Bipolaron in Spherical Quantum Dots

الجـامعــــــــــة اإلســـــالميــة بغــزة The Islamic University of Gaza

عمادة البحث العلمي والدراسات العليا Deanship of Research and Graduate Studies

كـليــــــــــــــــــــة العـلـــــــــــــــــــــوم Faculty of Science

مــــاجستيــــــــــر الفيــــــــزيـــــــــــاء Master of Physics

Environmental Health Bipolaron in Spherical Quantum Dots

البوالرون الثنائي في نقاط كمية كروية

By Heba S. Abukhousa

Supervised by

Prof. Dr. Bassam H. Saqqa Professor of Solid State Physics

A thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Physics

August/2019

إقــــــــــــــرار

أنا الموقع أدناه مقدم الرسالة التي تحمل العنوان:

Bipolaron in Spherical Quantum Dots

البوالرون الثنائي في نقاط كمية كروية

أقر بأن ما اشتملت عليه هذه الرسالة إنما هو نتاج جهدي الخاص، باستثناء ما تمت اإلشارة إليه حيثما ورد، وأن هذه الرسالة

ككل أو أي جزء منها لم يقدم من قبل االخرين لنيل درجة أو لقب علمي أو بحثي لدى أي مؤسسة تعليمية أو بحثية أخرى.

Declaration

I understand the nature of plagiarism, and I am aware of the University’s policy on this.

The work provided in this thesis, unless otherwise referenced, is the researcher's own work and has not been submitted by others elsewhere for any other degree or qualification.

هبة سعيد أبوخوصة :Student's name اسم الطالب: Date: التاريخ: Signature: التوقيع:

Abstract

The problem of the polaron in spherical quantum dots is studied using the strong coupling approximation. It is found that the energy of the polaron increases as the degree of confinement increases. The effect of the electron-phonon interaction is found to be enhanced as the degree of confinement increases. We have also considered the problem of bipolaron formation in the same confining potential by using the Landau–Pekar variational method in the limit of strong electron- phonon coupling. Special attention is paid to the stability of the bipolaronic state as a function of the degree of confinement and the average distance between the two electrons. The degree of confinement, represented by the parameter  is found to add another contribution to the problem in addition to the other parameters. Its contribution affects the problem in an efficient way due to its possible role in determining which of the forces, the Coulomb repulsive force, or the electron- phonon interaction force, will dominate over the other. Which of these two forces will be affected more due to the degree of confinement cannot be answered easily due to the interrelated dependence of all the parameters affecting the problem. It was found that for the bipolaron state to be stable the ratio between the Coulomb amplitude and the polaronic constant must be less than a critical value. This critical value is found to be dependent largely on the degree of confinement. Moreover, it is found that the large values of 푟0 (larger than the polaron radius) weaken the attraction between the two electrons by the phonon field.

III

ملخص الدراسة

تم دراسة مسألة البوالرون في النقاط الكمية الكروية باستخدام نظرية االقتران القوي. لقد وجد أن طاقة البوال رون تزداد

كلما زادت درجة الحصر. ووجد أن تأثير تفاعل االلكترون مع الفونونات يزداد كلما زادت درجة الحصر.

لقد درسنا أي ًضا مشكلة تكون البوالرون الثنائي في نفس جهد الحصر باستخدام طريقة النداو-بيكار التباينية في حدود

االقت ارن القوي بين االلكترون والفونونات. أعطينا اهتمام خاص الستقرار تكون البوالرون الثنائي كدالة في درجة الحصر

ومتوسط المسافة بين اإللكترونين. درجة الحصر، ممثلة في المعامل Ω , وجدت أنها تضيف مساهمة أخرى إلى

المسألة باإلضافة إلى العوامل األخرى . إن درجة الحصر تؤثر في المسألة بطريقة فعالة نظًار لدورها المحتمل في

تحديد أي من القوتين، قوة كولوم التنافرية بين االلكترونيين، أو قوة التجاذب بين االلكترون والفونونات، ستسود على

األخرى . أي من هاتين القوتين سوف تتأثر أكثر بسبب درجة الحصر ال يمكن اإلجابة عليها بسهولة بسبب االعتماد

المتداخل لجميع المعامالت التي تؤثر على المسألة. وقد وجد أنه لكي تكون حالة البوالرون الثنائي مستقرة ، يجب أن

تكون النسبة بين مقدار كولوم والثابت القطبي أقل من القيمة الحرجة. وقد وجد أن هذه القيمة الحرجة تعتمد بدرجة

كبيرة على درجة الحصر. إضافة لذلك فقد وجد أن القيم الكبيرة من 푟0 ) أكبر من نصف قطر البوالرون( تضعف

تجاذب االلكترونين مع مجال الفونونات.

Dedication

my role model in this life, my father may Allah have mercy on him,

my great mother,

my beloved sisters and brothers,

my fiance and his family, and to all those who challenge the difficulties to make of themselves something worthy of pride.

Acknowledgment

First of all, thanks to Allah for his kindness, generosity, and guidance to us on the straight path.

Then I would like to express my deep gratitude to my advisor Prof. Dr. Bassam Saqqa for all the patience and time invested in this thesis and above all for the fresh ideas in the moments when they were most needed. I have honored to work with him and I hope that one day I would become as good an advisor to my students as he has been to me.

Also, I would like to thank my family that grew up with their support and love for me.

All my love for my friend Yosra Matar, my sister and my study partner.

I will never forget everyone who taught me in the physics department through my study of both bachelor's and master's degrees especially Prof. Dr. Sofyan Taya and Prof. Dr. Nasser Farahat.

Table of Contents

Declaration…………………………………………………………………………...... I Judgment Result…………………………………………………………………...... II Abstract……………………………………………………………..…………………. III Abstract in Arabic……………………………………………………..…………….….IV Dedication………………………………………………………………………………..V Acknowledgment…………………………………………………………….…….…...VI Table of Contents………………………………………………………….…….……..VII List of Tables……………………………………………………………………....….VIII List of Figures………………………………………………...…………………….….IIX Chapter 1 Introduction………………………………………………………….……...1 1.1 Basic Concepts………...……………………………………………………….….2 1.1.1 The Concept of Polaron. ……………………………………………………....3 1.1.2 Types of Polarons…………..…..……..…………………………………….…5 1.2 The Concept of Bipolaron………….……………………………………….……..6 1.2.1 Frö lich Bipolaron ………….……………………………………………..…...8 1.2.2 Bipolaron and High-Temperature Superconductivity …………………….…10 1.3 Methods of Solving the Hamiltonian of Bipolaron………………………………11 1.4 Aim of the Work...………………………………..………………………………13 Chapter 2 Single Polaron Confined in Isotropic Quantum Dots……...………...... 15 2.1 Theory and calculations…………………………………………...…………...... 16 2.2 Results and Discussions……….………………………………………….….…..24 Chapter 3 Bipolaron Confined in Spherical Quantum Dots….………………….....26 3.1 Theory and calculations…………………………………………………...... 27 3.3 Results and Discussions……………………………………..…………...... 39 Conclusions.…………………………………..……..………………………………...47 Suggestions for Future Studies…………….....……..………………………………...50 References …………...……………………………………………...………………...52

VII

List of Tables

Table (3.1): The ground state energy of the two polarons compared with twice that of a single polaron in three-dimensional configuration (0) and 휂 = 휂푐 …………..…(46)

VIII

List of Figures

Figure (1.1): The electron-phonon interaction via (a) absorption (b) emission phonon by an electron …………………………………………………………………………….....4

Figure (1.2): The states of two electrons (a) each in its own potential well (b) are localized in the same potential well……………………………………………………...8

Figure (1.3): The phase diagram between stable bipolaron and metastable case……….9

Figure (2.1): The binding energy of the polaron as a function of electron-phonon coupling α for two different values of the confinement constant Ω …………………………………………………………………………………………..24

Figure (2.2): The binding energy of the polaron as a function of the confinement constant Ω for three distinctive values of α…………………………………………....25

Figure (3.1): The binding energy of the bipolaron versus the degree of confinement Ω at two selected values of the coupling constant α for η = 0.1……………………….……40

Figure (3.2): The binding energy of the bipolaron versus the polaronic constant α at two distinct values Ω for η = 0.01………………………………………………...………..41

Figure (3.3): The binding energy of bipolaron versus the degree of confinement Ω at different values of the Coulomb strength in terms of the relative permittivity 휂 for 훼 = 5 …………………………….…………………………...………………………………..43

Figure (3.4): The critical dielectric constant as a function of the confinement parameter Ω for different values of phonon-electron interaction constant 훼 …...…..….44

Figure (3.5): The binding energy of the bipolaron as a function of the inter-polaron distance 푟0for two distinct values of the dielectric constant 휂 for 훼 =5 and Ω = 1 ……………………..…………………………………………………………………....45

Chapter 1 Introduction

Chapter 1

Introduction

1.1 Basic Concepts

The crystalline structure forming the solids is a periodic arrangement of atoms, ions, or molecules. It can be described as three dimensional ordered systems consisting of designating points, called "lattice points", which considered as a regular periodic arrangement of points in space. The other content of the structure is the basis that is an atom or a set of atoms associated with each lattice point. The lattice points are the equilibrium positions of atoms but don't essentially lie at the center of atoms. The behavior of a small volume of the crystal can suffice to give the properties of the whole region because of the crystal symmetry. Moreover, the crystal structure forms the foundation of understanding of the properties of the material such as electronic band structure and optical properties (Omar, 1993).

In a crystal, the atoms don't seem to be fixed however vibrate regarding their equilibrium positions. These atoms are connected to each other by chemical bonds, that behave like springs that stretch and compress repeatedly throughout the oscillating motion. The movement of one atom back and forth affects the rest of the atoms that successively reply to that movement. The scene becomes a group of atoms moving in perfect harmony, and this collective motion spreads throughout the crystal producing localized traveling waves that have a particular value of energy and momentum. In keeping with Quantum Mechanics aspect, these waves are often treated as particles, which take the name of phonons (House, 2018).

Phonon forms the cornerstone of understanding several of the physical properties in material systems, including its influence on thermal, electrical, optical, noise phenomena and other properties of semiconductors and insulators. It acquires its name from the Greek word φωνή (phonē), which means sound or voice as a result of long-wavelength phonons

that are responsible for producing sound. The idea of phonons was presented by Russian physicist Igor Tamm (1932). These phonons are quantized with the energy ћ휔 = ℎʋ and oscillate with the same frequency ʋ = 휔/2휋 (Balandin, Pokatilov and Nika, 2007).

In many cases, very low frequencies of characteristic phonons have an effect on the motion of the electrons within the crystal forming a potential well in which the electrons are going to be trapped. This potential binds the electrons with energy that can be considered stable when the lowering of the energy of electrons goes beyond the energy needed to displace the associated atoms (Emin, 2017).

1.1.1 The Concept of Polaron

The electron-phonon interaction (see Figure (1.1)) is one of the most important fundamental interactions of quasiparticles in solids along with the Coulomb interaction. It serves as the main function for a variety of physical phenomena. Particularly, low-energy electronic excitations in metals are strongly modified by the coupling to lattice vibrations, which in turn influences their transport and thermodynamic properties. Electron-phonon coupling provides an appropriate environment for the attraction between two electrons, which is the origin of the electron pairing “Cooper pairs” that gave the physical explanation of what happens within the phenomenon of superconductivity (Pavarini, Koch and Scalettar, 2017).

The electron under certain conditions is clothed by a cloud of virtual phonons as it moves in an ionic or polar semiconductor crystal. The crystal, in turn, corresponds physically to the electron pulling nearby positive ions towards it and pushing nearby negative ions away (Clemens and Berciu, 2017).

(a) (b) Figure (1.1): Electron-phonon interaction (a) absorption and (b) emission of a phonon by an electron (Pavarini, Koch and Scalettar, 2017).

The system of electron plus the associated phonons can be treated as new quasiparticle known as polaron. The interaction with phonons modifies the electron properties, as a result, the physical properties of polaron differ from that of a free electron. For instance, the self-energy is lowered, whereas the effective mass is increased. This can be justified by the fact that as the electron moves, it forces the lattice distortion to accompany it creating its inertia to be larger. Thus, polaron can be studied through its effective mass, binding energy and its behavior under the influence of an external electromagnetic field (Senger, 2000).

The conception of polaron was ﬁrst introduced by Landau in (1933) (Landau and Lifshitz, 1965). After that, Landau and Pekar calculated the self-energy and the effective mass of polaron (Devreese, 1996).

The above description is for an electron polaron; the hole polaron is defined in the same way with electron is replaced by hole. Both of them are normal polarons that result from the interaction of electrons with longitudinal-optical (LO) phonons, however the original idea of polaron has been circulated over the years to incorporate polarization ﬁelds apart from the LO-phonon ﬁeld such as the interaction of electron with acoustic phonons that forms a piezo polaron and the interaction with atomic magnetic moments that forms a spin polaron (Steven, 2008).

1.1.2 Types of Polarons

The polaron is classified into large and small polaron according to its radius and how it compares to the lattice constant. The interest in small-polaron properties has been expanded and deeply studied by (Mott, 1990) who could identify several cases of small- polaron transport, including mobility across a variable domain, for which electrons jump across different values of distances rather than the closest neighbors, and also the importance of small polarons in non-crystalline semiconductors (Devreese, 1996). Many scientists have had an active role in understanding the basic features of small polaron such as (Eagles, 1963; Holstein, 1959; Lang and Firsov, 1962; Sewell, Yamashita1952, and Kurosawa, 1958) , while large polaron was further studied very well by (Devreese, 1996; Feynman, 1955; Fr¨ohlich, 1954; Pekar, 1946; Rashba, 1957) (Devreese and Alexandrov, 2009).

The large polaron is formed when the radius of a polaron is greater than the lattice spacing (푟푝 ≫ 푎), then the displacement of the ion is large as well as the deformation of the lattice will be large, which makes the large polaron extends over multiple equivalent sites (Steven, 2008). The large polaron moves quite slowly but harmoniously in coordination with the vibratory movements of the atom. The limited frequency of phonon and the slow motion of the polaron will allow the ion polarizations to follow polaron motion; this means that large polarons with low speed are expected to diffuse across the lattice as free electrons but with larger effective mass (Devreese and Alexandrov, 2009). In contrast, Holstein polaron or small discrete lattice occurs when the polaron is trapped in a region of dimensions in the order of lattice spacing (푟푝 ≤ 푎) and unable to move, therefore it collapses to a single equivalent site (Holstein, 1959). This will be realized as the coupling of the phonons increases, leading to the participation of all momenta of the Brillouin zone with electron in the polaron wave function. In this system, it can be said that the polaron binding energy is greater than the half-bandwidth 퐷 (Devreese and Alexandrov, 2009).

In addition to the size of polaron, the two types can be distinguished by their response to the application of an external electric field. Large polarons tend to have intra-molecular

(band) transport, while small polarons usually have inter-molecular (hopping) transport, which is caused by the interaction of the localized electron (hole) with the lattice vibrations which induces the electron to jump from one atom (or ion) to an adjacent atom (Devreese, 1996). The small polaron moves incoherently through a series of thermally activated hops (Emin, 2015).

The mobility of polaron changes with varying the number of phonons which are governed directly by temperatures, for large polaron its mobility is proportional to the inverse of the number of phonons, this makes it move with low mobility as the temperatures rise. While the behavior of the small polaron is reversed, the increasing of the number of phonons will support its mobility, so that the mobility of small polaron is thermally activated (Devreese, 2018).

The properties of such polarons are measured mainly by a unitless parameter called the Fröhlich coupling constant, denoted by 훼 which is proportional inversely to the polaron radius (Steven, 2008).

1.2 The Concept of Bipolaron

After giving the main concepts of polaron theory in the previous sections we can now go to the main subject of this thesis, which is the bipolaron concept and its formulation.

When an electron moves in a polar or ionic crystal, it will create a distortion in the lattice and leaves behind itself a positively charged path which in turn attracts other electrons. This occurs because of electrons much faster than phonons, thus the phonon attraction stays for a longer time range even after the original electron has moved away. The phonon is an intermediary to bind two electrons together with an attractive force even though the presence of Coulomb repulsion between them. The system consists of the interaction of two electrons via a common cloud of phonons is understood as bipolaron, if this phonon mediated attractive interaction can overcome the mutual Coulomb repulsion then the electrons can form a bipolaronic bound state. This idea was ﬁrst introduced in the

polaron literature by Pekar in the early ﬁfties (Verbist, Smondyrev, Peeters and Devreese, 1991).

If the binding is so strong, the two-phonon clouds merge into one and the two electrons may localize at the same atomic site in the lattice leading to one-center bipolaron or 푆0 bipolaron that means a spin-singlet state. In this state, two polarons form an s-orbital conﬁguration, where the net angular momentum of two united polarons is zero. On the other hand, if the binding is weak, each polaron preserves its own cloud and the binding is caused by mutual visits to the other carrier's cloud. In this case, we have got the 푆1 bipolaron (spin- triplet state); this means that two polarons localized at different sites so that they will form two-centered bipolaron. It is important to notice that two unbound polarons are totally different from two-centered bipolaron, the latter has nonzero formation energy of the bound state (Adolphs and Berciu, 2014; Nina, 2018).

A far from the case of a bipolaron, a polaronic exciton is a pair of an electron polaron bound to a hole polaron. In other words, two polarons of the same charges can bind into a bipolaron, and two polarons of opposite charges can bind into a polaronic exciton (Steven, 2008). As with polarons, there are two distinct types of bipolarons, small and large bipolarons. The self-trapped electronic carriers of a small bipolaron collapse to a single site when they are governed by their short-range interactions with surrounding atoms. By contrast, large bipolarons extend over multiple sites when they are governed by long-range interactions with surrounding ions (Emin, 2017).

1.2.1 Fr퐨̈ lich Bipolaron

The formation of bipolarons depends on the values of the electron-phonon coupling constant 훼 and the Coulomb repulsion coefficient 푈. Therefore, there are critical values of the repulsive coefficient, resulting directly from Coulomb interaction, and the attractive forces that result from the electron-phonon interaction lead to the formation of bipolarons or not (Verbist, Smondyrev, Peeters and Devreese, 1991; Devreese, 1996). As in

Figure(1.2), in the event that the Coulomb repulsion wins over the coupling to the phonon ﬁeld, the lowest energy states are two well-separated polarons, otherwise, the two electrons should form a bound pair. In other words, for a stable bipolaron state, the energy of the bipolaron should be less than twice the energy of a single polaron. The difference between twice the energy of a single polaron and the energy of a bipolaron is termed as the binding energy of the bipolaron (Miyao and Spohn, 2007).

(a) (b)

Figure (1.2): bipolaron formation. a) Two separated polarons each in its own polarization well. b) Bipolaron case where two electrons are localized in the same potential well (Nina, 2018).

Even though the polarons repel each other at quite large distances, under some circumstances, two large polarons can be also bound into a large bipolaron by exchange interaction. In the formulation of large bipolarons, the continuum approximation must be valid so that such bipolarons are referred to as Fröhlich bipolarons. The ratio 휂 = 휀∞⁄휀0 of the high-frequency and static dielectric constants is important, whose values are set from zero to one (0 ≤ 휂 ≤ 1). Moreover, the Fröhlich coupling constant 훼 and the dimensionless parameter 푈, which is a measure for the strength of the Coulomb repulsion between the two electrons, are characteristic parameters which the Frohlich bipolaron energy depends on. The formula that relates 푈 and 훼 is (Erc¸elebi and Senger, 2002):

√2α 푈 = 1 − 휂 (1.3)

As shown in Figure (1.3), only values of 푈 that satisfying the inequality 푈 ≥ √2훼 have a physical meaning. It was shown that bipolaron formation is preferred by larger values of 훼 and by smaller values of 휂.

The conventional condition for bipolaron stability is 퐸푏푝 ≤ 2퐸푝. From this condition, it follows that the Frö lich bipolaron is stable if the electron-phonon coupling constant is larger than a certain critical value: 훼 ≥ 훼푐 and the dielectric constant is less than its critical value: 휂 ≤ 휂푐(Devreese and Alexandrov, 2009).

Bipolaron Region

RepulsionU 푈 = √2α

휶

Figure (1.3): The stability region for bipolaron formation in 3D. The dotted line 푈 = √2α separates the physical region (푈 ≥ √2α) from the non-physical (푈 ≤ √2α) (Devreese and Alexandrov, 2009).

Due to the significant impact of the geometric structures on the stability of bipolaron formation, many researchers have highlighted the study of bipolaron properties under the effect of these structures confinement. These studies revealed that in the strong coupling limit, the stability region for bipolaron formation is enhanced with decreasing dimensionality of a structure. Thus, the quasi zero-dimensional system presents certain

interest as structures with maximal polaron effects. These structures are called quantum dots, which are considered as semiconductor crystals with zero-dimensional systems where the wave functions of electrons are confined in three dimensions which leads to size quantum effects to which the dot acquires its name (Houtepen, 2007; Fai, Teboul, Monteil, Nsangou and Maabou, 2004). Most of the crystal dots properties depend, mainly on its exact size and its geometry. These properties made the quantum dot structures possess a considerable technical interest, especially their potential use in lasers, in quantum computing, as storage devices, or as single-photon sources. Improvements in experimental techniques currently also allow studies of the properties of individual dots instead of ensembles (Holstein, 1959). Analogously, a quantum wire wave function is confined in two dimensions while a quantum well has confinement in one dimension (Houtepen, 2007).

1.2.2 Bipolaron and High-Temperature Superconductivity

The theory of superconductivity is one of the most prestigious topics in the field of solid-state physics. The physical interpretation of this strange phenomenon was introduced by BCS (Bardeen–Cooper–Schrieffer) theory after nearly 50 years of the discovery of the phenomenon. At that time, scientists thought that the theory was basically completed, but this belief changed after the discovery of high temperatures superconductors by Muller (1986), which demonstrated that the BCS theory is not applicable in their case. This, in turn, requires a more detailed analysis of the concept of Cooper pairs. Discontent with the Cooper theory gave birth to many alternative theories of pairing that trigger a great technological revolution which led to the emergence of new research and explanations of the superconductivity mechanism. One of them is the bipolaron theory in the case of strongly bound in order to keep it from collapsing at high temperatures. The formation of bipolarons in physical structures has very essential consequences. For instance, it has been suggested that Bose-Einstein condensation of bipolarons might be responsible for superconductivity in some high-Tc materials (Adolphs and Berciu, 2014; Lakhno, 2016).

Since the large bipolaron moves coherently with the vibrations of atoms with mobility that falls with increasing temperature and the coherent motion of individual quasiparticles can be considered as a prerequisite for a superconductor's large-scale coherence, this leads to the conclusion that large bipolarons provide the reasonable explanation for bipolarons superconductivity (Emin, 2015).

Superconductivity occurs when mutually interacting mobile particles of a Bose condensation are subjected to a collective ground-state which supports non-dissipative flow and this suggests that the system carries zero entropy. Non-dissipative flow results when the excitation, which can give rise to resistance to the flow of the collective ground state, is suppressed. Bose-Einstein condensate is a state of matter which was predicted by Albert Einstein and Satyendra Nath Bose in 1925, the condensate in itself was produced 70 years later, in 1995, by Eric Cornell and Carl Wieman in a gas of rubidium cooled to nearly absolute zero. In a Bose-condensate all the particles move consistently. They form one quantum-mechanical wave and behave like one huge particle. All of them are located in one and the same place and at the same time, each of them is spread over the entire region of space (Emin, 2017; Akson Russian Science Communication Association, 2018).

In addition, bipolarons have distinct characteristics that have made them a source of interest and study, as well as, they occupy an exceptional place in the polaron world. Bipolaron models have been successfully applied to the study of the transport properties of conducting polymers and organic magnetoresistance. Recently, it has been shown that bipolaron model leads to a straightforward interpretation of the isotope effect (Nina, 2018).

1.3 Methods of Solving the Hamiltonian of Bipolaron

According to the existence of different regimes in the Fröhlich model and since polaron (bipolaron) Hamiltonian that derived by Fröhlich (1954) cannot be solved exactly, many mathematical approximation techniques have been developed to deal with each regime separately to find a solution to the problem of polaron as well as bipolaron. One of these techniques is the strong coupling regime. This technique is the first method used to

deal with the polaron problem which is based on adopting a wavefunction consisting of two separate functions: one for the electron and the other for the phonon since the fast electron follows the slow motion of phonons adiabatically. It was originally introduced by Landau and Pekar in 1951. According to this approach, the polaron wavefunction takes the form

|ᴪ⟩ = |휑(푟)⟩|푓푖푒푙푑⟩ (1.1)

Where |휑(푟)⟩ is the electron-wave function, while |푓푖푒푙푑⟩ represents the Bose field of phonons (Devreese, 1996).

In this regime with ( > 1), the screening cloud is so strong that the electron becomes effectively self-trapped in the resulting potential. The theory includes variational calculations that represent very different means of obtaining approximate energies and wavefunctions for quantum mechanical systems. It can be often used to compute the ground state as well as to compute the low lying excited states and includes a guess of a reasonable, parametric form for a trial ground state wavefunction.

A modification is proposed to this method by Devreese, such that starting with an ansatz depending on the standard displaced oscillator transformation of the Pekar-strong- coupling theory and then modifying the adiabatic polaron state by a variationally determined perturbative extension that provides the theory to interpolate in the overall range of the coupling constant (Erc¸elebi and Senger, 1996).

In the weak coupling regime (α<1), the polaron can be thought of as a quasi-free electron carrying a broadly bound screening cloud of phonons. It is based on the mean-field theory suggested by Lee, Low and Pines (LLP) as a variational method. This theory supposes that the Hamiltonian interaction between electron and phonons as a small perturbed quantity and using the method of the perturbation technique to ﬁnd the energy of polaron. In this case, the momentum of the polaron is a conserved quantity. To identify the conserved polaron momentum, a unitary transformation is applied which translates phonons by an amount chosen such that the electron is shifted to the origin of the new frame, this

allows to remove the electron coordinate, followed by the displacement transformation and write only the phonon part of the Hamiltonian.

So far we were only interested in the extreme systems of either strong or weak coupling, but how these two regimes are connected, and how the polarons look like at intermediate couplings. To deal with these questions, Richard Feynman applied his path- integral formalism and developed a variational all-coupling polaron theory. The starting point of the formalism is that the imaginary-time path-integral for the Fröhlich Hamiltonian, with a single impurity (Stringari, Ketterle and Inguscio, 2016).

1.4 Aim of the Work

In this work, we are going to study the problem of a bipolaron in a quantum dot confinement and to investigate the impact of all the parameters controlling the problem, namely, the electron-phonon coupling constant, the Coulomb repulsive force strength, the mean distance between the two bound polarons, and the degree of confinement. How each of these parameters will affect the other will also be investigated. Special attention will be paid to the effect of the degree of confinement on the energy of the bipolaron and the effect of this confinement on the formation of the bipolaron. Increasing the degree of confinement is expected to enhance the importance of the electron-phonon interaction due to its effect in making the electron to move more slowly and then can respond more effectively to the lattice distortion. On the other hand, increasing the degree of confinement should make the two electrons closer and thus increasing the Coulomb repulsion between them. Which interaction will be affected more and dominate over the other will be investigated also.

Chapter 1 is devoted to the introduction with some theoretical concepts of the problem and some literature review, as the bipolaron is defined as a two polarons bound together, in chapter 2 we display a detailed calculation to the problem of a single polaron in the same confinement geometry, we discuss the numerical results that obtained and deduce the effect of phonon-electron interaction and the potential confinement to the energy of such a single polaron. Chapter 3 interested in studying the main purpose of this work,

which is the bipolaron problem over a range uncovering both the bipolaronic bound state and the state of two unbound but nearby polarons, the goal of this chapter is to determine the region of bipolaron stability and to study the bipolaron binding energy as a function of basic parameters that governing the system. While the last part is devoted to the conclusions, in which we put the summary of the work in a comprehensive manner.

Chapter2

Single Polaron Confined in Isotropic Quantum Dots

Chapter 2

Single Polaron Confined in Isotropic Quantum Dots

Before going in deep to the discussion of the bipolaron problem we think it is worth to devote this chapter to the theory of polaron. In doing so we can give the reader introductory concepts of the polaron theory so that when going to the bipolaron theory we can more easily handle its own terminology and corresponding formulation. In this chapter, we study the problem of electron immersed in longitudinal optical phonons and confined within parabolic potential in spherical quantum dots, we start with the Hamiltonian of polaron then by using the variational approximation we get the ground state energy of polaron.

2.1 Theory and Calculations

As an example of the polaron theory, we are considering the problem of the quasi- zero dimensional analog of the standard optical polaron relevant to a spherical quantum dot. Special attention is being devoted to the quasi zero-dimensional geometry, where the electrons do not have any free spatial direction to expand indeﬁnitely. Previous studies about polaron have been proved that lowering the dimensions leads to an increase in the polaronic effects such as the value of polaron binding energy (Erc¸elebi and Senger, 2002).

In the domain of the adiabatic approximation we start with the Hamiltonian of the electron immersed in the ﬁeld of bulk (LO) phonons and confined in a symmetric quantum dot 1 with an external parabolic potential 푉 (푟) = Ω2푟2, in which the dimensionless 푐표푛푓 2 frequency Ω serves for the measure of the degree of confinement of the electrons, and the units(푚 = ћ = ωLO = 1), where 푚 is the effective mass of the electron.

The Hamiltonian of the polaron is given by (Devreese, 2018)

퐻 = 퐻푒 + 퐻푝ℎ + 퐻푒−푝ℎ (2.1)

Where 퐻푒 is the electronic part of the Hamiltonian, which takes the form

푝2 1 퐻 = + Ω2푟2 푒 2 2 (2.2)

Where 푟⃗ is the position coordinate operator of the electron, 푝⃗ is its canonically conjugate momentum operator, and

1 휕 휕 푝2 = −∇2= − (푟2 ) (2.3) 푟 푟2 휕푟 휕푟

represents the radial part in spherical coordinates.

The Hamiltonian of phonon is defined as (Devreese, 2018)

+ 퐻푝ℎ = ∑ 푎푄푎푄 (2.4) 푄 + Where 푎푄 , 푎푛푑 푎푄 are the phonon creation and annihilation operators, respectively, and 푄⃗⃗ is the wave vector of the phonons.

The last term of Equation (2.1) represents the electron-phonon interaction part of the Hamiltonian that given as (Devreese, 2018)

푖푄⃗⃗⃗⃗.푟⃗ + −푖푄⃗⃗⃗⃗.푟⃗ 퐻푒−푝ℎ = ∑ 푉푄 (푎푄푒 + 푎푄푒 ) (2.5) 푄

Where 푉푄 is the amplitude of the electron-phonon interaction which is related to phonon wave vector 푄⃗⃗ through

1⁄2 ћ휔 2 2휋훼 ћ 1⁄4 퐿푂 √ (2.6) 푉푄 = −푖 ( ) ( ) ( ) 푄 푉 푚휔퐿푂

In Equation (2.6), 푉 represents the volume of the crystal (which is taken as a unit),

ћ휔퐿푂is the energy of the phonons, and 훼 is the electron-phonon coupling, which affects dramatically in the polaron formation.

The last form of the Hamiltonian becomes (Erc¸elebi and Senger, 2002)

+ 푖푄⃗⃗.푟⃗ + −푖푄⃗⃗.푟⃗ 퐻 = 퐻푒 + ∑ 푎푄푎푄 + ∑ 푉푄(푎푄푒 + 푎푄푒 ) 푄 푄 (2.7)

To solve the problem in adiabatic approximation we take the trial wavefunction as in Equation (1.1) which is

|Ѱ푝⟩ = |휑푒⟩|휑푝ℎ⟩ (2.8)

Where |휑푒⟩ is the ground state wave for the electron, and |휑푝ℎ⟩ is the phonon wave function which can be written as

휑푝ℎ = 푆푄|0⟩ (2.9)

The ket |0⟩ is a vacuum state.

푆푄 is the unitary displacement operator which is given by (Devreese, 2018)

+ )2.10( 푆푄 = 푒푥푝 ∑ 푉푄푠푄{푎푄 − 푎푄 } 푄

Here 푠푄 will be considered as a variational function.

A variational ansatz is made for the electron wavefunction. Motivated by the idea that the effective potential due to the phonon cloud can localize the electron, we select a Gaussian form to approximate the polaron ground state as

휆2푟2 − )2.11( 휑푒(푟⃗) = 푁휆푒 2

Where 휆 is a variational parameter to be determined numerically, and 푁휆 is a normalization constant of 휑푒.

To get the value of this normalization constant we can apply the normalization condition to the above equation, that is,

2휋 휋 ∞ 2 −휆2푟2 2 푁휆 ∫ ∫ ∫ 푒 푟 푑푟 푠푖푛(휃) 푑휃 푑ɸ = 1 )2.12( 0 0 0

We will get 휑푒(푟⃗) as

1 3 2 휆2푟2 λ − 2 )2.13( 휑푒(푟⃗) = ( 3) 푒 휋2

The expectation value of the Hamiltonian that given in (2.1) in the state of Equations (2.8) and (2.9) gives the ground state energy of polaron such as

퐸 = ⟨0|⟨휑 |푆−1퐻푆 |휑 ⟩|0⟩ = ⟨0|⟨휑 |퐻´|휑 ⟩|0⟩ (2.13) 푝 푒 푄 푄 푒 푒 푒

Where

´ −1 퐻 → 푆푄 퐻푆푄 (2.14)

By substituting for the Hamiltonian (2.7) in (2.14), we get

´ −1 −1 + 퐻 = 푆푄 (퐻푒)푆푄 + 푆푄 (∑ 푎푄푎푄) 푆푄 푄

−1 푖푄⃗⃗.푟⃗ + −푖푄⃗⃗.푟⃗ +푆푄 (∑ 푉푄(푎푄푒 + 푎푄 푒 ))푆푄 푄 (2.15)

By calling the Baker Campbell-Hausdorﬀ formula

1 1 푒퐴̂퐵̂푒−퐴̂ = 퐵̂ + [퐴̂, 퐵̂] + [퐴̂, [퐴̂, 퐵̂]] + ⋯ 1! 2! (2.16)

And Equation (2.10) which explains the expression of 푆푄, we obtain the following results:

−1 푆푄 [퐻푒]푆푄 = 퐻푒 (2.17)

−1 + + + 2 2 푆푄 [∑ 푎푄푎푄] 푆푄 = ∑ 푎푄푎푄 − ∑ 푉푄푠푄(푎푄 + 푎푄) + ∑ 푉푄 푠푄 푄 푄 푄 푄 (2.18)

−1 푖푄⃗⃗.푟⃗ + −푖푄⃗⃗.푟⃗ 푖푄⃗⃗.푟⃗ + −푖푄⃗⃗.푟⃗ 푆푄 [∑ 푉푄(푎푄푒 + 푎푄푒 )] 푆푄 = ∑ 푉푄 (푎푄푒 + 푎푄푒 ) 푄 푄

2 푖푄⃗⃗.푟⃗ −푖푄⃗⃗.푟⃗ − ∑ 푉푄 푠푄 (푒 + 푒 ) (2.19) 푄 The transformed Hamiltonian is then

´ + + 2 2 퐻 = 퐻푒 + ∑ 푎푄푎푄 − ∑ 푉푄푠푄(푎푄 + 푎푄) + ∑ 푉푄 푠푄 푄 푄 푄

푖푄⃗⃗.푟⃗ + −푖푄⃗⃗.푟⃗ 2 푖푄⃗⃗.푟⃗ −푖푄⃗⃗.푟⃗ + ∑ 푉푄(푎푄푒 + 푎푄 푒 ) − ∑ 푉푄 푠푄 (푒 + 푒 ) 푄 푄 (2.20)

Take the expectation value of 퐻´according to Equation (2.13), the energy will be then

2 2 퐸푝 = ⟨휑푒|퐻푒|휑푒⟩ + ∑ 푉푄 푠푄 푄

2 풊푄⃗⃗.푟⃗ −풊푄⃗⃗.푟⃗ (2.21) − ∑ 푉푄 푠푄⟨휑푒| (푒 + 푒 ) |휑푒⟩ 푄

By minimizing Equation (2.21) with respect to sQ, we can obtain sQ as

1 푠 = ⟨휑 | (푒풊푄⃗⃗.푟⃗ + 푒−풊푄⃗⃗.푟⃗) |휑 ⟩ (2.22) 푄 푒 2 푒

Replacing the last term in Equation (2.21) by 푠푄, then we get the polaron ground state energy as

2 2 퐸푝 = ⟨휑푒|퐻푒|휑푒⟩ − ∑ 푉푄 푠푄 )2.23( 푄

This Equation can be rewritten to the form:

퐸푝 = 퐸푘 + 퐸Ω − 퐸푝ℎ )2.24(

The contribution of the electron in the ground state energy is given by

푝2 1 ⟨휑 |퐻 |휑 ⟩ = 〈휑 | |휑 〉 + 〈휑 | Ω2푟2|휑 〉 )2.25( 푒 푒 푒 푒 2 푒 푒 2 푒

The first term is the kinetic energy which is calculated by using the representation of the 2 2 operator 푝 = −∇푟

푝2 −∇2 3휆2 퐸 = ⟨휑 | |휑 ⟩ =⟨휑 | 푟 |휑 ⟩ = )2.26( 푘 푒 2 푒 푒 2 푒 4

While the second term calculates the external potential through which the electron is confined, and can be found as

1 3Ω2 퐸 = 〈휑 | Ω2푟2|휑 〉 = )2.27( Ω 푒 2 푒 4휆2

To attain the energy caused by phonons, return to (2.22),

±푖푄⃗⃗.푟⃗ 푠푄 = ⟨휑푒| (푒 ) |휑푒⟩ )2.28(

∞ 휋 2 2 −휆2푟2 푖푄푟푐표푠(휃) 푠푄 = 2π푁휆 ∫ ∫ 푟 푒 푒 푠푖푛(휃) 푑휃 푑푟 0 0

∞ 2 −푄2 4휋푁휆 2 2 = ∫ 푟푒−휆 푟 푠푖푛(푄푟) 푑푟 = 푒 4휆2 푄 (2.29) 0

Projecting out the 푄⃗⃗ -summation in (2.23) using the transformation

∞ 2휋 휋 ∞ 1 4휋 ∑ → ∫ ∫ ∫ 푠푖푛(휃) 푑휃 푑∅ 푄2 푑푄 = ∫ 푄2 푑푄 (2휋)3 (2휋)3 (2.30) 푄 0 0 0 0

Then the last term in Equation (2.24) can be obtained as

∞ ∞ 4휋 −푄2 2훼 −푄2 훼휆 2 2 2 2 퐸푝ℎ = ∫ 푉푄 푒 2휆 푄 푑푄 = ∫ 푒 2휆 푑푄 = (2휋)3 휋 √휋 )2.31( 0 0

The variational energy in Equation (2.24) thus becomes

3휆2 3Ω2 훼휆 퐸푝 = + − )2.32( 4 4휆2 √휋

The equation tells that the degree of confinement  puts another parameter to the problem such that its value together with the value of the electron-phonon coupling  play an interrelated role in the problem such that one affects the role of the other in the problem.

By setting Ω = 0 in Equation (2.32), the ground state energy will have the form

3휆2 훼휆 퐸푝 = − 4 √휋 (2.33)

Minimizing Equation (2.33) with respect to the variational parameter 휆

휕퐸푝 3휆 훼 (2.34) = − = 0 휕휆 2 √휋

The value of the variational parameter 휆 that minimizes the energy is

2훼 휆 = 3√휋 (2.35)

Substituting Equation (2.35) into Equation (2.33) to get the local minimum of the energy which is

훼2 퐸 = − 푝 3휋 (2.36)

It is quite clear from Equation (2.36) that we recover the result of the three-dimensional problem (bulk limit) in the strong coupling regime as 0, as one expects.

It is important to point out that the binding energy of polaron can be determined from the relation (Devreese, 2018)

3 (2.37) 퐵퐸 = Ω − 퐸 2 푝

2.2 Results and discussions

휶

Figure (2.1): The binding energy of polaron versus electron-phonon coupling constant 훼 for Ω = 0.5, 10.

In Figure (2.1), we plot the binding energy as a function of the coupling constant  for two different values of . The Figure reveals that the importance of the polaronic effect becomes more pronounced by increasing the confinement parameter. This because, in our opinion that increasing the confinement of the electron makes it move more slowly, and hence responding to the lattice distortion in a more effective way. Also, the effect of the degree of confinement is more pronounced for large values of the coupling constant and this explains why the two lines in the figure approach each other for small values of . This enhances our previous claim about the fact that the two parameters do not enter the problem independently, but rather in an interrelated manner.

Ω

Figure (2.2): The binding energy of polaron versus the degree of confinement Ω for different values of α.

Figure (2.2) illustrates the binding energy versus the degree of confinement Ω for three distinct values of the coupling constant . Again the polaronic effect enhances for more confinement and the effect becomes significant as  increases as revealed from this figure.

Chapter 3 Bipolaron Confined in Spherical Quantum dots

Chapter 3

Fröhlich Bipolaron Confined in Spherical Quantum Dots

In this chapter, we study the bipolaron formation and its stability under the effect of many parameters such as the electron-phonon interaction, dielectric material constant and the confinement constant. Moreover, we introduce the polaron-polaron distance which makes the domain of stability more flexible. These factors have an effective role in determining the state of the two electrons.

3.1 Theory and Calculations

The model we are dealing with is a pair of electrons interact with LO phonons in spherical quantum dots and trapped within an isotropic parabolic potential box. Fröhlich introduced a model of Hamiltonian for the large bipolaron through which its dynamics are treated quantum mechanically.

The usual Fröhlich bipolaron Hamiltonian consists of three parts, the Hamiltonian of electron, the Hamiltonian of phonon, and the Hamiltonian of electron-phonon interaction, that is, (Erc¸elebi and Senger, 2002)

퐻 = 퐻푒 + 퐻푝ℎ + 퐻푒−푝ℎ (3.1)

The first term represents the electron Hamiltonian which has the form(Erc¸elebi and Senger, 2002)

1 2 2 2 푈 퐻푒 = ∑ (푝푗 + Ω 푟푗 ) + (3.2) 2 |푟⃗1 − 푟⃗2| 푗=1,2

In Equation (3.2), 푟⃗⃗𝑗⃗(j=1,2), 푝⃗푗(푗 = 1,2) are the positions and the momenta of the electrons respectively, and Ω represents the strength of the quantum dot potential that serves for the measure of the degree of confinement of the electrons, which is given by

1 2 푘 (3.3) Ω = ( 2 ) 푚푤퐿푂

In which 푘 denotes the force constant, and (ћ = 푚 = 휔퐿푂 = 1), where 푚 is the effective mass of the electron, 휔퐿푂 is the frequency of the longitudinal optical phonons (Devreese, 2018).

푈 is the dimensionless constant of the Coulomb interaction between the two electrons. Large values of 푈 mean that the repulsive force between the two electrons forming the bipolaron is strong and this makes the formation of a bipolaronic state less likely.

퐻푝ℎ represents the phonon Hamiltonian which is written as (Devreese, 2018)

+ 퐻푝ℎ = ∑ 푎푄푎푄 (3.4) 푄 + where 푎푄and 푎푄 represent ladder operators and have the eﬀect of lowering or raising the energy of the state, and 푄⃗⃗ is the wave-vector of the phonons.

The Hamiltonian of the electron-phonon interaction is (Erc¸elebi and Senger, 2002)

푖푄⃗⃗.푟⃗푗 + −푖푄⃗⃗.푟⃗푗 퐻푒−푝ℎ = ∑ ∑ 푉푄(푎푄푒 + 푎푄푒 ) (3.5) 푗=1,2 푄

where 푉푄 is the amplitude of the electron-phonon interaction which is given as (Devreese, 2018)

1⁄2 2√2휋훼 푉 = ( ) (3.6) 푄 푄2

The strength of the electron–phonon interaction is expressed by a dimensionless coupling constant 훼, which is deﬁned as (Devreese, 2018)

푒2 푚 1 1 훼 = √ ( − ) (3.7) ћ 2ћ휔퐿푂 휀∞ 휀0

In this deﬁnition, 휀∞ and 휀0 are, respectively, the high-frequency and the static dielectric constants of the material.

The dimensionless constants of the Coulomb interaction 푈 and of the electron–phonon coupling α are related by the equation (Erc¸elebi and Senger, 2002)

푒2 √2훼 푈 = = 휖∞ 1 − 휂 (3.8) in which the parameter 휂 is the ratio of the high-frequency and static dielectric constants of the material, that given by(Devreese, 2018)

휖∞ 휂 = < 1 (3.9) 휖표

It is clear from Equation (3.8) that increasing the parameter  leads to an increase in the Coulomb potential amplitude U and so an increase to the repulsive electron-electron interaction. The two parameters  and  play an important role in the problem since their values relative to each other determine whether we can have a bipolaronic state or not. The polaronic coupling should dominate over the Coulomb interaction to have a stable bipolaronic state. It should be noted that these two parameters do not inter the problem independently, but in an interrelated manner, as cited in Equation (3.8).

The last form of the Hamiltonian becomes

+ 푖푄⃗⃗.푟⃗푗 + −푖푄⃗⃗.푟⃗푗 퐻 = 퐻푒 + ∑ 푎푄 푎푄 + ∑ ∑ 푉푄(푎푄푒 + 푎푄푒 ) (3.10) 푄 푗=1,2 푄

In the framework of the adiabatic theory, we choose to use the strong-coupling polaron approximation, since verification of bipolarons needs large values of the electron- phonon coupling constant so as to preserve the phonon-mediated binding and stand up to the strong repulsive Coulomb interaction.

As mentioned earlier, we adopt the variational approximation by using the trial ansatz of the adiabatic theory which divides the total wavefunction into two parts: one contains the particle coordinate only, and the other attached to the phonon variables, i.e.

Ѱ푏푝 = 휑(푟⃗1, 푟⃗2; 푟⃗0)휑푝ℎ (3.11)

where 휑(푟⃗1, 푟⃗2; 푟⃗0) is the ground state wavefunction for the two electrons, 푟⃗0is the distance between the centers of the two electrons, and the phonon wavefunction is given by

휑푝ℎ = 푆푄|0⟩ (3.12) with

+ 푆푄 = 푒푥푝 ∑ 푉푄푠푄{푎푄 − 푎푄 } (3.13) 푄

Where |0⟩ is the phonon vacuum state, and the displaced-oscillator transformation 푆푄 yields the most efficient polarization ﬁelds around the electrons through the variational term sQ.

The electronic part formula of the trial wavefunction is chosen to suit all cases of the two electrons, which makes it sufficient to describe different ground state configurations of the two polaron system at once. An acceptable approach may be to manipulate two electrons, namely, the bipolaronic bound state, the state of two close polarons that are not in the bipolaronic bound state but yet confined within the same dot, and the state of two individual polarons. Compatible with the adiabatic approximation and the form of the confinement potential we write the two electrons wave function in terms of Gaussian functions as (Erçelebi and Senger, 2002)

1 1 휑 (푟⃗ , 푟⃗ ; 푟⃗ ) = 푔(|푟⃗ − 푟⃗ |)퐺 (푟⃗ − 푟⃗ ) 퐺 (푟⃗ + 푟⃗ ) (3.14) 푒 1 2 0 1 2 푎 1 2 0 푏 2 2 0

where 푔(|푟⃗1 − 푟⃗2|) is the Coulomb correlation function of Jastrow type. 퐺푎 and 퐺푏 are 1 1 Gaussian functions centered symmetrically around the origin at + 푟⃗ and− 푟⃗ , where a 2 0 2 0 and b are variational parameters to be determined numerically.

To calculate the energy of bipolaron, return to Equation (3.11), take the wavefunction of the bipolaron as

Ѱ푏푝 = 휑푒푆푄|0⟩ (3.15)

The expectation value of Equation (3.15) yields to the ground state energy of bipolaron

−1 퐸푏푝 = ⟨0|⟨휑푒|푆푄 퐻푆푄|휑푒⟩|0⟩ (3.16)

Making use of the Baker Campbell-Hausdorﬀ formula:

1 푒퐴̂퐵̂푒−퐴̂ = 퐵̂ + [퐴̂, 퐵̂] + [퐴̂, [퐴̂, 퐵̂]] + ⋯ (3.17) 2 with using of the standard commutator relation

[퐴̂, 퐵̂퐶̂] = [퐴̂, 퐵̂]퐶̂ + 퐵̂[퐴̂, 퐶̂] (3.18)

and Equation (3.13) which explains the expression of 푆푄, one can get the ground state energy of bipolaron as

2 2 2 푖푄⃗⃗.푟⃗푗 −푖푄⃗⃗.푟⃗푗 퐸푏푝 = ⟨휑푒|퐻푒|휑푒⟩ + ∑ 푉푄 푠푄 − ∑ ∑ 푉푄 푠푄⟨휑푒| (푒 + 푒 ) |휑푒⟩ (3.19) 푄 푗=1,2 푄

The minimization of 퐸푏푝 with respect to 푠푄 leads to

1 ⃗⃗ ⃗⃗ 푠 = ⟨휑 | ∑ (푒푖푄.푟⃗푗 + 푒−푖푄.푟⃗푗) |휑 ⟩ 푄 푒 2 푒 (3.20) 푗=1,2 substituting back in Equation (3.19) for 푠푄 to eliminate the summation over 푗, then we obtain

2 2 퐸푏푝 = ⟨휑푒|퐻푒|휑푒⟩ − ∑ 푉푄 푠푄 (3.21) 푄 transforming to a representation in the center of mass of both positions and momenta,

1 푅⃗⃗ = (푟⃗ + 푟⃗ ) , 푃⃗⃗ = 푝⃗ + 푝⃗ (3.22) 2 1 2 1 2 as well as relative coordinates of them are

1 푟⃗ = 푟⃗ − 푟⃗ , 푝⃗ = (푝⃗ − 푝⃗ ) (3.23) 1 2 2 1 2

The electronic wave function in Equation (3.14) corresponds to a form that can be separated into the center of mass and relative coordinates, each part having a distinctive oscillator-type waveform, given by (Erc¸elebi and Senger, 2002)

휑푒(푟⃗, 푅⃗⃗; 푟⃗0) = g(r)휑(푅⃗⃗)휑(푟⃗) (3.24)

푎2푅2 − (3.25) 휑(푅⃗⃗) = 푁푎푒 2

푏2(푟⃗−푟⃗ )2 − 0 (3.26) ᴪ(푟⃗) = 푁푏푟푒 2

Our choice for a Gaussian-type electronic wavefunction is quite understandable. Since we are interested in the bipolaronic state, the electron-phonon interaction is expected to be dominant over the Coulomb interaction and this makes a Gaussian-type wavefunction to be fit for such a situation over the other possible choices. Such a selection is also compatible with the quadratic barrier of the confined potential adopted in the calculation.

In our calculations, we set 푔(푟⃗) = 푟 to yield a comparatively lower variational upper bound to the ground-state energy for the bipolaron. Moreover, this ensures 휑푒 = 0 for 푟 = 0, so that the electrons are repulsively set separated (Erc¸elebi and Senger, 2002).

푁푎 and 푁푏 are normalization constants which can be determined by setting the integration all over the space to unity such as

∞ 휋 2휋 2 2 2 푁훾 ∫ ∫ ∫ 휑훾 푟 푠푖푛(휃) 푑∅ 푑휃 푑푟 = 1 (3.27) 0 0 0

The transformed forms of the electronic part of the Hamiltonian and the 푠푄 terms are written as

1 1 푈 퐻 = 푃2 + 푝2 + Ω2푅2 + Ω2푟2 + (3.28) 푒 4 4 푟 And,

1 푠 = 〈ᴪ(푟⃗)|2푐표푠 ( 푄⃗⃗. 푟⃗) |ᴪ(푟⃗)⟩⟨휑(푅⃗⃗)|푒±푖푄⃗⃗.푅⃗⃗|휑(푅⃗⃗)〉 (3.29) 푄 2

In a complete form, the ground state energy of the composite system of two polarons is evaluated as

퐸푏푝 = 퐸푅 + 퐸푟 + 푉푐표푛푓(푅⃗⃗) + 푉푐표푛푓(푟⃗) + 퐸푐 − 퐸푝ℎ (3.30) Where

1 −1 3 퐸 = 〈휑(푅⃗⃗)| 푃2|휑(푅⃗⃗)〉 = 〈휑(푅⃗⃗)| 훻2|휑(푅⃗⃗)〉 = 푎2 (3.31) 푅 4 4 푅 8

1 휕 휕 ∇2 = − (푅2 ) is the radial part in spherical coordinate. 푅 푅2 휕푅 휕푅

2 2 퐸푟 = ⟨ᴪ(푟⃗)|푝 |ᴪ(푟⃗)⟩ = ⟨ᴪ(푟⃗)| − ∇푟|ᴪ(푟⃗)⟩

2 휋푁푏 = [√휋(1 + Erf[푏푟 ])(4푏4푟4 + 20푏2푟2 + 7) 4푏3 0 0 0

2 2 −푏 푟0 3 3 (3.32) +푒 (4푏 푟0 + 18푏푟0)]

3Ω2 푉 (푅⃗⃗) = 〈휑(푅⃗⃗)|Ω2푅2|휑(푅⃗⃗)〉 = (3.33) 푐표푛푓 2푎2

and the confining potential in terms of the relative coordinate is derived as

Ω2푟2 푉 (푟⃗) = 〈ᴪ(푟⃗)| |ᴪ(푟⃗)〉 푐표푛푓 4

2 2 휋Ω 푁푏 = [√휋(1 + Erf[푏푟 ])(8푏6푟6 + 60푏4푟4 + 90푏2푟2 + 15) 16푏7 0 0 0 0 2 2 −푏 푟0 5 5 3 3 (3.34) +푒 (8푏 푟0 + 56푏 푟0 + 66푏푟0)]

The Coulomb force between the two electrons is 2 2 푈 휋Ω 푈푁푏 퐸 = 〈ᴪ(푟⃗)| |ᴪ(푟⃗)〉 = [√휋(1 + Erf[푏푟 ])(2푏3푟3 + 3푏푟 ) 푐 푟 푏4 0 0 0

2 2 −푏 푟0 2 2 +2푒 (푏 푟0 + 1)] (3.35) where Erf(x) refers to the Error Function.

The last term in Equation (3.30) represents the phonons contribution to the ground state energy of bipolaron. It is considered an important foundation and a pivot to understand the nature of bipolaron. In order to get it, we have to do long computational work that will be explained in details later in this chapter.

2 2 퐸푝ℎ = ∑ 푉푄 푠푄 (3.36) 푄

Rewrite Equation (3.29) as

푠푄 = 푠푄(푅⃗⃗)푠푄(푟⃗) (3.37)

The first part in the above equation corresponding to the variational function in terms of the center of mass and can be evaluated as

∞ 휋 ±푖푄⃗⃗.푅⃗⃗ 2 2 −푎2푅2 푖푄⃗⃗.푅⃗⃗ 푠푄(푅⃗⃗) = ⟨휑(푅⃗⃗)|푒 |휑(푅⃗⃗)⟩ = 2휋푁푎 ∫ ∫ 푅 푒 푒 푠푖푛(휃) 푑휃 푑푅 0 0

−푄2 = e4푎2 (3.38)

In the same manner, 푠푄(푟⃗) has been derived to get the form

푄⃗⃗. 푟⃗ 푠 (푟⃗) = ⟨ᴪ(푟⃗)|2푐표푠 ( ) |ᴪ(푟⃗)⟩ 푄 2 ∞ 휋 2 2 푄⃗⃗. 푟⃗ = 4휋푁2 ∫ ∫ 푟4푒−푏 (푟−푟0) cos ( ) 푠푖푛(휃) 푑휃 푑푟 푏 2 0 0

2 2 2 2 2 푖푄 2휋푁 −푏 푟0 −푄 푖푄푟0 −2푏 푟0 − 푏 2 2 = [−푖푒 2 푒32푏 훤(4) (푒 4 퐷−4 ( ) 푄푏4 √2푏

2 푖푄 −푖푄푟 −2푏 푟 + 0 0 2 − 푒 4 퐷−4 ( ))] (3.39) √2푏

where 퐷푛(푧) are the parabolic cylinder functions which can be expanded at 푛 = −4 as

푧2 푒 4 휋 휋 퐷 (푧) = [2 − 3√ 푧 + 3푧2 − √ 푧3] (3.40) −4 훤(4) 2 2

After a series of manipulation for Equation (3.39) in terms of (3.40), the final form of 푠푄(푟⃗) can be obtained as

2 −푄2 2휋푁푏 푄푟0 3 푄푟0 푠 (푟⃗) = e16푏2 [(푐 − 푐 푄2). 푠푖푛 ( ) + (푐 푄 − 푐 푄 ) . 푐표푠 ( )] (3.41) 푄 푄푏4 1 2 2 3 4 2

where

2 2 3 3 푐1 = 4 + 6√휋푏푟0 + 12푏 푟 + 4√휋푏 푟 0 0 (3.42)

3 3푟 푐 = + 0 2 4푏2 4푏 (3.43)

3√휋 푐 = + 6푟 + 3√휋푏푟2 (3.44) 3 2푏 0 0 √휋 푐 = 4 16푏3 (3.45)

Substituting (3.38) and (3.41) in (3.37), then squaring the result

2 2 −푄2 −푄2 2 2 2 2 2휋푁푏 푐1 푄푟0 푄푟0 푠 = ( ) e 8푏2 e2푎2 [( − 푐 푄) (푠푖푛 ( )) + (푐 − 푐 푄2)2 (푐표푠 ( )) 푄 푏4 푄 2 2 3 4 2

푐1 푄푟0 푄푟0 + 2 ( − 푐 푄) (푐 − 푐 푄2) 푠푖푛 ( ) 푐표푠 ( )] (3.46) 푄 2 3 4 2 2

Projecting out the 푄⃗⃗-summation in Equation (3.36) using the transformation

∞ 2휋 휋 ∞ 1 4휋 ∑ → ∫ ∫ ∫ 푠푖푛(휃) 푑휃 푑∅ 푄2 푑푄 = ∫ 푄2 푑푄 (2휋)3 (2휋)3 푄 0 0 0 0

Then Equation (3.36) becomes

∞ √2훼 퐸 = ∫ 푠2 d푄 (3.48) 푝ℎ 휋 푄 0 Setting

4√2 휋푁4훼 (3.49) 푐 = 푏 5 푏8

To get the desired Equation (3.48), integrate Equation (3.46) over Q term by term

∞ 2 2 −푄2 −푄2 푐1 푄푟0 푇 = ∫ 푐 ( − 푐 푄) e8푏2 e2푎2 (푠푖푛 ( )) 푑푄 1 5 푄 2 2 0

2 2 2 2 3 2 3 3 −2푟0 푎 푏 −푐 √푎2 + 4푏2 푐 푐 푎푏 2푐 푏 푎 2 2 1 2 1 2 = 푐5√2휋 [1 − 푒 푎 +4푏 ] [ − + ] 8푎푏 √푎2 + 4푏2 3√푎2 + 4푏2 2 2 2 −2푟0 푎 푏 2 2 5 5 푎2+4푏2 1 2 √2푟0푎푏 8푐5푐2 √2휋푟0 푎 푏 푒 (3.50) + 푐5푟0휋푐1 Erf [ ] + 4 √푎2 + 4푏2 5√푎2 + 4푏2

∞ −푄2 −푄2 푐1 푄푟0 푄푟0 푇 = 2 ∫ 푐 ( − 푐 푄) (푐 − 푐 푄2)e8푏2 e2푎2 푠푖푛 ( ) 푐표푠 ( ) 푑푄 2 5 푄 2 3 4 2 2 0

2 2 2 3 7 7 3 3 5 5 −2푟0 푎 푏 −32푐 푐 푟 푎 푏 2푟 푎 푏 (푐 푐 + 푐 푐 ) 24푐 푐 푟 푎 푏 2 2 2 4 0 0 1 4 2 3 2 4 0 = [2푐5√2휋푒 푎 +4푏 ] [ − + ] 7√푎2 + 4푏2 3√푎2 + 4푏2 5√푎2 + 4푏2

1 √2푟0푎푏 + 휋푐5푐1푐3Erf [ ] (3.51) 2 √푎2 + 4푏2

∞ 2 −푄2 −푄2 푄푟0 푇 = ∫ 푐 (푐 − 푐 푄2)2 (푐표푠 ( )) e8푏2 e2푎2 푑푄 3 5 3 4 2 0

2 2 2 −2r0a b c √2πabe a2+4b2 3 = [ 5 ] [ c2a8b4 + 8c2b2a6 + 48c2b4a4 + 128c2b6a2 + 3c2a6b6 9 8 4 3 3 3 4 (a2 + 4b2)2

1 +6c2a4b8 − 3c2a8b6r2 − 12c2a6b8r2 − c c a8b2 − 6c c a6b4 − 24c c a4b6 4 4 0 4 0 2 4 3 4 3 4 3 1 −32c c a2b8 + 2c4r4a8b8 + 16c c r2a6b6 + 32c c r2a4b8 + 128c2b8 + c2a8] 4 3 4 0 4 3 0 4 3 0 3 2 3

푐 √2휋 1 + [ 5 ] [ 푐2푎13푏 + 1920푐2푎5푏9 + 3072푐2푎3푏11 + 2048푐2푎푏13 13 2 3 3 3 3 (푎2 + 4푏2) 2 2 7 7 2 11 3 2 9 5 2 13 5 2 11 7 2 7 11 +64푐3 푎 푏 + 12푐3 푎 푏 + 120푐3 푎 푏 + 0.375푐4 푎 푏 + 6푐4 푎 푏 + 96푐4 푎 푏 2 5 13 3 13 11 5 9 7 7 9 +96푐4 푎 푏 − 512푐4푐3푎 푏 − 10푐4푐3푎 푏 − 80푐4푐3푎 푏 − 320푐4푐3푎 푏

5 11 1 13 3 −640푐4푐3푎 푏 − 푐4푐3푎 푏 ] 2 (3.52)

The resultant of (3.50), (3.51), and (3.52) gives rise to the energy of phonons which can be written as

퐸푝ℎ = 푇1 + 푇2 + 푇3 (3.53)

Up to now, we have executed Equation (3.30) analytically, the obtaining of the variational parameters 푎 and 푏 have done numerically. As that the minimization of the variational energy with respect to these parameters provides an upper bound of the ground state energy of bipolaron as a function of the inter-polarons distance, 푟0.

The stability of the bipolaron state is achieved when the energy of the bipolaron system be less than twice the energy of a single polaron system. Defining the binding energy of the problem as the difference between the ground state energy of bipolaron state and the ground state energy of two single polarons separately, that is

퐸푏푖푛 = 2퐸푝 − 퐸푏푝 (3.54)

Where 퐸푏푝 is the ground state energy of the bipolaron problem given by Equation (3.30), and 퐸푝 is the ground state energy for a single polaron derived in the previous chapter 2 and given by Equation (2.32). From our definition of Equation (3.54), it is clear now that the condition to have a stable bipolaronic state is that the binding energy 퐸푏푖푛 should be positive. The function 휂푐(훼, Ω) describing the boundary of the bipolaron stability region, is found from the equation

퐸푏푖푛 = 0 (3.55)

3.2 Results and Discussions

The degree of confinement Ω is expected to add another contribution to the problem in addition to the parameters , , and 푟0. Its contribution affects the problem in an efficient way due to its possible role in determining which of the forces will dominate over the other: increasing  means an increase in the repulsive interaction between the two electrons and this, in turn, results in an increase in the energy of the system. On the other hand, the large values of 훼 mean strong attractive interaction between the two electrons via the polaronic coupling and this leads to a decrease in the energy of the system and this, in turn, provides the bipolaron binding energy to a certain extent. If the Coulomb force is so strong such that it dominates over the polaronic force, which holds the two electrons together, the bipolaronic state is likely to break down. In this context, the degree of confinement of the potential, represented by the parameter  is also expected to affect both the Coulomb potential and the polaronic coupling. This because increasing the confinement on the problem put further squeezing in the two electrons making them respond to the lattice deformation in a different complex way. The average distance between the two electrons 푟0 plays a complementary role with all the other parameters and its value will depend largely on them in a complex way. There is a certain range of 푟0 that giving a stable bipolaron while for other values the system of bipolaron collapse into two separated polarons. It was found that for the bipolaron state to be stable the ratio between

the Coulomb amplitude and the polaronic constant must be less than a critical value. This critical value is found to be dependent largely on the degree of confinement. According to this condition, the energy of the bipolaron is expected to be lower than the sum of the energies of each of the two polarons separately.

In Figure (3.1), we plot the binding energy of the bipolaron as a function of the degree of confinement  for two different values of the coupling constant (=5 and =10) and a given value of 휂. The figure reveals that the effect of the polaronic coupling is less pronounced for small  and the binding energy decreases with increasing  as it should be expected since the degree of confinement puts an additional contribution to the confinement of the two electrons and so makes the polaronic effect more

important.

nergy

BindingE

The

Ω

Figure (3.1): The binding energy of the bipolaron versus the degree of confinement Ω at two selected values of the coupling constant α for 휂 = 0.1.

To study in more details the - dependence of the problem we, in Figure (3.2), plot the binding energy versus  for two distinct values of  and a given value of 휂. The figure shows that the binding energy first increases until attains its maximum value then decreases as  increases. The reason about this feature, in our opinion, is because for small electron-phonon coupling, the dominant force will be the Coulomb interaction force where increasing the degree of confinement makes the two electrons more close and so the Coulomb force increases thus, in total, makes the formation of the bipolaronic state less possible. The value of  at which we have a favorable bipolaronic state is less for large

values of .

nergy nergy

Type equation here. equation Type

(

TheBinding

Figure (3.2): The binding energy of the bipolaron versus the polaronic constant α at two distinct values Ω for 휂 = 0.01.

In this figure, we note that the maximum binding energy occurs when the distance between the two electrons was in the order of the polaron radius, otherwise an increase or

a decrease from this value will decrease the binding energy. This can be explained as follows: when the distance between the two electrons is large in comparison with the polaron radius, such that each electron moves in a separate potential well, the Coulomb repulsion obviously exceeds the phonon-mediated attraction. In the opposite case, when the distance between the two electrons is smaller than the polaron radius, the repulsion also exceeds the attraction since the Coulomb potential diverges.

In Figure (3.3), we plot the binding energy against the degree of confinement for two different values of the dielectric constant  for a given value of 훼. It is obvious from the graph that the binding energy decreases with increasing the degree of confinement. The variation is extremely fast above a certain value of  where above this critical value the bipolaron becomes unstable. The reason for such a behavior is that at such a degree of confinement a bipolaron breaks down into two individual polarons. The increase in the degree of confinement shortens the average distance between the two electrons forming the bipolaron and thus increasing the average Coulomb repulsion between the electrons. When this Coulomb repulsive force reaches a value such that it dominates over the polaronic forces, responsible for the attraction between the two electrons, the formation of stable bipolarons becomes not possible. However, for small values of , the binding energy of the problem does not change much with the degree of confinement and hence the stability of the bipolaron becomes more independent of  due to the dominance of the polaronic force in this limit, which is consistent for the bulk limit. Furthermore, the figure reveals that the binding energy decreases with increasing . The reason behind this feature is a quite reasonable since a decrease in  means that the electron-electron repulsion force is weak and thus the polaronic interaction prevails. Therefore, the formation of a stable bipolaronic state is enhanced by reducing the dielectric constant .

It should be noted that the parameters , , , and 푟0 do not enter the problem independently, but rather in an interrelated way such that each parameter affects the other implicitly. In Figure (3.4), we display this critical value 푐 as a function of the degree of confinement  for distinct polaronic constants . As it is clear from the figure, the values

of 푐decrease with increasing the degree of confinement. This can be explained as follow. The degree of confinement enhances the importance of the polaronic interaction and so this interaction can dominate over the Coulomb repulsive force for fewer values of . The increasing in 훼 makes the value of the critical  to occur at high confinement “small quantum dot size” as one should expect.

nergy nergy

TheBinding

Ω

Figure (3.3): The binding energy of the bipolaron versus the degree of confinement Ω at different values of the Coulomb strength in terms of the relative permittivity 휂 for 훼 =5.

It is clear from Figure (3.5) that an increase in the Coulomb strength 휂 inversely affects the formation of the bipolaron by decreasing the binding energy as well as the chance of bonding between the electrons will be at lower separation distance. This can be justified by the inverse relationship between Coulomb force and the distance between the electrons. As shown in the graph if the distance between the two particles is greater than

0.1 and 0.4 for each curve of 휂 =0.3 and 0.01 respectively, the bipolaron will dissociate into two separated polarons. And this because the attractive interaction between the polarons due to the overlap of their phonon ﬁelds rapidly decreases as the inter-polaron

distance gets larger than the polaron size. In other words, the large values of 푟0 (larger than the polaron radius) weaken the attraction between the two electrons via the phonon field and this indicates that the Coulomb repulsive force is great pushing the two electrons apart.

휼풄

Ω

Figure (3.4): The critical dielectric constant as a function of the confinement parameter Ω for different values of phonon-electron interaction constant 훼.

In Table (3.1), we give explicit numerical values of the binding energy of the “two- polaron” system compared with that of inﬁnitely separated non-interacting polarons. We

found that at 휂 = 휂푐 and Ω = 0 the ground state energy of the bipolaron is proportional to 훼2 according to the formula

2 퐸 ≅ −0.2121훼2 = − 훼2 푏푝 3휋 (3.56)

This value is the same as obtained in the literature (Senger, Kozal, Chatterjee and Erc¸elebi, 2010). In view of the table and by referring to Equation (2.36) in the previous chapter we find that the local minimum of two separated polarons is twice of the value that obtained for a single polaron. Equation (3.56) tells that in order to achieve the state 2 of bipolaron, the ratio between 퐸푏푝 and 훼 must be less than−0.2121 and this occurs when

휂 < 휂푐 as we mentioned previously.

nergy

TheBinding

풓ퟎ

Figure (3.5): The binding energy of the bipolaron as a function of the inter-polaron distance at two distinct values of the dielectric constant 휂 for 훼 =5 and Ω =1.

훼 7 9 10 15 20 퐸푝 -5.1991 -8.5943 -10.6103 -23.8732 -42.4413 2 퐸푝⁄훼 -0.1061 -0.1061 -0.1061 -0.1061 -0.1061 퐸푏푝 -10.5182 -17.1731 -21.55 -47.7536 -84.8386 2 퐸푏푝⁄훼 -0.2146 -0.21201 -0.2115 -0.2122 -0.2121

Table (3.1): The ground state energy of the two polarons compared with twice that of

a single polaron in three-dimensional configuration (0) and 휂 = 휂푐.

Conclusions

In this thesis, we have reviewed the problem of polaron in a quantum dot confining using the strong coupling approximation. Even though the problem is well known where it was investigated in details in the literature using different approaches, we displayed the problem in chapter 2 to be the reference for the main theme of the work, the bipolaron.

The main finding in this chapter is the effect of the degree of confinement on the problem. It is found that the polaronic effect becomes more pronounced by increasing the confinement parameter. This because, in our opinion, the increase of the confinement on the electron makes it move more slowly, and hence responding to the lattice distortion in a more effective way. Also, the effect of the degree of confinement is more efficient for large values of the coupling constant . The binding energy of the polaron is found to increase with increasing the degree of confinement and this dependence becomes more significant as  increases. This ensures the fact that the two parameters , and the degree of confinement  do not enter the problem independently, but rather in an interrelated manner.

The problem of the bipolaron is addressed in chapter 3 where the stability of the bipolaronic state is studied as a function of all the parameters controlling the problem, namely, the electron-phonon coupling constant , the Coulomb field strength , the degree of confinement , and the mean distance between the two electrons forming the bipolaron 푟0. The criterion for which a stable bipolaron state takes place when the ground state energy of the pair of the composite polarons making up the bipolaron be lower than twice the energy of single polaron. It was found that for the bipolaron state to be stable the ratio between the Coulomb amplitude and the polaronic constant must be less than a critical value. This critical value is expected to depend largely on the degree of confinement due to the argument we have already discussed before. According to this

condition, the energy of the bipolaron is expected to be lower than the sum of the energies of each of the two polarons separately.

We conclude that the parameters 휂, 훼, and Ω which characterize the system do not enter the problem in an independent way but all together take part in the binding energy in connected and somewhat involved manners, opposing the effect of one against the other. The extra parameter Ω adds more to the complexity in the balance between the two forces, Coulomb force and the attractive force via the phonon field, of the problem. A decrease in 휂 would lead to an enhancement in the bipolaron binding energy favouring the formation of stable bipolarons, and the large values of 훼 mean strong attractive interaction between the two electrons via the polaronic coupling and this also enhances the bipolaronic effect; this means that the bipolaron binding energy is an increasing function of α and a decreasing function of 휂.

One of the main results we have in this work is that the attractive interaction between the polarons due to the overlap of their phonon ﬁelds rapidly decreases as the inter-polaron distance gets larger than the polaron size. In another word, when the distance between the two electrons is large in comparison with the polaron radius, such that each electron moves in a separate potential well, the Coulomb repulsion obviously exceeds the phonon-mediated attraction. In the opposite case, when the distance between the two electrons is smaller than the polaron radius, the repulsion also exceeds the attraction since the Coulomb potential diverges.

Suggestions for Future Studies

The exact dependence of the degree of confinement on the problem and to what extent it will affect the other parameters is still to be answered and left as an open question to be addressed in future studies.

We also suggest for future works to study the problem of two electrons immersed in longitudinal optical (LO) phonons and confined in symmetric or asymmetric quantum dots by parabolic potential box under the influence of an external magnetic field, this will add another confinement to the pair of electrons and so will affect the stability of bipolaron in an efficient way.

References

References:

Adolphs, C. P. J., & Berciu, M. (2014). Strongly bound yet light bipolarons for double- well electron-phonon coupling. Physical Review B, 90, 085149.

Akson Russian Science Communication Association. (2018). Theory gives free rein to superconductivity at room temperature. Russia: Advances in Condensed Matter Physics.

Alexandrov, A. S., & Devreese, J. T. (2009). Superconductivity; Strongly Correlated Electrons, Fr¨ohlich polaron and bipolaron: recent developments. Reports in Progress in Physics, 72, 131.

Balandin, A., Pokatilov, E. P., & Nika, D. L. (2007). Phonon Engineering in Hetero- and Nanostructures. Journal of Nanoelectronics and Optoelectronics, 2, 140–170.

Clemens, P. J., & Berciu, M. (2017). Strongly bound yet light bipolarons for double-well electron-phonon coupling. Quantum Matter Institute. Canada.

Devreese, J. T. (1996). Statistical Mechanics: Strongly Correlated Electrons. Encyclopedia of Applied Physics, 14, 383 – 409.

Devreese, J. T. (2018). Fr¨ohlich Polarons: Lecture course including detailed theoretical derivations. (8th Ed.). Antwerpen, Belgium. 337.

Emin, D. (2015). In-plane conductivity of a layered large-bipolaron liquid. Philosophical Magazine, 95, 918-934.

Emin, D. (2017, 28 January – 2 February). Large bipolarons, and oxide superconductivity. Paper presented at OPTO, the optoelectronics, photonic materials and devices Conference, San Francisco, California, United States.

Erc¸elebi, A., & Senger, R. T. (1996). Ground-state description of quasi-one-dimensional polarons with arbitrary electron-phonon coupling strength. Physical Review B, 53, 11008.

Erc¸elebi, A., & Senger, R. T. (2002). On the stability of Fr¨ohlich bipolarons in spherical quantum dots. Journal of Physics: Condensed Matter, 14, 5549–5560.

Fai, L. C., Teboul, V., Monteil, A., Nsangou, I., & Maabou, S. (2004). Polaron in cylindrical and spherical quantum dots. Condensed Matter Physics, 7, 157–166.

House, J. E. (2018). Fundamentals of Quantum Mechanics. (3rd Ed.). USA: Elsevier Inc.372.

Holstein, T. (1959). Studies of polaron motion: Part II. The "small" polaron. Annals of Physics, 8, 343-389.

Houtepen, A. J. (2007). Charge injection and transport in quantum confined and disordered systems. Universiteit Utrecht, Holland.

Lakhno, V. (2016). Cooper pairs and bipolarons. Modern Physics Letters B, 30.

Landau, L. D., & Lifshitz, E.M. (1965). Quantum Mechanics: Non-relativistic theory. Oxford: Pergamum. Addison Wesley Publishing Company Inc. 3.

Miyao, T., & Spohn, H. (2007). The bipolaron in the strong coupling limit. Annales Henri Poincaré, 8, 1333–1370.

Nina, B. (2018). Theoretical studies of lattice- and spin-polarons. Acta Universitatis

Upsaliensis, Uppsala, Sweden. 103.

Omar, M. A. (1993). Elementary Solid State Physics: Principles and Applications. USA: Addison Wesley Publishing Company.

Pavarini, E., Koch, E., & Scalettar, R. (2017). The Physics of Correlated Insulators, Metals, and Superconductors. Schriften des Forschungszentrums Jülich Reihe Modeling and Simulation: Institute for Solid State Physics. Retrieved March 12, 2019, from: https://www.cond-mat.de/events/correl17/.

Senger, R. T. (2000). Phonon mediated electron-electron interaction in confined media: low dimensional bipolarons (Unpublished Doctoral Thesis). Bilkent University, Ankara, Turkey. 102-111.

Senger, R. T., Kozal, B., Chatterjee, A., & Erc¸elebi, A. (2010). Hartree-Fock approximation of bipolaron state in quantum dots and wires. The European Physical Journal B, 78, 525-529.

Steven, J.F. (2008). Basic theory and phenomenology of polarons. Retrieved March 27, 2019, from: http://www.researchgate.net.

Stringari, S. Ketterle, W., & Inguscio, M. (2016). Quantum Matter at Ultralow

Temperatures. Italy: Societa Italiana di Fisica. 191, 63-141.

Verbist, G., Smondyrev, M. A., Peeters, F. M., & Devreese J.T. (1991). Strong-coupling analysis of large bipolarons in two and three dimensions. Phys. Rev. B, 45, 5262.