonon interaction The authors have confirmed that both the Coulomb interaction and the electron – phonon coupling act together in binding the electron – hole pairs and establishing the excitonic condensation state However, B Zenker has studied only at zero temperature 1.3.2 Experimental observation In strongly correlated electronic systems, it is difficult to observe excitonic condensation state However, increasing of experimental observations on some materials has confirmed the existence of the EI state which is theoretically predicted For example, in semiconductor Ta2 NiSe5 or in transition metal dichalcogenide 1T -TiSe2 , ARPES shows the flattening of the valence peak at low temperature, this only is explained by the formation of an EI state In a narrow band SC TmSe0.45 Te0.55 , 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 be created These excitons condense into an EI superfluid state at sufficiently low temperatures CHAPTER 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 operators aν and bµ , respectively Let us assume that only interactions between different kind of particles are important By relapcing the pairing operators a†ν aν with their average values and a small correction Neglecting the constant contribution, Hamiltonian is written a b HM F = HM F + HM F , (2.8) where εaν a†ν aν + a HM F = ν (2.9) Vµν,µ ν a†ν aν b†µ bµ (2.10) µµ εbµ b†µ bµ + b HM F = Vνµ,ν µ b†µ bµ a†ν aν , µ νν a(b) where HM F can be considered as Hamiltonian describing the a(b) particles moving in the mean field caused by b(a) particles Obviously HM F contains only single-particle operators Thus, the multi-particle system problem has been replaced by the one-particle system problem and easily gives accurate results 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 on Hamiltonian Thus, the MF approximation gives a physical significance result to the study of the interaction systems, in which the correlations are less important The choice of the meanfield 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 different particles system, we applied the approximation to the interaction term so-called the Hartree approximation However, for the like particles, Hamiltonian not only contains the Hartree term but 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 Fock Hartree HHF = H0 +Vint +Vint , (2.21) where Hartree = Vint Fock Vint = ± Vνµ,ν µ nµµ c†ν cν + 2 Vνµ,ν µ nνµ c†µ cν ± Vνµ,ν µ nνν c†µ cµ − 2 Vνµ,ν µ nµν c†ν cµ ∓ Vνµ,ν µ nνν nµµ , (2.22) Vνµ,ν µ nνµ nµν , (2.23) where nνν = c†ν cν and nνµ = c†ν cµ with c†ν , cν are particle creation and annihilation operators with the quantum number ν , respectively The (+) mark applies to the boson particle system, and the (−) mark applies to the fermion system 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-zero expectation value of some macroscopic quantities which have a symetry lower than the original Hamiltonian, it is called spontaneous breaking of symmetry Those quantities are called order parameters that indicate the phase transition For the mean field theory, we select the finite mean field through order parameters, then we derive a set of self-consistent equations determining the order parameters 2.3.2 The Heisenberg model of ionic ferromagnets Applying the MF theory to the Hamiltonian of Heisenberg ferromagnetic model, we obtain MF Hamiltonian which is diagonalized in the site index HM F = −2 mSi + mN Sz (2.28) i We can easily derive the equation α = tanh(bα), (2.31) where α = m/nJ0 b = nJ0 β This equation can be numerically solved and the result given the temperature dependence of the magnetization m 2.3.3 The Stoner model of metallic ferromagnets Applying HFA to the metallic ferromagnet model, based on the Hubbard model, the MF Hamiltonian becomes F † εM kσ ckσ ckσ − HM F = kσ UV nσ n−σ + σ UV n2σ , (2.39) σ where F εM ¯, kσ = εk + U (n↑ + n↓ − nσ ) = εk + U nσ with nσ = V k (2.40) c†kσ ckσ is the spin density From this Hamiltonian, we can find self-consistent equations for the spin density Then we find the solutions of the model 2.3.4 BCS theory One of the most striking examples of symmetry breaking is the superconducting phase transition c†kσ and ckσ are creation and annihilation operators with momentum k and spin σ =↓, ↑, respectively, BCS Hamiltonian in HFA is εk c†kσ ckσ − MF HBCS = kσ (∆k c†k↑ c†−k↓ + H.c.), (2.51) k where ∆k = − Vkk c−k ↓ ck ↑ , (2.52) k is called the gap equation This Hamiltonian is solved by the Bogoliubov transformation de† termining new fermionic operators αk↑ and α−k↓ which are called creation and annihilation quasiparticle operators αk↑ = u∗k ck↑ + vk c†−k↓ , † α−k↓ = −vk∗ ck↑ + uk c†−k↓ (2.56) where u2k + vk2 = Finally, BCS Hamiltonian can be diagonalized in a form † † Ek (αk↑ αk↑ + αk↓ αk↓ ), MF HBCS = (2.59) k where Ek = ε2k + |∆k |2 Using this Hamiltonian, we can find solutions of the gap equation Then we get the BCS prediction that the ratio of gap to critical temperature which agrees qualitatively with extracted data from experiments 2.3.5 The excitonic insulator – EI Applying MF approximation to the electronic system in the two-band model with Coulomb interaction between them Similar to the superconducting state survey in BCS theory, excitonic condensation state is characterized by quantity c†k fk = In HFA, neglecteing constants 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 ε˜ck c†k ck + HM F = ε˜fk fk† fk + k k (∆k fk† ck + H.c.), (2.71) k where Vk−k c†k fk ∆k = (2.72) k acts as an energy gap, or EI state order parameter ε˜ck and ε˜fk are the dispersion energies of c electrons and f electrons having contribution of Hartree-Fock energy shift In order to diagonalize the Hamiltonian, we use the Bogoliubov transformation to define the new fermion operators αk and βk The Hamiltonian of the system in the MF approximation will be completely diagonalized Ekα αk† αk + MF HEI = k Ekβ βk† βk , (2.79) k where α/β Ek ε˜ck + ε˜fk ∓ = ξk2 + |∆k |2 (2.80) with ξk = 21 [˜ εck − ε˜fk ] and Ek2 = ξk2 + |∆k |2 This Hamiltonian allows us to determine all expectation values At T = 0, ∆k is determined by the gap equation ∆k = Vk−k k ∆k 2Ek (2.81) This equation is similar to the gap equation of superconducting in BCS theory ∆k = indicates the hybridization between electrons in the valence band and the conduction band Therefore, the system turn into the excitonic insulator state CHAPTER 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 interaction can be written εck c†k ck + H= k εfk fk† fk + ω0 k q g b†q bq + √ N [c†k+q fk b†−q + bq + H.c.], (3.1) kq where c†k (ck ); fk† (fk ) and b†q (bq ) are creation (annihilation) operators of c, f electrons carrying momentum k and phonons carrying momentum q, respectively; g is a electron – phonon coupling constant; N is the number of the lattice sites c,f εc,f − tc,f γk − µ, k =ε (3.2) where εc,f are the on-site energies; tc,f are the nearest-neighbor particle transfer amplitudes In a 2D square lattice, γk = (cos kx + cos ky ) and µ is the chemical potential At sufficiently low temperature, the bound pairs with finite momentum Q = (π, π) might condense, indicated by a non-zero value of dk = c†k+Q fk and d= N ( c†k+Q fk + fk† ck+Q ), (3.4) k These quantities express the hybridization between c and f electrons so they are called the order parameters of the excitonic condensate The order parameter is nonzero representing the system stabilize in excitonic condensation state 3.2 Applying mean field theory Applying MF theory with mean fields g ∆ = √ b†−Q + b−Q , N g h = c†k+Q fk + fk† ck+Q , N (3.9) (3.10) 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 ) and the phononic part (Hph ) are as follows HHF = He + Hph , (3.11) where εck c†k ck + He = k εfk fk† fk + ∆ k (c†k+Q fk + fk† ck+Q ), (3.12) k √ N h(b†−Q + b−Q ), b†q bq + Hph = ω0 (3.13) q The phononic part is diagonalized by defining a new phonon operator Bq† = b†q + √ N h δq,Q ω0 (3.14) Meanwhile, the electronic part can be diagonalized itself by using a Bogoliubov transformation with the new quasi-particle fermionic operators C1k and C2k Then finally, we are led to a completely diagonalized Hamiltonian † † Ek1 C1k C1k + Ek2 C2k C2k + ω0 Hdia = Bq† Bq , (3.17) q k where the electronic quasiparticle energies read as Ek1,2 = εfk + εck+Q ∓ sgn(εfk − εck+Q ) Wk , (3.18) 0.0 1.5 T=0 T=0.1 T=0.2 T=0.3 d, xQ 1.0 d -0.1 T=0 T=0.1 T=0.2 T=0.3 0.5 -0.2 0.0 0.5 1.0 1.5 2.0 2.5 -0.5 0.0 3.0 0.1 0.2 0.3 0.4 0.5 0.6 g 0 Fig 3.7: The order parameter d as functions of Fig 3.8: The order parameter d (filled sym- the phonon frequency ω0 for different values bols) and the lattice displacement xQ (open of temperature at εc − εf = and g = 0.5 symbols) as functions of g for some values of T at εc − εf = and ω0 = 0.5 1.0 0.8 g 0.6 0.4 0.2 T=0 T=0.1 T=0.2 T=0.3 0.0 1.0 0.8 EI/CDW g 0.6 0.4 0.2 0.0 0.1 0.5 1.0 1.5 2.0 2.5 3.0 0.1 0.5 1.0 1.5 2.0 2.5 3.0 0 0 Fig 3.9: The phase diagram of the model in the (ω0 , g) plane at εc − εf = for some values of temperature The excitonic condensation phase is indicated in orange when the electron – phonon coupling is larger than a critical value gc Fig 3.9 shows the phase diagram in the (ω0 , g) plane when εc − εf = for some values of temperature The larger phonon frequency, the greater critical value gc for phase transition of the excitonic condansation state The higher temperature, the narrower condensation region 11 3.0 3.0 T=0 2.0 2.0 1.5 1.5 1.0 1.0 EI/CDW 0.5 0.0 0.0 T=0.1 2.5   2.5 0.5 EI/CDW 0.5 1.0 1.5  - c 2.0 2.5 3.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0  - c f f Fig 3.13: The phase diagram of the excitonic condansation state of the model in the (εc − εf , ω0 ) plane at g = 0.5 for T varies The excitonic condensation phase is indicated in orange Fig 3.13 shows the relationship of the phonon frequency and the c and f bands overlap (the external pressure) when T changes at g = 0.5 The diagram shows that if the temperature increases, the critical value ω0c decreases and the exciton condensation regime shrinks Fig 3.15: The dependence of the order parameter |dk | on the momentum and the temperature along the (k, k) direction in the first Brillouin zone for some values of ω0 at εc − εf = and g = 0.5 The Fermi momenta are indicated white dashed lines Fig 3.15 shows the nature of the excitonic condensation state in the system, indecating the dependence of the order parameter |dk | on T for some values of ω0 at g = 0.5 and εc −εf = in the first Brillouin zone At below the critical temperature Tc , |dk | is strongly peaked at momenta close to the Fermi momentum kF (described by the white dashed lines) which shows that excitons condense in the BCS-type Increasing ω0 , |dk | decreases and Tc also decreases The influence of the temperature and the phonon frequency on the excitonic condensation state in the model is shown on the phase diagram (ω0 , T ) for two values of the electron – phonon coupling g = 0.5 (Fig 3.16a) and g = 1.0 (Fig 3.16b) at εc − εf = The excitonic 12 condensation regime is expanded when increasing electron – phonon coupling constant a) b) 1.0 g=0.5 0.8 1.0 g=1.0 0.8 0.6 T T 0.6 0.4 0.4 W CD EI/ 0.2 0.0 0.0 EI/CDW 0.2 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.0 0.5 1.0 1.5 0 2.0 2.5 3.0 0 Fig 3.16: The phase diagram of the excitonic condansation state of the model in the (ω0 , T ) plane at εc − εf = for g = 0.5 (Fig a) and g = 1.0 (Fig b) The excitonic condensation phase is indicated in orange 0.3 1.0 g=0.2 g=0.4 g=0.5 0.8 0.1 0.4 d, xQ d, xQ 0.6 0=0.5 0.2 0.0 0=2.5 -0.1 0.0 (b) -0.2 (a) -0.2 0.0 g=1.0 g=1.1 g=1.2 0.2 0.1 0.2 0.3 0.4 0.5 -0.3 0.0 0.1 0.2 0.3 0.4 0.5 T T Fig 3.17: The order parameter d (filled symbols) and the lattice displacement xQ (open symbols) as functions of T at ω0 = 0.5 (Fig a) and ω0 = 2.5 (Fig b) for some values of g with εc − εf = Fig 3.17 shows that d and xQ are still intimately related For a given ω0 and g , d and xQ is only non-zero when the temperature is smaller than the critical temperature value Tc The temperature dependence of the lattice displacement agrees qualitatively with extracted data from neutron diffraction experiments at low temperatures as follows Tc The temperature dependence of the order parameter is similar to the superconducting parameter This once again reminds us of a similar relevance to the BCS theory of the superconductivity where Cooper pairs are formed Then, the phase diagram of the model in the (g, T ) plane when fixing εc − εf = for the phonon frequency ω0 = 0.5 (the adiabatic regime) and ω0 = 2.5 (the anti-adiabatic regime) is shown in Fig 3.19 When the temperature increases, a large thermal fluctuation destroys the bound state of c − f electrons, the excitonic condensation state thus is weakened The diagram also shows that, when increasing the phonon frequency from the 13 adiabatic limit (Fig.a) to the anti-adiabatic limit (Fig.b), the critical value of the electron – phonon coupling constant also increases The excitonic condensation regime thus narrows a) b) 1.0 1.0 0.8 0=0.5 0.8 0=2.5 0.6 0.6 T T EI/CDW 0.4 0.4 0.2 0.2 EI/CDW 0.0 0.0 0.4 0.8 1.2 0.0 0.0 1.6 0.4 0.8 1.2 1.6 g g Fig 3.19: The phase diagram of the excitonic condansation state of the model in the (g, T ) plane at εc − εf = for ω0 = 0.5 (Fig a) ω0 = 2.5 (Fig b) The excitonic condensation phase is indicated in orange a) b) 1.0 1.0 g=0.5 0.8 g=0.7 0.8 0.6 T T 0.6 0.4 0.4 0.2 EI/CDW 0.2 EI/CDW 0.0 0.0 0.5 1.0 1.5  - e 2.0 2.5 3.0 0.0 0.0 0.5 1.0 1.5  - h e 2.0 2.5 3.0 h Fig 3.21: The phase diagram of the excitonic condansation state of the model in the (εc − εf , T ) plane at ω0 = 0.5 and g = 0.5 (Fig a) or g = 0.7 (Fig b) The excitonic condensation phase is indicated in orange Finally, the phase diagram of the model in the (εc −εf , T ) plane for the electron – phonon coupling constant g = 0.5 (Fig a) and g = 0.7 (Fig b) at ω0 = 0.5 is shown in Fig 3.21 The phase diagram shows that for each given value of g , we always find the EI/CDW state (indicated by the orange regime) below the critical temperature Tc This critical value Tc decreases as εc − εf increases, thus the excitonic condensation regime shrinks Our results the temperature dependence of the excitonic condensation state in the system fit quite well with the recent experimental observation of C.Monney et al The results also confirm the important influence of temperature and phonon on excitonic condensation state 14 The excitonic condensation state is only formed when the system is at low temperatures and the electron – phonon interaction is large enough CHAPTER EXCITONS CONDENSATE IN THE EXTENDED FALICOV-KIMBALL MODEL INVOLVING ELECTRON – PHONON INTERACTION 4.1 The extended Falicov-Kimball model involving electron – phonon interaction The Hamiltonian for the extended Falicov-Kimball model involving electron – phonon interaction can be written H = H0 + Hint , (4.1) where H0 discribes the non-interacting part of electron – phonon system εck c†k ck + H0 = εfk fk† fk + ω0 (4.2) q k k b†q bq here c†k (ck ); fk† (fk ) and b†q (bq ) are creation (annihilation) operators of c, f electrons carrying momentum k and phonons carrying momentum q, respectively The c(f ) electronic excitation energies are still given by equation (3.2) The interacting part Hamiltonian reads Hint = U N g c†k+q ck fk† −q fk + √ N k,k ,q [c†k+q fk (b†−q + bq ) + H.c.], (4.4) kq where U is the Coulomb interaction and g is the electron – phonon coupling constant 4.2 Applying mean field theory Using Hartree-Fock approximation is similar to chapter 3, and diagonalizing Hamiltonian, we have a completely diagonalized Hamiltonian † Ek+ α1k α1k + Hdia = k † Ek− α2k α2k + ω0 Bq† Bq , (4.10) q k † † where α1k (α1k ) and α2k (α2k ) are the Bogoliubov quasi-particle fermionic creation (annihi- lation) operators, respectively, with the electronic quasiparticle energies Ek± = εfk + εck+Q ∓ sgn(εfk − εck+Q ) Γk , (4.11) here Γk = (εck+Q − εfk )2 + 4|Λ|2 , (4.12) and the electronic excitation energies now have acquired Hartree shifts f /c εk f /c = εk + U nc/f , (4.7) with nc(f ) is the c(f ) electron density; Λ also acts as the order parameters of the excitonic condensation state which is given by g U Λ = √ b†−Q + b−Q − N N 15 c†k+Q fk k (4.9) We also obtain the system of self-consistently equations from the average values nck+Q = nfk = c†k+Q ck+Q = u2k nF (Ek+ ) + vk2 nF (Ek− ), (4.13) fk† fk = vk2 nF (Ek+ ) + u2k nF (Ek− ), (4.14) Λ c†k+Q fk = − nF (Ek+ ) − nF (Ek− ) sgn(εfk − εck+Q ) , (4.15) Γk √ Nh δq,Q, , (4.16) = − ω0 nk = b†q where nF (Ek± ) is the Fermi-Dirac distribution function; uk and vk are the prefactors of the Bogoliubov transformation which satisfy u2k + vk2 = The lattice displacement and the singleparticle spectral functions of c and f electrons are therefore also determined by 1 h xQ = √ √ b†−Q + bQ = − ω0 N 2ω0 , ω0 (4.19) + − Ack (ω) = u2k−Q δ ω − Ek−Q + vk−Q δ ω − Ek−Q , (4.23) Afk (ω) = vk2 δ ω − Ek+ + u2k δ ω − Ek− (4.24) 4.3 Numerical results and discussion For the two-dimensional system consisting of N = 150 × 150 lattice sites, the numerical results are obtained by solving self-consistently Eqs (4.7) – (4.9) and (4.13) – (4.16) starting from some guessed values for b†Q and nk with a relative error 10−6 In what follows, all energies are given in units of tc and we fix tf = 0.3; εc = 0; ω0 = 2.5 The chemical potential µ has to be adjusted such that the system is in the half-filled band state, i.e., nf + nc = 4.3.1 The momentum dependence of the quasiparticle energies and the order parameter Fig 4.1 and Fig 4.2 show the momentum dependence along the (k, k) direction in the U=0 U=1.0 U=1.5 E+k, E-k -2 -2 0.0 0.0 -0.2 -0.2 U=0 U=1.0 U=1.5 -0.4 -1.0 nk|nk| nk |nk| E+k, E-k -0.5 0.0 U=3.5 U=3.8 U=4.2 0.5 U=3.5 U=3.8 U=4.2 -0.4 1.0 -1.0 k/ -0.5 0.0 0.5 1.0 k/ Fig 4.1: The quasiparticle energies Ek+ (solid Fig 4.2: The quasiparticle energies Ek+ (solid lines); Ek− (dash lines) and |nk | for small val- lines); Ek− (dash lines) and the order parame- ues of U at g = 0.6; T = ter |nk | for large values of U at g = 0.6; T = first Brillouin zone of the quasiparticle energy bands Ek+ ; Ek− and the order parameter |nk | for 16 some values of U in the weak and strong interaction limit at g = 0.6, εf = −2.0 in the ground state In Fig 4.1, the Fermi surface plays an important role to form the condensation state of excitons We affirm that excitons in system condensate in the BCS-type, like the Cooper pairs in superconductivity BCS theory Fig 4.2 shows that large Coulomb interaction binds an electron in the conduction band and an electron in the valence band in a tightly bound state Therefore, |nk | has a maximum value at zero momentum, this confirms that excitons condensate in BEC-type, like normal bosons The investigation similarly the momentum dependence of the quasiparticle energies and the order parameter when g or T changes The results confirmed that the condensation state is only formed when the temperature is low enough and the electron – phonon coupling constant and Coulomb interaction are large enough 4.3.2 The EI order parameter and the lattice displacement Fig 4.5: Λ (solid lines) and Fig 4.6: Λ (solid lines) and Fig 4.8: Λ (solid lines) and xQ (dash lines) as functions xQ (dash lines) as functions xQ (dash lines) as functions of U for different g at εf = of U for different εf at g = of T for different g at U = −2.0; T = 0.6; T = 1.5; εf = −2.0 In Fig 4.5, the EI order parameter Λ and the lattice displacement xQ are shown as functions of U for some values of g at T = and εf = −2.0 And Fig 4.6 shows Λ and xQ as a function of U at zero temperature when g = 0.6 for different values of εf The results confirm that excitonic condensation state exists only in a limited range of Coulomb interactions In presence of the electron – phonon interaction, we observe the EI/CDW state Fig 4.8 shows Λ and xQ depending on T when changing g At g is greater than the critical value gc , Λ always exists simultaneously with xQ At T ≤ Tc , both are nonzero and the system exists in excitonic condensation state with a finite lattice distortion Increasing g , the EI transition temperature Tc increases The temperature dependence of the lattice displacement fits quite well with experimental results obtained from neutron diffraction experiments at low temperatures or the recent experimental observation in the quasi-two-dimensional 1T -TiSe2 4.3.3 The nature of excitonic condensation state in the model Fig 4.10 shows the momentum dependence of the excitonic condensation order parameter |nk | in the ground state for some values of U at g = 0.6 and εf = −2.0 in the first Brillouin 17 Fig 4.10: The order parameter |nk | depending on momentum k in the first Brillouin zone for some values of U at g = 0.6; εf = −2.0; T = The Fermi momenta are determined by the white dashed lines Fig 4.11: The order parameter |nk | depending on momentum along the (k, k) direction and Coulomb interaction in the first Brillouin zone for g = 0.6 and εf = −2.0 at T = zone The excitons with low Coulomb interaction condense in the BCS-type in which the Fermi surface plays an important role in the formation and condensation of excitons The ex18 Fig 4.12: The order parameter |nk | depending on momentum k in the first Brillouin zone for different temperatures U = 1.5 (left panels) and U = 3.7 (right panels) at g = 0.6 and εf = −2.0 The Fermi momenta are determined by the white dashed lines citons with strong Coulomb interaction will condense in the BEC-type The value U = 3.39 can be called the critical value for the BCS-BEC crossover of the excitonic condensation for the set of parameters chosen in Fig 4.10 The excitonic condensation state is only established when the Coulomb interaction is in between Uc1 and Uc2 as shown in Fig 4.11 Fig 4.12 shows in detail the nature of the excitonic condensation state in the model influenced by Coulomb interaction and temperature Increasing temperature, thermal fluctuations diminish the c − f electron coupling, which is illustrated by a decrease of the |nk | amplitude If the temperature is higher than the critical temperature, the large thermal fluctuation com- 19 pletely destroys the exciton bond and the system exists in the plasma state of the electrons In Fig 4.12, we also find that the EI transition temperature in the strong Coulomb interaction limit is higher than in the weak Coulomb interaction limit 4.3.4 The single-particle spectral functions of electrons Fig 4.16 shows the variation of Ack (ω) (left) and Afk (ω) (right) for different temperature (a) (d) T=0 Afk() k  Ack() (b) (e) k  T = 0.1 (c) (f) k  T = 0.2 -2  -2  Fig 4.16: The single-particle spectral functions of c-electron (left) and f -electron (right) along the (k, k) direction at εf = −2.0; U = 1.5; g = 0.6 for different T Red lines indicate the spectral function at Fermi momentum T At low temperatures, the gap feature opened at the Fermi level, the excitons are formed and 20 condense in the BCS-type The gap disappears at high temperatures indicate that the bound state of the excitons is completely broken The system settles into the plasma state of the electrons The spectral functions dependence on momentum when U and g change also elucidates the results presenting in the previous section 4.3.5 Phase diagram of the excitonic condensation state The following phase diagrams give a comprehensive picture of the role of Coulomb interaction, electron – phonon interaction and temperature on the excitonic condensation state in the EFKM model involving the electron – phonon interaction The excitonic condensation phase typifying either BCS- or BEC-type is indicated, respectively, by blue or red The SM (SC) disordered state is indicated by the green (orange) regime Fig 4.17 shows the phase 1.0 0.8 g 0.6 0.4 0.2 f=-2.6 f=-3.0 0.0 1.0 0.8 BEC 0.6 g BCS 0.4 SM SC 0.2 f=-1.8 f=-2.0 0.0 4 U U Fig 4.17: Ground-state phase diagram of the EFKM with additional electron – phonon coupling in the (U, g) plane at different εf diagram of excitonic condensation state at T = in the (U, g) plane for different values of εf At a given εf , one always finds the excitonic condensate regime in between two critical Coulomb interactions Uc1 and Uc2 Increasing the electron – phonon coupling, Uc1 decreases while Uc2 increases, the excitonic condensate window thus increases Moving up the one-site 21 energy of the f -electron level, both SM and SC regimes are weakened and the excitonic condensate window is expanded The BCS – BEC crossover shifts to a larger U value The phase diagram addressed here is similar to the phase diagram of B Zenker et al However, in our case, the phase structure of the excitonic condensate with lattice displacement under the influence of both Coulomb interaction and the electron – phonon interaction in the ground state is examined in more detail, especially involving the change of εf 0.8 g=0 g=0.6 T 0.6 0.4 0.2 0.0 0.8 g=0.8 g=0.85 SC 0.6 T SM 0.4 BEC BCS 0.2 0.0 70 U U Fig 4.18: The excitonic condensation phase and the BCS – BEC crossover scenario of the EFKM with additional electron – phonon coupling in the (U, T ) plane for different g at εf = −2.0 Fig 4.18 shows the phase diagram of the excitonic condensation state in the (U, T ) plane for different values of g at εf = −2.0 At low temperatures, the excitonic condensation regime is always found in between the two critical values of the Coulomb interaction, Uc1 and Uc2 for any value of g Increasing g , Uc1 decreases while Uc2 increases, the excitonic condensate window is therefore expanded as seen in Fig 4.17 and the BCS – BEC crossover shifts to a larger U value If the electron – phonon interaction is large enough, for example g > 0.8, the excitonic condensation state can be found even at zero Coulomb interaction In all cases, when increasing temperature, the thermal fluctuations destroy the bound state and the excitonic condensate order parameter decreases If the temperature is larger than the critical value of 22 the excitonic condensate transition temperature, all the bound states of excitons are completely destroyed and the system settles in the plasma liquid of electrons Our phase diagram is similar to the phase diagram of EFK model discussing by other authors However, in our case, the phase structure of the excitonic condensate under the influence of both Coulomb interaction and the electron – phonon interaction in the ground state is examined in more detail Thus, numerical results show that the Coulomb interaction and the electron – phonon coupling act together in establishing the excitonic condensate phase with lattice distortion At a sufficiently low temperature, exciton condensation is found if the electron – phonon coupling is sufficiently large and the Coulomb interaction is in between two critical values If the Coulomb interaction is small that condensation typifies the BCS-type In contrast, the condensation typifies the BEC-type if the Coulomb interaction is strong enough CONCLUSIONS AND RECOMMENDATIONS In the framework of the thesis, we have developed the MF theory applying to the twoband model including electron – phonon interaction and the extended Falicov-Kimball model involving electron – phonon interaction to investigate the excitonic condensation phase transition in SM – SC transition systems The physical scenario obtained is entirely consistent with recent experimental results on some materials and is similar in both adiabatic limit and anti-adiabatic limit For the two-band model including electron – phonon interaction, numerical results show the important influence of temperature and electron – phonon interaction on the excitonic condensation state in the model The excitonic condensation state is only formed when the tmperature is low enough and the electron – phonon interaction is large enough The results also showed that the EI stability – BCS-type and the lattice displacement are intimately related The phase diagram (g, T ) shows that, for each given value of the phonon frequency, one always find the excitonic condensation regime above the critical value gc of the electron – phonon coupling constant and below critical value Tc of EI/CDW transition temperature On the other hand, the phase diagram (ω0 , T ) confirms that the excitonic condensation regime is expanded when increasing electron – phonon interaction or the critical value of the phonon frequency increases when increasing the electron – phonon coupling constant For the extended Falicov-Kimball model involving electron – phonon interaction, our numerical results show that the Coulomb interaction and the electron – phonon interaction act together in establishing the excitonic condensate phase with lattice distortion The excitonic condensation state is only found if the electron – phonon interaction is sufficiently large and the Coulomb interaction is in between two critical values when the temperature is low enough If the Coulomb interaction is small that condensation typifies the BCS-type In contrast, the condensation typifies the BEC-type if the Coulomb interaction is strong enough The phase 23 diagrams (U, g) and (U, T ) show that, at low temperatures, the excitonic condensation regime is always found in between the two critical values of the Coulomb interaction for any value of g When increasing the electron – phonon interaction, the excitonic condensate with lattice distortion window increases and the BCS – BEC crossover shifts to a larger value of U In particular, if the electron – phonon coupling constant is greater than the critical value, the excitonic condensation state can be found even in the absence of the Coulomb interaction However, the problem has only been solved by the MF approximation In order to investigate carefully the nature of the excitonic condensation state, it is necessary to expand the problem when involving the contribution of electronic correlations, such as applying PR method These are the next research of the PhD Student after completing the thesis NEW CONTRIBUTIONS OF THE THESIS The mean-field theory has been successfully applied to the 2D two-band electronic model once the electron – phonon interaction is taken into account and to the 2D extended Falicov-Kimball model involving the electron – phonon interaction A program solving numerically the mean-field self-consistent equations has been established The effects of phonon, the temperature and the Coulomb interaction on the excitonic condensation state then are discussed Depending on the interactions, excitons might condense in either the BCS-type or the BEC-type state The spectral structure of electrons in the excitonic condensation state has been addressed The phase diagrams of the excitonic condensation state in the models under the influence of temperature, the phonon frequency, the electron – phonon coupling and the Coulomb interaction are constructed Signature of the BCS – BEC crossover of the excitonic condensation is also mentioned and discussed in detail The thesis contributes to the development of the excitonic condensation field in Vietnam LIST OF WORKS PUBLISHED Thi-Hong-Hai-Do, Huu-Nha-Nguyen, Thi-Giang-Nguyen and Van-Nham-Phan, Temperature effects in excitonic condensation driven by the lattice distortion, Physica Status Solidi B 253, 1210, 2016 Thi-Hong-Hai-Do, Dinh-Hoi-Bui and Van-Nham-Phan, Phonon effects in the excitonic condensation induced in the extended Falicov-Kimball model, Europhysics Letters 119, 47003, 2017 Thi-Hong-Hai Do, Huu-Nha Nguyen and Van-Nham Phan, Thermal Fluctuations in the Phase Structure of the Excitonic Insulator Charge Density Wave State in the Extended Falicov-Kimball Model, Journal of Electronic Materials 48, 2677, 2019 24 Do Thi Hong Hai and Phan Van Nham, Excitonic condensate phase transition in transition metal dichalcogenides, Journal of Science and Technology, Duy Tan university (25), 17–21, 2017 Do Thi Hong Hai and Phan Van Nham, BCS and BEC excitonic condensations in transition-metal dichalcogenides, Journal of Science and Technology, Duy Tan university (25), 30–35, 2017 Do Thi Hong Hai, Nguyen Thi Hau, Ho Quynh Anh, Temperature effect on the excitonic condensation state in the extended Falicov-Kimball model including electron – phonon interaction, Journal of Military Science and Technology, CBES2, 204–209, 2018 Do Thi Hong Hai and Phan Van Nham, Spectral properties in the extended FalicovKimball model involving the electron-phonon interaction: Excitonic insulator state formation, DTU Journal of Science and Technology (31), 89–94, 2018 Do Thi Hong Hai and Phan Van Nham, Phase diagram of excitonic condensation state in the extended Falicov-Kimball model involving the electron-phonon interaction, DTU Journal of Science and Technology (31), 95–100, 2018 Do Thi Hong Hai and Phan Van Nham, Influence of phonon frequency on the excitonic insulator state, DTU Journal of Science and Technology (34), 87–92, 2019 10 Do Thi Hong Hai and Phan Van Nham, Excitonic condensation in two-band model involving electron – phonon interaction, DTU Journal of Science and Technology (34), 106–111, 2019 11 Do Thi Hong Hai and Phan Van Nham, Effects of phonons in the excitonic insulator in the 2D extended Falicov-Kimball model, 41th National Conference on Theoretical Physics, Nha Trang, – August 2016 12 Do Thi Hong Hai and Phan Van Nham, Excitonic condensation phase diagram in the extended Falicov-Kimball model with electron-phonon interaction, 42th National Conference on Theoretical Physics, Can Tho, 31 July – August, 2017 13 Do Thi Hong Hai and Phan Van Nham, Phase diagram of excitonic condensation state in transition metal dichalcogenides, 43th National Conference on Theoretical Physics, Quy Nhon, 30 July – August, 2018 25 ... the thesis are presented in chapters and CHAPTER EXCITON AND EXCITONIC CONDENSATION STATES 1.1 The concept of excitons 1.1.1 What is an exciton? An exciton is a bound state of an electron in conduction... larger U value The phase diagram addressed here is similar to the phase diagram of B Zenker et al However, in our case, the phase structure of the excitonic condensate with lattice displacement... the excitonic condensation induced in the extended Falicov-Kimball model, Europhysics Letters 119, 47003, 2017 Thi-Hong-Hai Do, Huu-Nha Nguyen and Van-Nham Phan, Thermal Fluctuations in the Phase