VIETNAM ACADEMY OF SCIENCE
MINISTRY OF EDUCATION 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 TmSe0.45Te0.55, transition-metal dichalcogenide 1T -TiSe2, semiconductor Ta2NiSe5, layer double graphene,... which have increased the studies of the excitonic condensation state both the theoretical 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 Ta2NiSe5 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.
• Developing mean-field theory for a 2D two-band model including electron – phonon
In detail:
interaction and the extended Falicov-Kimball model involving electron/hole – phonon
• Studying the properties of electronic systems in EI state through investigating the above
interaction.
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
p and c†
p are hole creation operators in valence
Considering a two-band model with f †
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 √
(1.17)
δk,p+p(cid:48)ϕn(q)c†
pf † p(cid:48).
a† k,n =
V
p,p(cid:48)
(cid:88)
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
Ta2NiSe5 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 TmSe0.45Te0.55, 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 be
created. These excitons condense into an EI superfluid state at sufficiently low temperatures.
CHAPTER 2. MEAN FIELD THEORY
2.1. The basic concepts
νaν(cid:48) with their average values and a small correction.
2.1.1. Mathematical representation of mean-field theory
(2.8)
HM F = Ha
M F + Hb
M F ,
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† Neglecting the constant contribution, Hamiltonian is written
where
(2.9)
Ha
Vνµ,ν(cid:48)µ(cid:48)(cid:104)b†
µbµ(cid:48)(cid:105)a†
νaν(cid:48),
νa† εa
νaν +
M F =
µµ(cid:48) (cid:88)
ν (cid:88)
(2.10)
Hb
Vµν,µ(cid:48)ν(cid:48)(cid:104)a†
νaν(cid:48)(cid:105)b†
µbµ(cid:48).
µb† εb
µbµ +
M F =
µ
νν(cid:48)
(cid:88) (cid:88)
where Ha(b) M 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.
,
(2.21)
HHF = H0+V Fock
int +V Hartree int
The mean-field Hamiltonian in HFA is written in the form
where
=
(2.22)
Vνµ,ν(cid:48)µ(cid:48)nµµ(cid:48)c†
Vνµ,ν(cid:48)µ(cid:48)nνν(cid:48)c†
Vνµ,ν(cid:48)µ(cid:48)nνν(cid:48)nµµ(cid:48),
νcν(cid:48) +
µcµ(cid:48) −
V Hartree int
1 2
1 2
1 2
(cid:88) (cid:88) (cid:88)
= ±
Vνµ,ν(cid:48)µ(cid:48)nνµ(cid:48)c†
Vνµ,ν(cid:48)µ(cid:48)nνµ(cid:48)nµν(cid:48), (2.23)
Vνµ,ν(cid:48)µ(cid:48)nµν(cid:48)c†
νcµ(cid:48) ∓
µcν(cid:48) ±
V Fock int
1 2
1 2 νcµ(cid:48)(cid:105) with c†
1 2 νcν(cid:48)(cid:105) and nνµ(cid:48) = (cid:104)c† where nνν(cid:48) = (cid:104)c† ν, cν are particle creation and annihilation op- erators with the quantum number ν, respectively. The (+) mark applies to the boson particle
(cid:88) (cid:88) (cid:88)
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
(2.28)
mSi + mN (cid:104)Sz(cid:105).
HM F = −2
i
(cid:88)
(2.31)
α = tanh(bα),
We can easily derive the equation
where α = m/nJ0 và b = nJ0β. 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
HM F =
(2.39)
nσn−σ +
n2 σ,
kσ c† εM F
kσckσ −
U V 2
U V 2
σ
σ
kσ
(cid:88) (cid:88) (cid:88)
(2.40)
εM F kσ = εk + U (n↑ + n↓ − nσ) = εk + U n¯σ,
where
k(cid:104)c†
kσckσ(cid:105) is the spin density. From this Hamiltonian, we can find self-consistent
(cid:80)
with nσ = 1 V 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 kσ and ckσ are creation and annihilation operators with momentum k and spin
transition. c† σ =↓, ↑, respectively, BCS Hamiltonian in HFA is
HM F
(2.51)
εkc†
(∆kc†
BCS =
kσckσ −
k↑c†
−k↓ + H.c.),
kσ
k
(cid:88) (cid:88)
where
(2.52)
Vkk(cid:48)(cid:104)c−k(cid:48)↓ck(cid:48)↑(cid:105),
∆k = −
k(cid:48)
(cid:88)
(2.56)
kck↑ + vkc† −k↓, kck↑ + ukc†
αk↑ = u∗ α† −k↓ = −v∗
−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.
k + v2
k = 1. Finally, BCS Hamiltonian can be diagonalized in a form
where u2
(2.59)
HM F
Ek(α†
BCS =
k↑αk↑ + α†
k↓αk↓),
k
(cid:88)
where Ek = (cid:112)ε2 k + |∆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
operator of electron is replaced with the creation operator of hole and vice versa.
interaction between them 1. Similar to the superconducting state survey in BCS theory, exci- tonic condensation state is characterized by quantity (cid:104)c† kfk(cid:105) (cid:54)= 0. In HFA, neglecteing constants 1Note that, the electronic representation is completely equivalent to the hole representation by electronic transformation – hole. Then the annihilation
6
we can rewrite Hamiltonian
(2.71)
HM F =
(∆kf †
kc† ˜εc
kck +
kf † ˜εf
kfk +
kck + H.c.),
k
k
k
(cid:88) (cid:88) (cid:88)
where
(2.72)
∆k =
Vk−k(cid:48)(cid:104)c†
k(cid:48)fk(cid:48)(cid:105).
k(cid:48)
k and ˜εf
k are the dispersion energies of c
(cid:88)
acts as an energy gap, or EI state order parameter. ˜εc 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
HM F
(2.79)
EI =
k α† Eα
kαk +
k β† Eβ
kβk,
k
k
(cid:88) (cid:88)
where
k
∓
(2.80)
=
ξ2 k + |∆k|2.
Eα/β k
k + ˜εf ˜εc 2
(cid:113)
k − ˜εf
k = ξ2
k + |∆k|2.
2[˜εc
k] and E2
with ξk = 1
This Hamiltonian allows us to determine all expectation values. At T = 0, ∆k is deter-
mined by the gap equation
.
(2.81)
∆k =
Vk−k(cid:48)
∆k(cid:48) 2Ek(cid:48)
k(cid:48)
(cid:88)
This equation is similar to the gap equation of superconducting in BCS theory. ∆k (cid:54)= 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 √
[c†
H =
(3.1)
+ H.c.],
b† qbq +
b† −q + bq
kc† εc
kf † εf
kfk + ω0
kck +
k+qfk
N
q
k
k
kq
k (ck); f †
k (fk) and b†
(cid:16) (cid:17) (cid:88) (cid:88) (cid:88) (cid:88)
where c† 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
(3.2)
εc,f k = εc,f − tc,f γk − µ,
coupling constant; N is the number of the lattice sites.
7
k+Qfk(cid:105) and
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 = (cid:104)c†
d =
((cid:104)c†
(3.4)
k+Qfk(cid:105) + (cid:104)f †
kck+Q(cid:105)),
1 N
k
(cid:88)
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
g √
∆ =
(cid:104)b†
(3.9)
−Q + b−Q(cid:105),
N (cid:88)
(3.10)
h =
(cid:104)c†
k+Qfk + f †
kck+Q(cid:105),
g N
k
Applying MF theory with mean fields
act as the order parameters which specify to spontaneous broken symmetry, Hamiltonian in
(3.11)
HHF = He + Hph,
(3.1) is reduced to Hamiltonian Hartree-Fock involving two parts, the electronic part (He) and the phononic part (Hph) are as follows
where
(c†
(3.12)
He =
kc† εc
kck +
kf † εf
kfk + ∆
k+Qfk + f †
kck+Q),
k
k
k
√
(cid:88) (cid:88) (cid:88)
N h(b†
(3.13)
Hph = ω0
b† qbq +
−Q + b−Q),
q
(cid:88)
√
B†
N
(3.14)
δq,Q.
q = b†
q +
h ω0
The phononic part is diagonalized by defining a new phonon operator
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
(3.17)
B†
Hdia =
qBq,
kC† E1
kC†
1kC1k + E2
2kC2k + ω0
q
k
(cid:88) (cid:88)
where the electronic quasiparticle energies read as
k+Q
k+Q)
∓
(3.18)
E1,2
Wk,
k =
εf k + εc 2
k − εc 2
sgn(εf
8
(εc
k+Q − εf
k)2 + 4|∆|2. The quadratic form of Eq. (3.17) allows us to compute all
(cid:113)
(3.22)
knF (E1 k) + ξ2
,
k+Q = (cid:104)c† nc k = (cid:104)f † nf dk = (cid:104)c†
k+Q)
knF (E2 k) + η2 k), knF (E2 k), k)] sgn(εf k) − nF (E2
k − εc
k+Qck+Q(cid:105) = ξ2 knF (E1 kfk(cid:105) = η2 k+Qfk(cid:105) = −[nF (E1
∆ Wk
k ) are the Fermi-Dirac distribution functions; ξk and ηk are the prefactors of the k = 1. The lattice displacement in the EI state
k + η2
with Wk = expectation values, resulting in
1 √
√
(cid:104)b†
,
(3.24)
xQ =
−Q +bQ(cid:105) = −
here nF (E1,2 Bogoliubov transformation which satisfy ξ2 at momentum Q
h ω0
N
1 2ω0
(cid:114) 2 ω0
3.3. Numerical results and discussion
For the two-dimensional system consisting of N = 150 × 150 lattice sites, the numerical
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 εc − εf = 1 in the ground state.
bols) and the lattice displacement xQ (open symbols) as functions of εc − εf for some val-
ues of ω0 at g = 0.5, T = 0.
results are obtained by solving self-consistently Eqs. (3.9), (3.10), (3.22) and (3.24) starting from some guessed values for (cid:104)b† Q(cid:105) 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
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
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.
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).
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 ωc 0. 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
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 of temperature at εc − εf = 1 and g = 0.5.
bols) and the lattice displacement xQ (open symbols) as functions of g for some values of T at εc − εf = 1 and ω0 = 0.5.
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
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
0 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.
(the external pressure) when T changes at g = 0.5. The diagram shows that if the temperature increases, the critical value ωc
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
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.
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.
condensation regime is expanded when increasing electron – phonon coupling constant.
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 –
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.
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.
phonon coupling constant also increases. The excitonic condensation regime thus narrows.
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
(4.1)
H = H0 + Hint,
interaction can be written
(4.2)
H0 =
b† qbq.
kc† εc
kck +
kf † εf
kfk + ω0
q
k
k
k (ck); f †
k (fk) and b†
(cid:88) (cid:88) where H0 discribes the non-interacting part of electron – phonon system (cid:88)
here c† 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
g √
(4.4)
Hint =
−q + bq) + H.c.],
k+qck(cid:48)f † c†
k(cid:48)−qfk +
k+qfk(b† [c†
U N
N
k,k(cid:48),q
kq
(cid:88) (cid:88)
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
B†
(4.10)
Hdia =
qBq,
k α† E+
1kα1k +
k α† E−
2kα2k + ω0
q
k
k
2k (α2k) are the Bogoliubov quasi-particle fermionic creation (annihi-
1k (α1k) and α†
(cid:88) (cid:88) (cid:88)
k+Q)
k+Q
E±
∓
(4.11)
Γk,
k =
sgn(εf
k − εc 2
where α† lation) operators, respectively, with the electronic quasiparticle energies εf k + εc 2
here
(εc
(4.12)
Γk =
k+Q − εf
k)2 + 4|Λ|2,
(cid:113)
(4.7)
k + U nc/f ,
k = εf /c εf /c
and the electronic excitation energies now have acquired Hartree shifts
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 √
Λ =
(cid:104)b†
(cid:104)c†
(4.9)
−Q + b−Q(cid:105) −
k+Qfk(cid:105).
U N
N
k
(cid:88)
15
(4.13)
(4.14)
knF (E+ k ) + u2
, (4.15)
k+Q)
k − εc
knF (E− k ) + v2 k ), knF (E− k ), k )(cid:3) sgn(εf k ) − nF (E−
Λ Γk
k+Q = (cid:104)c† nc k+Qck+Q(cid:105) = u2 nf k = (cid:104)f † knF (E+ kfk(cid:105) = v2 k+Qfk(cid:105) = − (cid:2)nF (E+ nk = (cid:104)c† √
(4.16)
(cid:104)b†
δq,Q,,
q(cid:105) = −
N h ω0
k ) is the Fermi-Dirac distribution function; uk and vk are the prefactors of the k = 1. The lattice displacement and the single-
k + v2
We also obtain the system of self-consistently equations from the average values
1 √
√
(cid:104)b†
,
(4.19)
xQ =
h ω0
N
1 2ω0 (cid:16)
−Q + bQ(cid:105) = − (cid:17)
where nF (E± Bogoliubov transformation which satisfy u2 particle spectral functions of c and f electrons are therefore also determined by
Ac
ω − E+
,
(4.23)
k−Q
(cid:16) (cid:17) (cid:114) 2 ω0 ω − E−
(4.24)
k−Q (cid:1) + u2
+ v2 k−Qδ kδ (cid:0)ω − E−
k−Qδ kδ (cid:0)ω − E+
k
k
k (ω) = u2 Af k (ω) = v2
(cid:1) .
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 (cid:104)b† Q(cid:105) 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: The quasiparticle energies E+ lines); E−
Fig. 4.2: The quasiparticle energies E+ lines); E−
k (solid k (dash lines) and |nk| for small val-
ues of U at g = 0.6; T = 0.
k (solid k (dash lines) and the order parame- ter |nk| for large values of U at g = 0.6; T = 0.
Fig. 4.1 and Fig. 4.2 show the momentum dependence along the (k, k) direction in the
k ; E−
k and the order parameter |nk| for
first Brillouin zone of the quasiparticle energy bands E+
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.
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 of U for different g at εf =
xQ (dash lines) as functions of U for different εf at g =
−2.0; T = 0.
0.6; T = 0.
xQ (dash lines) as functions of T for different g at U = 1.5; εf = −2.0.
4.3.2. The EI order parameter and the lattice displacement
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
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 influ-
enced 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.
k(ω) (right) for different temperature
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
Fig. 4.16 shows the variation of Ac 4.3.4. The single-particle spectral functions of electrons k(ω) (left) and Af
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 elec-
trons. 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 in-
teraction, 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
Fig. 4.17: Ground-state phase diagram of the EFKM with additional electron – phonon coupling in the (U, g) plane at different εf .
(SC) disordered state is indicated by the green (orange) regime. Fig. 4.17 shows the phase
diagram of excitonic condensation state at T = 0 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 con-
densate 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 in-
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.
fluence of both Coulomb interaction and the electron – phonon interaction in the ground state is examined in more detail, especially involving the change of εf .
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 two-
band model including electron – phonon interaction and the extended Falicov-Kimball model
involving electron – phonon interaction to investigate the excitonic condensation phase tran-
sition 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 in-
vestigate 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 estab-
lished. 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 inter-
action 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
1. Thi-Hong-Hai-Do, Huu-Nha-Nguyen, Thi-Giang-Nguyen and Van-Nham-Phan, Tem-
perature effects in excitonic condensation driven by the lattice distortion, Physica Status
Solidi B 253, 1210, 2016.
2. 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.
3. 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
4. Do Thi Hong Hai and Phan Van Nham, Excitonic condensate phase transition in tran-
sition metal dichalcogenides, Journal of Science and Technology, Duy Tan university 6
(25), 17–21, 2017.
5. Do Thi Hong Hai and Phan Van Nham, BCS and BEC excitonic condensations in
transition-metal dichalcogenides, Journal of Science and Technology, Duy Tan univer-
sity 6 (25), 30–35, 2017.
6. 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.
7. Do Thi Hong Hai and Phan Van Nham, Spectral properties in the extended Falicov-
Kimball model involving the electron-phonon interaction: Excitonic insulator state for-
mation, DTU Journal of Science and Technology 6 (31), 89–94, 2018.
8. 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 6 (31), 95–100, 2018.
9. Do Thi Hong Hai and Phan Van Nham, Influence of phonon frequency on the excitonic
insulator state, DTU Journal of Science and Technology 3 (34), 87–92, 2019.
10. Do Thi Hong Hai and Phan Van Nham, Excitonic condensation in two-band model in-
volving electron – phonon interaction, DTU Journal of Science and Technology 3 (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, 1 – 4 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 Con-
ference on Theoretical Physics, Can Tho, 31 July – 3 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 – 2 August, 2018.
25
