Summary of phycics doctoral thesis: Excitonic condensation in semimetal – semiconductor transition systems

Chia sẻ: _ _ | Ngày: | Loại File: PDF | Số trang:27

Thêm vào BST

Báo xấu

15
lượt xem 4
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

Thesis 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.

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: Summary of phycics doctoral thesis: Excitonic condensation in semimetal – semiconductor transition systems

MINISTRY OF EDUCATION VIETNAM ACADEMY OF SCIENCE AND TRAINING 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 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 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 investi- gating 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 vari- ational 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 charac- teristics 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 su- 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 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 in- teraction 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 to- gether in binding the electron – hole pairs and establishing the excitonic condensation state. 1
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 mod- els 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 in- volving 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. 2. 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. 3. 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 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 the thesis are presented in chapters 3 and 4. 2
CHAPTER 1. 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 † 1 X † ak,n = √ δk,p+p0 ϕn (q)c†p fp0 . (1.17) V p,p0 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 tempera- ture 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 3
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 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 operators aν and bµ , re- spectively. Let us assume that only interactions between different kind of particles are impor- tant. By relapcing the pairing operators a†ν aν 0 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 X X a HM F = εaν a†ν aν + Vνµ,ν 0 µ0 hb†µ bµ0 ia†ν aν 0 , (2.9) ν µµ0 X X b HM F = εbµ b†µ bµ + Vµν,µ0 ν 0 ha†ν aν 0 ib†µ bµ0 . (2.10) µ νν 0 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 4
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 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 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 1X 1X 1X Hartree Vint = Vνµ,ν 0 µ0 nµµ0 c†ν cν 0 + Vνµ,ν 0 µ0 nνν 0 c†µ cµ0 − Vνµ,ν 0 µ0 nνν 0 nµµ0 , (2.22) 2 2 2 1X 1X 1X Fock Vint = ± Vνµ,ν 0 µ0 nνµ0 c†µ cν 0 ± Vνµ,ν 0 µ0 nµν 0 c†ν cµ0 ∓ Vνµ,ν 0 µ0 nνµ0 nµν 0 , (2.23) 2 2 2 where nνν 0 = hc†ν cν 0 i and nνµ0 = hc†ν cµ0 i with c†ν , cν are particle creation and annihilation op- erators 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 orig- inal 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 X HM F = −2 mSi + mN hSz i. (2.28) i We can easily derive the equation α = tanh(bα), (2.31) where α = m/nJ0 và b = nJ0 β . This equation can be numerically solved and the result given the temperature dependence of the magnetization m. 5
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 X F † UV X UV X 2 HM F = εM kσ ckσ ckσ − nσ n−σ + nσ , (2.39) 2 2 kσ σ σ where εM F kσ = εk + U (n↑ + n↓ − nσ ) = εk + U nσ ¯, (2.40) 1 P † with nσ = V k hckσ ckσ i 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 − (∆k c†k↑ c†−k↓ + H.c.), X X MF HBCS = (2.51) kσ k where X ∆k = − Vkk0 hc−k0 ↓ ck0 ↑ i, (2.52) k0 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 = 1. Finally, BCS Hamiltonian can be diagonalized in a form † † X MF HBCS = Ek (αk↑ αk↑ + αk↓ αk↓ ), (2.59) k p 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 1 . Similar to the superconducting state survey in BCS theory, exci- tonic condensation state is characterized by quantity hc†k fk i = 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. 6
we can rewrite Hamiltonian ε˜ck c†k ck + ε˜fk fk† fk + (∆k fk† ck + H.c.), X X X HM F = (2.71) k k k where Vk−k0 hc†k0 fk0 i. X ∆k = (2.72) k0 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 + Ekβ βk† βk , X X MF HEI = (2.79) k k where ε˜ck + ε˜fk q α/β Ek = ∓ ξk2 + |∆k |2 . (2.80) 2 εck − ε˜fk ] and Ek2 = ξk2 + |∆k |2 . with ξk = 21 [˜ This Hamiltonian allows us to determine all expectation values. At T = 0, ∆k is deter- mined by the gap equation X ∆k0 ∆k = Vk−k0 . (2.81) 2Ek0 k0 This equation is similar to the gap equation of superconducting in BCS theory. ∆k 6= 0 in- dicates the hybridization between electrons in the valence band and the conduction band. Therefore, the system turn into the excitonic insulator state. 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 inter- action can be written g X † X † εfk fk† fk + ω0 [ck+q fk b†−q + bq + H.c.], X X H= εck ck ck + b†q bq + √ (3.1) k k q N kq where c†k (ck ); fk† (fk ) and b†q (bq ) are creation (annihilation) operators of c, f electrons carry- ing momentum k and phonons carrying momentum q, respectively; g is a electron – phonon coupling constant; N is the number of the lattice sites. εc,f k =ε c,f − tc,f γk − µ, (3.2) 7
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 ) 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 = hc†k+Q fk i and 1 X † d= (hck+Q fk i + hfk† ck+Q i), (3.4) N 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 ∆ = √ hb†−Q + b−Q i, (3.9) N g X † h = hck+Q fk + fk† ck+Q i, (3.10) N 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 + εfk fk† fk + ∆ (c†k+Q fk + fk† ck+Q ), X X X He = (3.12) k k k √ N h(b†−Q + b−Q ), X Hph = ω0 b†q bq + (3.13) q The phononic part is diagonalized by defining a new phonon operator √ h Bq† = b†q + N δq,Q . (3.14) ω0 Meanwhile, the electronic part can be diagonalized itself by using a Bogoliubov transforma- tion with the new quasi-particle fermionic operators C1k and C2k . Then finally, we are led to a completely diagonalized Hamiltonian † † X X Hdia = Ek1 C1k C1k + Ek2 C2k C2k + ω0 Bq† Bq , (3.17) k q where the electronic quasiparticle energies read as εfk + εck+Q sgn(εfk − εck+Q ) Ek1,2 = ∓ Wk , (3.18) 2 2 8
q with Wk = (εck+Q − εfk )2 + 4|∆|2 . The quadratic form of Eq. (3.17) allows us to compute all expectation values, resulting in nck+Q = hc†k+Q ck+Q i = ξk2 nF (Ek1 ) + ηk2 nF (Ek2 ), nfk = hfk† fk i = ηk2 nF (Ek1 ) + ξk2 nF (Ek2 ), (3.22) ∆ dk = hc†k+Q fk i = −[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 the Bogoliubov transformation which satisfy ξk2 + ηk2 = 1. The lattice displacement in the EI state at momentum Q r 1 1 h 2 xQ = √ √ hb†−Q +bQ i = − , (3.24) N 2ω0 ω0 ω0 3.3. 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. (3.9), (3.10), (3.22) and (3.24) starting from some guessed values for hb†Q i and dk with a relative error 10−6 . In what follows, all energies are given in units of tc and we fix tf = 0.3 to consider the half-filled band case, i.e. nc + nf = 1. The chemical potential µ has to be adjusted such that this equation is satisfied. 3.3.1. The ground state 0.0 1.0 g=0.2 -0.1 g=0.4 0.8 =0.5 g=0.5 g=0.6 0.6 =1.0 =1.5 d, xQ d -0.2 0.4 =2.0 0.2 0.0 -0.3 -0.2 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 0 - c f Fig. 3.2: The order parameter d as functions of Fig. 3.5: The order parameter d (filled sym- phonon frequency ω0 for different values of g at bols) and the lattice displacement xQ (open εc − εf = 1 in the ground state. symbols) as functions of εc − εf for some val- ues of ω0 at g = 0.5, T = 0. Fig. 3.2 shows the dependence of the excitonic condensate order parameter d at T = 0 on the phonon frequency ω0 for different values of g at εc − εf = 1. For a given value of the coupling constant, the order parameter decreases when increasing phonon frequency. This is 9
also shown in Fig. 3.5 the dependence of the order parameter d and the lattice displacement xQ on εc − εf for some values of ω0 at g = 0.5, T = 0. The diagram shows that d and xQ are intimately related. When increasing ω0 , both d and xQ decrease significantly, indicating a weakened condensation state. d and xQ are non-zero, the systems thus stabilize in the excitonic condensation state with the charge density wave state (EI/CDW). 3.0 2.5 g=0.4 g=0.5 2.0 1.5  1.0 0.5 0.0 3.0 2.5 g=0.6 g=0.7 2.0 1.5  1.0 EI/CDW 0.5 0.0 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  - c f  - c f Fig. 3.6: Ground-state phase diagram of the model in the (εc − εf , ω0 ) plane for different values of g . The excitonic condensation phase is indicated in orange. Fig. 3.6 shows the phase diagram of the model in the (εc −εf , ω0 ) plane in the ground state for different g . If g 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 value increases when increasing g . The excitonic condensation regime is narrowed if decreasing the two 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 parameter d on the phonon frequency ω0 when varying the temperature at εc − εf = 1 and g = 0.5. For a given value of temperature, the value of the order parameter decreases rapidly when increasing the phonon frequency. The dependence of the order parameter d the lattice displacement xQ on the electron – phonon interaction when the temperature changes for εc − εf = 1 and ω0 = 0.5 are shown in Fig. 3.8. d and xQ are always closely related, they are non-zero i.e. the system exists in EI/CDW state 10
0.0 1.5 T=0 T=0 T=0.1 T=0.1 1.0 T=0.2 -0.1 T=0.2 T=0.3 T=0.3 d, xQ 0.5 d -0.2 0.0 -0.5 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 g 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 = 1 and g = 0.5. symbols) as functions of g for some values of T at εc − εf = 1 and ω0 = 0.5. 1.0 0.8 0.6 g 0.4 0.2 T=0 T=0.1 0.0 1.0 0.8 EI/CDW 0.6 g 0.4 0.2 T=0.2 T=0.3 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 = 1 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 = 1 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 2.5 T=0 2.5 T=0.1 2.0 2.0 1.5 1.5   1.0 1.0 0.5 EI/CDW 0.5 EI/CDW 0.0 0.0 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  - c f  - c 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 = 1 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 = 1 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 = 1. The excitonic 12
condensation regime is expanded when increasing electron – phonon coupling constant. a) b) 1.0 1.0 g=0.5 g=1.0 0.8 0.8 0.6 0.6 T T 0.4 0.4 EI/CDW EI/ CD 0.2 0.2 W 0.0 0.0 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 0 0 Fig. 3.16: The phase diagram of the excitonic condansation state of the model in the (ω0 , T ) plane at εc − εf = 1 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=1.0 g=0.2 0.2 g=1.1 0.8 g=0.4 g=1.2 g=0.5 0.6 0.1 0.4 0.0 d, xQ d, xQ 0=0.5 0.2 -0.1 0=2.5 0.0 -0.2 (b) -0.2 (a) -0.3 0.0 0.1 0.2 0.3 0.4 0.5 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 = 1. 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 = 1 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=0.5 0=2.5 0.8 0.8 0.6 0.6 EI/CDW T T 0.4 0.4 EI/CDW 0.2 0.2 0.0 0.0 0.0 0.4 0.8 1.2 1.6 0.0 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 = 1 for ω0 = 0.5 (Fig. a) và ω0 = 2.5 (Fig. b). The excitonic condensation phase is indicated in orange. a) b) 1.0 1.0 g=0.5 g=0.7 0.8 0.8 0.6 0.6 T T 0.4 0.4 EI/CDW 0.2 0.2 EI/CDW 0.0 0.0 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  - e h  -e 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 (indi- cated 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 4. 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 X † X f † X H0 = εck ck ck + εk fk fk + ω0 b†q bq . (4.2) k k q 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 U X † g X † Hint = ck+q ck0 fk†0 −q fk + √ [ck+q fk (b†−q + bq ) + H.c.], (4.4) N N k,k0 ,q 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 Hamilto- nian, we have a completely diagonalized Hamiltonian † † X X X Hdia = Ek+ α1k α1k + Ek− α2k α2k + ω0 Bq† Bq , (4.10) k k q † † where α1k (α1k ) and α2k (α2k ) are the Bogoliubov quasi-particle fermionic creation (annihi- lation) operators, respectively, with the electronic quasiparticle energies εfk + εck+Q sgn(εfk − εck+Q ) Ek± = ∓ Γk , (4.11) 2 2 here q Γk = (εck+Q − εfk )2 + 4|Λ|2 , (4.12) and the electronic excitation energies now have acquired Hartree shifts f /c f /c εk = 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 X † Λ = √ hb†−Q + b−Q i − hck+Q fk i. (4.9) N N k 15
We also obtain the system of self-consistently equations from the average values nck+Q = hc†k+Q ck+Q i = u2k nF (Ek+ ) + vk2 nF (Ek− ), (4.13) nfk = hfk† fk i = vk2 nF (Ek+ ) + u2k nF (Ek− ), (4.14) Λ nk = hc†k+Q fk i = − nF (Ek+ ) − nF (Ek− ) sgn(εfk − εck+Q ) , (4.15) Γk √ Nh hb†q i = − δq,Q, , (4.16) ω0 where nF (Ek± ) is the Fermi-Dirac distribution function; uk and vk are the prefactors of the Bogoliubov transformation which satisfy u2k + vk2 = 1. The lattice displacement and the single- particle spectral functions of c and f electrons are therefore also determined by r 1 1 h 2 xQ = √ √ hb†−Q + bQ i = − , (4.19) N 2ω0 ω0 ω0 + − Ack (ω) = u2k−Q δ ω − Ek−Q 2 + 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 hb†Q i 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 = 1. 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 8 6 U=3.5 U=0 6 U=3.8 4 U=1.0 U=1.5 U=4.2 4 E+k, E-k E+k, E-k 2 2 0 0 -2 -2 0.0 0.0 nk |nk| -0.2 -0.2 nk|nk| U=0 U=3.5 -0.4 U=1.0 -0.4 U=3.8 U=1.5 U=4.2 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 k/ 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 = 0. ter |nk | for large values of U at g = 0.6; T = 0. 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 conden- sate 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 con- firmed 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. 0.6; T = 0. 1.5; εf = −2.0. In Fig. 4.5, the EI order parameter Λ and the lattice displacement xQ are shown as func- tions of U for some values of g at T = 0 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 parame- ter |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 = 0. 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 = 0. 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 ex- 18