الجـامعــــــــــة اﻹســـــﻻميــة 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 البوﻻرون 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 البوﻻرون الثنائي في نقاط كمية كروية أقر بأن ما اشتملت عليه هذه الرسالة إنما هو نتاج جهدي الخاص، باستثناء ما تمت اﻹشارة إليه حيثما ورد، وأن هذه الرسالة ككل أو أي جزء منها لم يقدم من قبل اﻻخرين 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: I 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 البوﻻرون في النقاط الكمية الكروية باستخدام نظرية اﻻقتران القوي. لقد وجد أن طاقة البوﻻ رون تزداد كلما زادت درجة الحصر. ووجد أن تأثير تفاعل اﻻلكترون مع الفونونات يزداد كلما زادت درجة الحصر. لقد درسنا أي ًضا مشكلة تكون البوﻻرون الثنائي في نفس جهد الحصر باستخدام طريقة ﻻنداو-بيكار التباينية في حدود اﻻقت ارن القوي بين اﻻلكترون والفونونات. أعطينا اهتمام خاص ﻻستقرار تكون البوﻻرون الثنائي كدالة في درجة الحصر ومتوسط المسافة بين اﻹلكترونين. درجة الحصر، ممثلة في المعامل Ω , وجدت أنها تضيف مساهمة أخرى إلى المسألة باﻹضافة إلى العوامل اﻷخرى . إن درجة الحصر تؤثر في المسألة بطريقة فعالة نظًار لدورها المحتمل في تحديد أي من القوتين، قوة كولوم التنافرية بين اﻻلكترونيين، أو قوة التجاذب بين اﻻلكترون والفونونات، ستسود على اﻷخرى . أي من هاتين القوتين سوف تتأثر أكثر بسبب درجة الحصر ﻻ يمكن اﻹجابة عليها بسهولة بسبب اﻻعتماد المتداخل لجميع المعامﻻت التي تؤثر على المسألة. وقد وجد أنه لكي تكون حالة البوﻻرون 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 البوﻻرون( تضعف تجاذب اﻻلكترونين مع مجال الفونونات. IV Dedication To 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. V 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. VI 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 IX 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