Whispering Gallery Modes of Electrons In Quantum Dots

Shahab Ramezanpour

Whispering Gallery Modes of Electrons In Quantum Dots

This thesis shows that the dimension of the QD including its height has a critical effect on the QD's energy spectra, therefore it cannot be considered as a 2D harmonic oscillator. The Coulomb interaction is comparable to the energy of interband transition which cause intermixing energy bands even for zero magnetic field. It shows resonances along the height of the lateral dot which are quite unusual, and studies perturbation and topological effects on surface states in these dots and also in spherical and triangular ones.

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Вуз: Санкт-Петербургский национальный исследовательский университет информационных технологий, механики и оптики (ФГБОУ ВПО «СПбНИУ ИТМО»)

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Министерство науки и высшего образования Российской Федерации ФЕДЕРАЛЬНОЕ ГОСУДАРСТВЕННОЕ АВТОНОМНОЕ ОБРАЗОВАТЕЛЬНОЕ УЧРЕЖДЕНИЕ ВЫСШЕГО ОБРАЗОВАНИЯ “САНКТ-ПЕТЕРБУРГСКИЙ НАЦИОНАЛЬНЫЙ ИССЛЕДОВАТЕЛЬСКИЙ УНИВЕРСИТЕТ ИНФОРМАЦИОННЫХ ТЕХНОЛОГИЙ, МЕХАНИКИ И ОПТИКИ” ВЫПУСКНАЯ КВАЛИФИКАЦИОННАЯ РАБОТА МОДЫ ШЕПЧУЩЕЙ ГАЛЕРЕИ ЭЛЕКТРОНОВ В КВАНТОВОЙ ТОЧКЕ Автор Рамезанпур Шахаб _______________ (Фамилия, Имя, Отчество) Направление подготовки (специальность) (Подпись) 12.04.03 (код, наименование) Фотоника и оптоинформатика Квалификация магистр (бакалавр, магистр) Руководитель ВКР Богданов А.А. к.ф.-м.н. (Фамилия, И., О., ученое звание, степень) ______________ (Подпись) К защите допустить Руководитель ОП Белов П.А., д.ф.-м.н. _____________ (Фамилия, И.О., ученое звание, степень) (Подпись) ―_____‖__________________ 20 ____г. Санкт-Петербург, 2019 г.

Студент Рамезанпур Ш. Группа Z4240 Факультет ФТФ (Фамилия, И. О.) Направленность (профиль), специализация «Метаматериалы» _______________________________________________________________________________ Консультант (ы): а) ____________________________________________________________ _____________ (Фамилия, И., О., ученое звание, степень) (Подпись) б) ______________________________________________________________ _____________ (Фамилия, И., О., ученое звание, степень) (Подпись) ВКР принята ―____‖________________________20 ____г. Оригинальность ВКР ______________% ВКР выполнена с оценкой _______________________________ Дата защиты ―____‖________________________20 ____г. Секретарь ГЭК ______________________________________________ __________________ (ФИО) (подпись) Листов хранения ___________________________________ Демонстрационных материалов/Чертежей хранения _________________________________

Ministry of Science and Higher Education FEDERAL STATE AUTONOMOUS EDUCATIONAL INSTITUTION OF HIGHER PROFESSIONAL EDUCATION “Saint Petersburg State University of Information Technologies, Mechanics and Optics” GRADUATION THESIS WHISPERING GALLERY MODES OF ELECTRONS IN QUANTUM DOT Author Ramezanpour Shahab _______________ (full name) Subject area (signature) 12.04.03 Photonics and optoinformatics (code, name of program track) ____________________________________________________________________________ Degree level master (Bachelor, Master) Thesis supervisor Bogdanov A.A., PhD (surname, initials, academic title, degree) _____________ (signature) Approved for defense Head of program Belov P.A., PhD, D.Sc _____________ (surname, initials, academic title, degree) (signature) ―_____‖__________________ 20 ____ St. Petersburg, 2019

Student Ramezanpour Sh. Group Z4240 Faculty PhE (Surname, initials) Subject area, program/major «Metatmaterials» _______________________________________________________________________________ Consultant(s): a) ____________________________________________________________ _____________ (surname, initials, academic title, degree) (signature) b) ______________________________________________________________ _____________ (surname, initials, academic title, degree) (signature) Thesis received ―____‖________________________20 ____ Originality of thesis: ______________% Thesis completed with the grade: _______________________________ Date of defense ―____‖________________________20 ____ Secretary of State Exam Commission_________________________ (ФИО) __________________ (подпись) Number of pages ___________________________________ Number of supplementary materials/Blueprints _________________________________

6 Introduction----------------------------------------------------------------------------------- 8 1 Overview of the research field------------------------------------------------------- 10 1.1 Energy Spectra of a few Electrons Lateral Quantum Dot------------------ 10 1.1.1 Shell Filling and Spin Effect------------------------------------------- 10 1.1.2 Direct Coulomb and Exchange Interaction--------------------------- 12 1.1.3 Capacitance-Voltage Traces of InAs Dot----------------------------- 13 1.1.4 Perturbation Approach for the Coulomb interactions between electrons------------------------------------------------------------------- 15 1.2 Creating Whispering Gallery Modes------------------------------------------- 16 1.2.1 Graphene------------------------------------------------------------------- 16 1.2.2 Oligothiophene nano-rings--------------------------------------------- 18 1.3 WGM as Superpersistent Current---------------------------------------------- 19 1.4 Fabrication of Lateral Quantum Dot------------------------------------------- 20 1.5 Wigner Localization and Conductance Anomalies------------------------- 21 2 Theoretical approach------------------------------------------------------------------- 23 2.1 Schrodinger equation in effective mass approximation------------------- --- 23 2.2 Evaluating the Effect of Coulomb energy in energy spectra---------------- 24 2.3 Spin effect of the electrons in Coulomb interaction-------------------------- 25 2.4 Fock-operator---------------------------------------------------------------------- 27 3 Methods and Results-------------------------------------------------------------------- 30 3.1 4 meV QD-------------------------------------------------------------------------- 30 3.1.1 Energy Spectra------------------------------------------------------------ 30 3.1.2 Wavefunction------------------------------------------------------------- 31 3.1.3 WGMs--------------------------------------------------------------------- 31 3.2 2 meV QD-------------------------------------------------------------------------- 32 3.2.1 Energy Spectra and Wavefunctions----------------------------------- 32 3.3 Effect of Coulomb Interaction-------------------------------------------------- 34 3.4 Variational Method (Hartree-Fock approach)---------------------------------- 37

7 3.5 Magnetic Field Effect-------------------------------------------------------------- 41 3.6. Experimental Result--------------------------------------------------------------- 44 3.7 Surface Modes in Lateral QD--------------------------------------------------- 45 3.7.1 WGMs--------------------------------------------------------------------- 45 3.7.2 Resonance across the Height-------------------------------------------- 46 3.8 Perturbation------------------------------------------------------------------------- 48 3.8.1 Lateral QD----------------------------------------------------------------- 48 3.8.2 Spherical QD--------------------------------------------------------------- 49 3.8.3 Triangular QD------------------------------------------------------------- 50 Conclusion------------------------------------------------------------------------------------- 52 References------------------------------------------------------------------------------------- 53

8 Introduction Semiconductor Quantum Dot (QD) can be modeled as an artificial atom, and reveal discrete energy levels similar to an atom. It is shown in refs. [1-2] that by measuring current versus gate voltage, a QD has similar properties of atoms, as filling the shell structures in a 2D harmonic oscillator. It is discussed that electrons tend to fill the shell with parallel spin (ferromagnetic filling), according to the Hund's law. However, by applying a magnetic field, degenerate energy levels with plus/minus angular momentum would be split (Zeeman effect), which can change ground states of the QD. Besides, by introducing direct and exchange coulomb interaction, filling of the shells can be antiparallel (antiferromagnetic). It is shown in ref. [3] that energy levels of InAs QD are as s -, p - and d -shell by using high-resolution capacitance spectroscopy and detecting maximums of the capacitance. Furthermore, it shows that applying magnetic field causes intermixing of p- and d -shells. In order to model interband spectroscopy, in ref. [4], Coulomb interaction between electrons and also electrons and holes are calculated and treated as a perturbation, where confining potential is considered parabolic and two-dimensional. Quantum Dots can support states with high angular momentum. These states behave as Whispering Gallery Modes (WGMs) in optical resonator. The distribution of wavefunction associated to WGM is confined to the edges of QD, which can have applications in electronic lenses and resonators. It is shown in ref. [5] that WGM can be created in Graphene by an induction from scanning tunneling probe. The size of the circular trap can be tuned by the voltage of the back-gated Graphene device. It is shown in ref. [6] that WGM can be created in Oligothiophene nano-rings which can act as electronic resonator. It is shown in ref. [7] that WGMs can be used as a superpersistent currents in Dirac materials which has application in Qubits and Topological Insulators (TI) [8]. In a few electron QD, electrons can create a crystalization called Wigner Molecules (WM). In these dots, excited electrons can reveal different modes including WGM. Structural and optical properties of InP/GaInP QDs are studied in ref. [9], while in [1011], it is shown that WGMs can be in the resonance of Wigner Molecules (WMs) ground state. Furthermore, zero-bias conductance anomalies (having conductance at zeo

9 bias voltage) in Quantum Point Contact (QPC) is explained by Wigner localization and alternating equilibrium and non-equilibrium of Kondo screening of different spin states [12-13]. Therefore, evaluating the energy spectra of the QD is challenging, since many parameters can affect it such as Coulomb interaction between the electrons, shape of the QD, magnetic field, etc. Furthermore, literature usually consider lateral QD with a zero thickness, however, our calculation reveals that even very small thickness of the QD can have significant effect on the energy spectra. Besides, due to the application of some of the modes in the QD such as lasing, lenses, electronic resonator, ..., we study WGMs in the lateral QD and its energy levels. We evaluate the energy spectra of QD with both perturbation approach and variational method. Although, perturbation approach is simpler, it suits well for our structure, since we consider the QD which contains a few amount of electrons in the ground state and one or two electrons in the excited state. Although, the variational method is much more time-consuming, it takes into account the effect of the wavefunctions on each other. We could attain almost the same result from these two methods. This thesis shows that the dimension of the QD including its height has a critical effect on the QD's energy spectra, therefore, it cannot be considered as a 2D harmonic oscillator. The Coulomb interaction is comparable to the energy of interband transition which cause intermixing energy bands even for zero magnetic field. It shows resonances along the height of the lateral dot which are quite unusual, and studies perturbation and topological effects on surface states in these dots and also in spherical and triangular ones.

10 1 OVERVIEW OF THE RESEARCH FIELD 1.1 Energy Spectra of a Few Electrons Lateral Quantum Dot 1.1.1 Shell Filling and Spin Effect Reference [1] represents that adding an electron to a semiconductor quantum dot, ―addition energy‖ is required which is similar to the real atom. However, this addition energy is greater than the interband energy, due to the coulomb interaction between the electrons. Vertical quantum dots are like a disk with a diameter around 10 times of its thickness. They can be modeled by two dimensional (2D) harmonic oscillators, since their lateral potential can be considered to have a cylindrical symmetry with soft walls. The artificial shells can be filled completely by the number of electrons 2, 6, 12, ..., which are considered as magic numbers. The addition energy is usually larger when the electron numbers are equal to the magic numbers. To study the magnetic field dependence, at a sufficiently small magnetic field (B< 0.4 T), spin filling obeys Hund‘s rule, while at higher magnetic fields (B> 0.4 T), the filling of states are with successive spin-up and spin-down. Figure 1(a) shows the current at drain voltage V  150 V as a function of gate voltage Vg for a dot with diameter D  0.5 m , where the picks are related to adding one electron to the dot. Figure 1(b) depicts the addition energy versus the electrons number N, for two different devices.

11 Figure 1 - (a) current versus gate voltage at B  0 T for a D  0.5  m dot. (b) Addition energy versus electron number for two different dots with D  0.5 m and 0.44 m [1] The energy spectrum in a B field for a 2D harmonic oscillator can be obtained analytically as: 1 1 En,l  (2n | | 1) ( c2  02 )1/2  4 2 c with a radial quantum number n ( 0,1, 2, ...) and angular momentum quantum number ( 0, 1, 2,...) , while 0 is the electrostatic confinement energy and c is the cyclotron energy. Figure. 2(a) shows En,l versus B calculated for 0  3meV . A singleparticle state with a positive angular momentum shifts lower while the state with a negative one shifts higher energies, respectively, with increasing B. Figure. 2(b) shows the B-field dependence of the fifth, sixth, and seventh current peaks. From this figure, one can observe that the fifth and sixth peaks form a pair. At 1.3 T , the maximum and minimum of the sixth and seventh peaks, respectively, can be attributed to the crossing of the third and fourth energy curves at 1.3 T in Fig. 2(a).

12 Figure 2 - (a) Calculated single-particle energy versus magnetic field for a parabolic potential with 0  3meV . (b) Evolution of the fifth, sixth, and seventh current peaks with magnetic field for D  0.5m dot [1] 1.1.2 Direct Coulomb and Exchange Interaction [2] By applying magnetic field, ref. [2] shows that spin-configurations can be explained in terms of two-electron singlet and triplet states. Figure. 3(b) shows two, spin-degenerate single-particle states with energies Ea and Eb crossing each other at B = B0. For two electrons we can distinguish four possible configurations with either total spin S = 0 (spin-singlet) or S = 1 (spin-triplet). The corresponding energies, Ui(2, S) for i = 1 to 4, are given by: U1(2, 0) = 2Ea + Caa (two elctrons in Ea state with different spins),U2(2, 0) = 2Eb + Cbb (two electrons in Eb state with different spins), U3(2, 1) = Ea+Eb+Cab−|Kab| (one electron in Ea and another in Eb

13 state with same spin), U4(2, 0) = Ea +Eb+Cab (one electron in Ea and another in Eb state with different spin), where Cij and Kij are direct and exchange Coulomb interaction, respectively. Figure 3 - (a) Scanning electron micrograph of the semiconductor quantum dot device. (b) Schematic diagram of two single particle states with energies Ea and Eb crossing each other at a magnetic field B = B0 [2] 1.1.3 Capacitance-Voltage Traces of InAs Dot Figure 4 shows the essential layer sequence and a sketch of the conduction band edge [3]: Figure 4 - (a) Layer sequence of the devices. The InAs dots are distributed within the plane sandwiched between two GaAs layers. (b) Sketch of the conduction-band edge Ec with respect to the Fermi level EF along the growth direction for gate voltages at which no electrons are in the InAs dots. The indicated distances define the lever arm according to ttot /tb (in our case equal 7) which converts voltage into energy differences

14 Figure 5(a) shows the capacitance versus gate voltage for different applied magnetic fields between 0 and 23 T. It shows two degenerate maxima and four degenerate maxima, which can be attributed to the s and p shells, respectively. Figure 5(b) shows the dependence of these maximas to the magnetic field, which depicts that the maximas related to the s shell is almost unaffected (because of zero angular momentum), while two of the maximas related to the p shell decrease (because of positive angular momentum) and two of them increase (due to the negative angular momentum). Figure 5 - (a) Differential capacitance of the layered structure at different magnetic fields (b) Magnetic-field dependence of each maxima [3]

15 1.1.4 Perturbation Approach for the Coulomb interactions between electrons In magnetic field, single particle energies have the following energy levels: ' s, 1 c 2 1 p  2  ' c 2 d   3  ' c p  2  ' d0  3  ' (1) d   3  ' c where  '   2  c2 / 4 (2) is effective frequency and c  eB / m * is cyclotron frequency. For instance the s-state wavefunction is  se  1  le exp(r 2 / 2le2 ) (3) where le is effective length of electrons le  m* (4) while for holes effetive length is: lh  m h * h (5)

16 On the other hand, direct and exchange Coulomb interaction can be obtained from Eqs. (6) and (7), respectively: Eij  Eij  e2 4 0 r e2 4 0 r   | ie (r1 ) |2 |  ej (r2 ) |2 | r1  r2 | dr1dr2  ie (r1 )* ej (r2 )* ie (r2 ) ej (r1 ) | r1  r2 | (6) dr1dr2 (7) For example, direct Coulomb interaction between two electrons in s-state is calculated: Eij  e2  1 4 0 r 2 le (8) The ground state energy for N electrons can then be obtained from: EN  E sp ( N )  E c ( N )   1 NeVg (9) where Esp(N) is the sum of the single-particle energies, EC(N) the matrix element of the Coulomb interaction, and -λ-1NeVg expresses the electrostatic energy due to the electric field between gate and back contact. 1.2 Creating Whispering Gallery Modes 1.2.1 Graphene In [5], Whispering Gallery Mode (WGM) is created in Graphene by creating pn junction, induced by a scanning tunneling probe (Fig. 6). The size of the resonator can be tuned by back-gated graphene device. It demonstrates an entirely different approach, inspired by the peculiar acoustic phenomena in whispering galleries. This type of resonators can be used in quantum electron-optics such as electronic lenses and resonators. Therefore, scanning tunneling microscopy (STM) probe is utilized for both probing electronic states and also creating pn junction which serves as confining potential for electrons. Two types of WGM are detected in the structure (Fig. 7). First type is called WGM‖ which occur in conventional energy states ε𝜈=μ0+𝑒𝑉b. However, the tip bias variation causes the Fermi level beneath the tip to move through system energy levels ε𝜈, which create another type of WGM (WGM‘) at Fermi energy level ε𝜈=μ0.

17 Figure 6 – (A) The rings are induced by the STM tip voltage bias (𝑉b) and back-gate voltage (𝑉g) is adjusted to reverse the carrier polarity beneath the tip relative to the ambient polarity. The cavity radius and the local carrier density are tunable by both Vb and Vg. (B) Spatial profile of WGM resonances. The confinement is stronger for the larger angular momentum m values [5] Figure 7 – (A) Differential tunneling conductance, 𝑑𝐼/𝑑𝑉b, map for a single-layer graphene device as a function of sample bias, 𝑉b and back gate voltage, 𝑉g. (B)

18 Interference features in 𝑑𝐼/𝑑𝑉𝑏 calculated from the relativistic Dirac model. The boundaries of WGM‘ (WGM‖) regions are marked by dashed (dotted) white lines [5] 1.2.2 Oligothiophene Nano-Rings In Whispering gallery Modes (WGMs), such as the dome of St Paul‘s Cathedral in London, waves travel along a curved path. For closed-loop galleries, wave resonances appear when an integer number of wavelengths equals the perimeter of the resonator. To have WGM, the coherence length of the waves must exceed the perimeter of the resonator, and the walls must efficiently reflect the waves. Reference [6] creates WGM in Oligothiophene nano-rings as shown in Fig. 8. This figure also contains wire topology excitement, which has application in atomic wire. Figure 8 – (Color online) (a) Topographic STM image (I = 100 pA and Vsample = 0.1 V, 6.6 × 2.4 nm2) and (b) constant height differential conductance spectra (set point: I = 5 pA and Vsample = 1 V) of a linear-[12]-thiophene. The grey and black spectra correspond to two positions of the tip on top of the wire (see grey and black arrows in (a)). Green lines are Gaussian function fits. (c) to (f) are constant height conductance maps acquired

19 at voltages corresponding to maxima in (b). The same data were acquired for a cyclo[12]-thiophene: (g) topographic STM image (2.8 × 2.8 nm2), (h) constant height differential conductance spectra acquired on top of the wire, and (i) to (k) constant height conductance maps acquired at voltages corresponding to maxima in (h) 1.3 WGM as Superpersistent Current In the presence of random scatterings, e.g., due to classical chaos, Persistent Currents (PCs), one of the most intriguing manifestations of the Aharonov-Bohm (AB) effect, vanish for Schrӧdinger particles [7]. However, relativistic Dirac quantum AB rings threaded by a magnetic flux are extremely robust (superpersistent currents (SPCs)). A striking finding is that the SPCs can be attributed to a robust type of relativistic quantum states, i.e., Dirac whispering gallery modes (WGMs) that carry large angular momenta and travel along the boundaries and can potentially be the base for a new class of relativistic qubit systems. By examining the eigenstates, we note that, for low energy levels, the Schrödinger particle is strongly localized throughout the domain, as shown in Figs. 9(a– c), since asymmetry in the domain geometry cause mixing angular momentum states and leading to localization of lower states in the entire domain region and vanishing AB oscillations. However, the Dirac fermion typically travels around the ring‘s boundaries, forming relativistic WGMs that persist under irregular boundary scattering due to chaos and are magnetic flux dependent, as shown in Fig. 9(d–f). Figure 9 – Probability distribution of the 10th eigenstate for (a–c) nonrelativistic and (d–f) relativistic AB chaotic billiard, for different chaos degee [7]

20 1.4 Fabrication of Lateral Quantum Dot Due to the vast application of self-organized InP quantum dots (QDs), especially in laser and single photon-source (SPS), they have been widely studied in the recent decades [9]. Self-organized InP quantum dots (QDs) grows on Gax In1x P lattice-matched to GaAs substrates (further denoted as InP/ GaInP QDs). Interestingly, InP/GaInP QDs have been shown to reveal Wigner Molecule (WM) states which make them an ideal candidate to be used in nano-electronics, quantum computing devices. Although, WM states could be observed in InSb nanowires, and carbon nanotubes. To this end, control of the QD‘s optical and structural properties is essential. For laser and SPS applications, the control of the properties of the QDs with sizes 40–70 nm have done, but not for larger InP/GaInP QDs (>100 nm) which is required for the optimization of WM structures. In [9], the QD sample structure is as follows: 500 μm in direction [1 0 0] GaAs substrate misoriented by 2° or 6° towards the [1 1 0] direction, 50 nm thick GaAs buffer layer, 50 nm Gax In1x P latticed matched to the GaAs grown at 725 °C, seven monolayers of InP at 725 °C to form the QDs, and a 60 nm Gax In1x P cap layer grown at either 650 or 725 °C. Representative plan-view Transmission Electron Microscopy (TEM) images of the four samples are shown in figure 10 (a) as well as the lateral size probability density functions (PDFs). The QD lateral sizes range from ~100–200 nm. The PDFs peaked at 140, 160, 100, and 130 nm for samples i, ii, iii and iv, respectively.

21 Figure 10 – Structural data of the samples i, ii, iii and iv: plan-view TEM images with the extracted lateral dot size probability distribution functions—(a) and cross-sectional TEM images and EDX scans—(b). The amount of Gax (Gax In1 x P) for each line scan (along the vertical lines indicated) is shown to the image‘s right [9] 1.5 Wigner Localization and Conductance Anomalies A Quantum Point Contact (QPC) is a constriction in the transverse direction which create a resistance for the electron motion where with applying voltage across the constriction, current can be induced. Hence, the QPC shows quantized conductance for different gate voltage, however, a shoulder like curve appear near the conductance 0.7G0=0.7*2e2/h which cannot be explained by single particle approach. This anamoly is called ‗0.7 anomaly‘ or zero-bias peak called ‗zero-bias anomaly‘ (ZBA). Reference [12] observes repetitive splitting of the zero-bias anaomy by changing the distance of the scanning gate microscope tip, also appearing of 0.7 anomaly, simultaneously (Fig. 11).

22 Figure 11 – Transport measurements. Base temperature is 20 mK. (a) Electron micrograph of the QPC gates. Scale bar, 300 nm. (b) Differential conductance G at zero bias versus split-gate voltage Vgate. The 0.7 anomaly is visible below the first plateau. (c) Differential conductance G versus source-drain bias for different gate voltage Vgate from -1.08 to -0.96 V This behavior is explained as existing of Wigner localization containing charges with different parities, in which spin states in the channel shows alternating equilibrium and nonequilibrium Kondo screenings (Kondo effect can be explained as hybridization of localized electrons to the conduction electrons at low temperature which creates a narrow band gap). This behavior can be interpreted in terms of alternating equilibrium and nonequilibrium Kondo screenings of different spin states localized in the channel. These alternating Kondo effects point towards the presence of a Wigner crystal containing several charges with different parities.

23 2 THEORETICAL APPROACH 2.1 Schrodinger Equation in Effective Mass Approximation Schrodinger equation in effective mass approximation can be written as:  2  1   .( )  V    E me  2  (10) Regarding that, our structure is symmetric in azimuthal (  ) direction, by separation of variables, we can write wavefuntion as:    (r, z)( ) On the other hand, in cylinderical coordinates, the operator .( .( (11) 1 ) is: me 1 1  1   1  1 2 )  ( r ) ( ) me r r me r z me z me r 2  2 (12) By inserting Eqs. (11) and (12) into (10):  where   1  1   1   h 2   2 ( r )  ( )    V   E    2 2 2 r  r m  r  z m  z 8  m r   e e e   h2 8 2 (13) h  . Deividing both sides of this equation by gives: 2 me r 2 me r 2 h2 1  1  1   1   h 2 1  2 2 ( r )  ( )  m r ( V  E )    e 8 2   r r me r z me z  8 2   2 (14) Therefore, we can equate both side of the above equation to a constant like h2 2  2 l which gives two independent equations as: 8 1  2  l 2   2 (15) and me r 2 h2 1  1  1   1   h2 2 2 ( r )  ( )  m r ( V  E )   l   e 8 2   r r me r z me z  8 2 (16) The solution of Eq. (15) is in the form:   exp(il ) (17)

24 where due to the periodic condition (  2 )  ( ) , l , the principle quantum number, should be integer. Furthermore, multiplying both sides of Eq. (16) with  and arranging the me r 2 terms gives:  h2 8 2   1   1   h2  h2 l 2 ( )  ( )   ( V  )  E   z me z  8 2 me r r 8 2 me r 2  r me r (18) which is in the form of a coefficient form partial differential equation (PDE): .(cu  u   )  au   u  dau (19) with: c h2 8 2 me ; r   h2 8 2 me r ; a V  h2 l 2 8 2 me r 2 ; da  1 ;   E (20) while other coefficients are zero. We note that in Eq. (19),   (   , ) is considered. r z 2.2 Evaluating the Effect of Coulomb Energy in Energy Spectra To calculate Coulomb interaction between two electrons, the wavefunctions associated with these two electrons can be written as:  (r1, r2 )  c11 (r1 ) 2 (r2 )  c21 (r2 ) 2 (r1 ) (21) where 1 (r ) and  2 (r ) are wavefunctions of two states, while r1 and r2 are positions of the electrons. The quantity | c1 |2 is the propabibility of the electron with state 1 (r ) being in the position r1 and electron with state  2 (r ) being in the position r2 . The similar definition stands for | c2 |2 . Therefore, c1 and c2 | c1 |2  | c2 |2  1 are equal to  1 , regarding that 2 and | c1 |2 | c2 |2 . Since electrons are fermions, the wavefunction of these two electrons should be antisymmetrized, hence, we choose c1  Therefore, from Eq. (21): 1 1 and c2   . 2 2

25  (r1 , r2 )  1 1  1 (r1 ) 2 (r2 )   1 (r2 ) 2 (r1 ) 2 2 (22) In fact, Eq. (22) is a representation of slater determinant, since this determinant can be 2 2 decomposed to determinants, and for studying two electrons, only the related 2 2 determinant would be kept and other components would be considered zero. On the other hand, the potential energy between two-charge distribution can be calculated from: dV  dq1dq2 4 0 r r12 1 (23) where dq1dq2  e2 | (r1 , r2 ) |2 dv1dv2  e2 (r1 , r2 )* (r1 , r2 )dv1dv2 (24) Therefore, V e2 4 0 r   * (r1 , r2 ) (r1 , r2 ) r12 dv1dv2 (25) Inserting  (r1, r2 ) from Eq. (22), one can obtain: V |  1 (r1 ) |2 |  2 (r2 ) |2  1* (r1 ) 2* (r2 ) 1 (r2 ) 2 (r1 ) dv dv  dv1dv2 1 2  4 0 r  r12 r12 e2 (26) where the first term: Vd  |  1 (r1 ) |2 |  2 ( r2 ) |2 dv1dv2 4 0 r  r12 e2 (27) is called direct coulomb energy and Ve    1* (r1 ) 2* (r2 ) 1 (r2 ) 2 (r1 ) r12 dv1dv2 is exchange coulomb energy. 2.3 Spin effect of the electrons in Coulomb interaction Let's define the wavefunction of two correlated electrons as: (28)

26  (r1, 1, r2 , 2 )   (r1, r2 ) (1, 2 ) (29) where r1 and r2 are spatial parameters and 1 and 2 are spin parameters. The function  (r1, 1, r2 , 2 ) should be antisymmetrized. If the spins of the electrons are same,  (1, 2 ) is symmetrized, therefore,  (r1 , r2 ) should be antisymmetrized (as Eq. 22). Therefore, by inserting this wavefunction in the Coulomb energy (Eq. 25), the exchange part of Coulomb energy would be negative. Vice versa, for different spin,  (r1 , r2 ) should be symmetrized and exchange coulomb interaction is positive. However, according to another approach, we assume the wavefunction of an electron as:  (r, )   (r ) () (30) Let's calculate the notation  i | hˆ |  j  :  i | hˆ |  j    drdi* (r ,  )hˆ j (r ,  ) (31) Inserting Eq. (30) into (31):  i | hˆ |  j    drdi* (r ) i* ( )hˆ j (r ) j ( ) (32) Separating spin function and spatial function gives:  i | hˆ |  j    d i* ( ) j ( )  dri (r )hˆ j (r ) (33) For same spin  i   j  d * i ( ) j ( )  1 (34) and for different spin, the integral is zero. Therefore, to calculate single particle energy, we can write  i | hˆ | i    d i* ( ) i ( )  dri (r )hˆ j (r ) (35)  i | hˆ | i    dri (r )hˆ j (r ) (36) Therefore, For exchange coulomb inetraction, we can write:

27 [  i  j |  j  i ]   dr1d1dr2 d2  i* (r1 , 1 )  j ( r1 , 1 ) 1 *  j ( r2 , 2 )  i ( r2 , 2 ) r12 (37) With separating spin part and spatial parts:   1 [ i  j |  j i ]    d1 i* (1 ) j (1 )    d2 *j (2 ) i (2 )    dr1dr2i* ( r1 ) j ( r1 )  *j (r2 )i (r2 )  r12   (38) Therefore, for different spin  i   j , the integral is zero. However, for direct coulomb interaction, we can write:   1 [ i i |  j  j ]    d1 i* (1 ) i (1 )    d2 *j (2 ) j (2 )    dr1dr2i* ( r1 )i ( r1 )  *j (r2 ) j (r2 )  r12   (39) Regardless of same or different spins, we can write: [  i  i |  j  j ]   dr1dr2i* (r1 )i (r1 ) 1 *  j (r2 ) j (r2 ) r12 (40) According to this approach, for same spin, exchange coulomb energy is negative, and for different spin, it is zero. In our calculations, we have used the second approach. 2.4 Fock-operator The energy associated to an electronic system can be defined as: E HF     i | h |  i   i 1  [ i i |  j  j ]  [ i  j |  j i ] 2 ij (41) where the first, second and third terms are single particle energy, direct coulomb energy and exchange coulomb energy. The coefficient 1/ 2 is due to the fact that in the sum of electron interactions, each electrons has considered twice. However, currently we do not care about coefficients and sign, since later, we can adjust them regarding the number and spin of the electrons in the orbitals. Let's assume wavefunction of orbital i th ( i ) changes a bit as: i  i  i (42) Langragian of this orbital can be defined as: L{i }  E HF {i }    ij ( i |  j    ij ) ij (43)

28 In fact Eq. (43) investigate the orthogonality of the orbitals. The unknown coefficients (Langragian multiplier)  ij can be found by differentiating of Eq. (43) and equating it to zero. The differentiation of the Langragian defined in Eq. (43) is:  L{i }   E HF {i }    ij  i |  j  (44) ij where the differentiation of the term  i |  j  is:   i |  j   i |  j    i |  j  (45) and differentiation of E HF from Eq. (41) is:   E HF   i | hˆ | i    i | hˆ | i  i   1 [i i | i i ]  [ ii | i i ]  [ i i | i i ]  [i i | ii ] 2 ij  1 [i  j |  j i ]  [ i j |  j i ]  [ i  j |  j i ]  [ i  j |  ji ] 2 ij (46) If we consider the integral form of Eq. (46), we can observe that some terms in this equation are equal to each other, and it can written as:   E HF    i | hˆ |  i     i | hˆ |  i  i    [ i  i |  i  i ]  [  i i |  i  i ]   [ i  j |  j  i ]  [  i j |  j  i ] ij ij (47) By inserting Eqs. (45) and (47) into (44), we can write:   L{ i }    i | hˆ |  i     i | hˆ |  i  i    [ i  i |  i  i ]  [  i i |  i  i ]   [ i  j |  j  i ]  [  i j |  j  i ] ij (48) ij   ij ( i |  j     i |  j  ) ij Some terms of Eq. (48) is complex conjugate of each other, therefore, this equation can be written as:

29  L{i }   i | hˆ | i    [i i | i i ]   [i  j |  j  i ]    ij  i |  j   complex.conjugate (49) i ij ij ij We can write Eq. (49) in the integral form as: 1 1 ˆ  h(r1 ) i (r1 )   i (r1 )  dr2  *j (r2 )  j (r2 )    j (r1 )  dr2  *j ( r2 ) i ( r2 )   r12 r12 j j  L{i }    dr1i* (r1 )    c.c  0 i    ij  j (r1 )   j  (50) In order that Eq. (50) being equal to zero, the expression in the bracket should be zero:     1 1 hˆ(r1 ) i (r1 )     dr2  *j (r2 )  j (r2 )  i (r1 )     dr2  *j (r2 ) i (r2 )   j (r1 )    ij  j (r1 ) r12 r12 j  j  j   (51) In Eq. (51), we can define operators Jˆ j and Kˆ j which are related to the direct and exchange coulomb interaction, respectively: 1 Jˆ j  i (r1 )   dr2  *j (r2 )  j (r2 )  i (r1 ) r12 1 Kˆ j  i (r1 )   dr2  *j (r2 )  i (r2 )  j (r1 ) r12 (52) (53) Therfore, Eq. (51) can be written in the operator form as: ˆ   h   Jˆ j   Kˆ j  i (r1 )    ij  j (r1 ) j j j   (54) We choose Langragian multiplier  ij to be diagonal (  ij  0 ; i  j ), therefore, Eq. (54) can be written as: ˆ   h   Jˆ j   Kˆ j  i (r1 )   ii i (r1 ) j j   (55)

30 3 METHODS AND RESULTS 3.1 4 meV QD 3.1.1 Energy Spectra The eigenvalues (eigenenergies) and eigenfunctions of Eq. (19) with coefficients defined in Eq. (20) can be evaluated by Finite Element Method by Comsol software, "Coefficient Form PDE" section. To obtain eigenenergies in eV unit, we can divide h and me in Eq. (20) by the charge of electron e . The Quantum Dot (QD) and surrounding areas are InP and GaAs with effective mass of electrons 0.08me and 0.067me , respectively while the applied potential on the QD is 0 and on the surrounding area is 0.2 eV . Therefore, In Comsol, two PDE are defined, one of them for QD and another one for sorrounding area. Although the height of QD is too small, it should be considered as a 3D structure in order to achieve a more accurate results, since some energy degeneracy would be created due to the non-zero height of the QD. In order to achieve a precise result, a dense meshing is considered for the structure as shown in Fig. 12 (a). Figure 12 (b) shows energy spectrum of a QD for which its dimensions is adjusted in order to achieve 4 meV energy splitting between two lowest energy levels, s and p . Figure 12 - (a) Meshing. (b) Energy Spectrum for QD with 4 meV s-p energy splitting

31 This figure shows some energy degeneracy for each azimuthal number ( m ). 3.1.2 Wavefunction The wavefuncions of the QD (Figure 13) shows that these degeneracy is due to the changing the quantum number associated to the height of the QD ( z -direction). It starts between 5th and 6th eigenvalues and the splitting increases with increasing the azimuthal number. Figure 13 - Eigenfunctions of 4 meV QD for azimuthal numbers m  0 3.1.3 WGMs The probability densities ( | |2 ) of some of the Whispering Gallery Modes (WGMs) is also depicted in Figure 15.

32 Figure 14 - Probability densities of WGMs for m  2 to m  7 3.2 2 meV QD 3.2.1 Energy Spectra and Wavefunctions With changing the size of the QD, its energy spectrum can be manipulated. Figure 15, shows energy spectrum of a QD for which its size is adjusted in order to create 2 meV energy splitting between s & p -states. It shows that energy spectrum is denser comparing to 4 meV QD, and its wavefunctions depicted in Fig. 16 shows that

33 energy degeneracy starts between 4th and 5th eigenvalues (instead of 5th and 6th in 4 meV QD). Figure 15 - Energy spectrum of 2 meV QD

34 Figure 16 - Eigenfunctions of 2meV QD for azimuthal numbers m  0 3.3 Effect of Coulomb Interaction We assume a QD which contain 5 electrons in ground state and also one electorn in the excited state or 4 electrons in the ground state and 2 electrons in the excited state. Therefore, to evaluate Coulomb interaction, different arrangements are considered (Fig. 18). py px wgmy wgmx py s px s (a) (b) wgmy wgmx py px wgmy wgmx py px s s (c) (d) Figure 17 - Different arrangement of 6 electrons in the QD In Fig. 17a, coulomb energy of one of the electron in px orbital is C1   px px    2 px p y  ( px p y )e    2spx  ( spx )e  (56) where px px is direct coulomb interaction between two electrons in px state, which can be evaluated from Eq. (27). The term px py is direct Coulomb interaction between one

35 electron in px state and two electrons in p y state (therefore, it has coefficient 2) and ( px p y )e is exchange Coulomb interaction between these electrons, since the electron under study in px has same spin with one of the electrons in ( px has  dependent as cos(m ) while p y has sin(m ) ). The same condition is considered for the other states. In Fig. 18b, coulomb energy of the electron in wgmx is C2   px wgmx  ( px wgmx )e    2 p y wgmx  ( p y wgmx )e    2swgmx  (swgmx )e  (57) The calculated Coulomb energy difference between Figs. 18a and 18b is C1  C2  5.93 meV . If we consider the arrangemnet of Fig. 18c (excited electron in wgmy ), the Coulomb energy difference between Figs. 18c and 18a is calculated 5.7 meV . For two electron in wgm orbital (Fig. 18d), the coulomb energy difference between Figs. 18d and 18a is calculated 15.15 meV , therefore, we can expect that each of the electrons in wgm has about 15.15 / 2  7.58 meV lower energy than the electron in p orbital. With excitation of an electron, a s - hole would be created in the structure, hence, the effect of exiton energy should also be added in the Coulomb interaction. The s -hole wavefunction is obtained by considereng effective mass of hole m  0.6me . The exiton energy beween s -hole and an electron in px orbital is calculated sh px  7.75 meV while for wgmx (m  2) , it is sh wgmx  6.3 meV , therefore, the value 7.75  6.3  1.45 meV should also be added to the coulomb energy difference between Figs. 18a and 18b. For the other states, the difference of the Coulomb interaction can be calculated, similarly. We assume first eigenstates of azimuthal number m  2, 3, ... as wgm , the second eigenstates of m  0,1, 2,... as xII , and third eigenstates of m  0,1, 2,... as xIII . Figure 19a shows, energy spectra for one electron in the excited state (Fig. 18b) with taking into account the Coulomb interaction between the electrons, while in Fig. 19b exitonic energy is also included.

36 (a) (b) Figure 18 - Energy spectra for one electron in the excited state: (a) without excitonic energy. (b) with exitonic energy Fig. 19 is related to the energy spectra for 2 electrons in the excited state (Fig. 17d).

37 (a) (b) Figure 19 - Energy spectra for 2 electron in the excited state: (a) without exitonic energy. (b) with exitonic energy 3.4 Variational Method (Hartree-Fock Approach) In this section we evaluate the result with variational (Hartree-Fock (HF)) method. Energy of an electron in state i can be calculated from Fock-operator as:

38 ˆ   h   Jˆ j   Kˆ j  i (r1 )   ii i (r1 ) j j   (58) The general concept is that the shape of orbitals deviate a bit from their single particle wavefunctions due to the interaction between the orbitals. Therefore, the new wavefunction can be written as a linear combination of all the orbitals (in order to take into account their interactions) and estimate their new energy and wavefunction by variational method. Equation (58) is an eigenvalue problem which gives wavefuncion and energy of i th orbital. First, we should choose eigenbasis, and write wavefunction of the orbitals as a linear combination of these eigenbasis. It is more convenient to use single particle wavefunctions as the eigenbasis, since they are currently available. If we consider single particle wavefunction, equivalent to atomic orbital and the wavefunction under study i , equivalent to molecular orbital, this method is called Molecular Orbital as a Linear Combination of Atomic Orbital (MO-LCAO) method. For our structure, we assume ground states have the wavefunction as their single particle states, but the excited state, as a linear combination of the single particle wavefunctions. For instance, for WGM with azimuthal number m  2 we can write: WGM  c1s  c2 p y  c3 px  c4 wgmx (59) By inserting Eq. (59), into (58): ˆ   h   Jˆ j   Kˆ j   c1s(r1 )  c2 p y (r1 )  c3 px (r1 )  c4 wgm(r1 )    ii  c1s (r1 )  c2 p y (r1 )  c3 p x (r1 )  c4 wgm(r1 )  j j   (60) where,  ii is the energy of orbital WGM . In this equation j  1, 2,3 is related to the orbitals s, py , px , respectively. The operators J1 and K1 are related to the direct and exchange coulomb interation of orbital s(r ) and WGM (r ) , and we can write:

39 J1 WGM (r1 )   2  dr2 .s (r2 )* s (r2 ) K1 WGM (r1 )     dr2 .s (r2 )* s (r1 ) 1 WGM (r1 ) r12 1 WGM (r2 ) r12 (61) (62) In Eq. (61), the coefficient 2 is due to the direct coulomb effect of 2 electrons in s orbital on WGM orbital. The minus sign in Eq. (62) is due to the one electron in s(r ) orbital which has same spin with the electron in WGM (r ) orbital. For the coulomb operator of p y (r ) orbital on WGM (r ) , we can write: J 2 WGM (r1 )   2  dr2 . p y (r2 )* p y (r2 ) 1 WGM (r1 ) r12 1 Kˆ 2 WGM (r1 )     dr2 . p y (r2 )* p y (r1 ) WGM (r2 ) r12 (63) (64) The coulomb operator of px (r ) orbital on WGM (r ) is: 1 Jˆ3 WGM (r1 )    dr2 . px (r2 )* px (r2 ) WGM (r1 ) r12 K 3 WGM (r1 )     dr2 . px (r2 )* px (r1 ) 1 WGM (r2 ) r12 (65) (66) regarding that there is one electron in px (r ) orbital with the same spin as the electron in WGM (r ) orbital. Since there are four unknown parameters c1, c2 , c3 , c4 , four equations are needed to calculate them. According to Roothan method, first equation can be obtained by multiplying both sides of Eq. (60) by  s(r1 ) | (multiplying it by s* (r1 ) and taking integral with respect to r1 ). Similarly, second, third and fourth equation can be obtained by multiplying it by  p y (r1 ) | ,  px (r1 ) | ,  wgm(r1 ) | , respectively. From these four equations,

40 we can arrange the terms with respect to the unknown coefficients c1, c2 , c3 , c4 , which yields the matrix form equation as:  F11 F  21  F31   F41 F12 F22 F32 F42 F13 F23 F33 F43 F14   c1   S11 S12  S   F24  c2  S 22   ii  21  S31 S32 F34   c3     F44  c4   S 41 S 42 S13 S 23 S33 S 43 S14   c1  S 24   c2  S34   c3    S 44   c4  (67) where due to the large expressions, it is shown parametrically. Equation (67) can be written as: S 1 FC   ii C (68) which is an eigenvalue problem. The eigenvalues and eigenvectors of the matrix S 1 F gives  ii and C , respectively (we note that if we choose the ground states as a linear combination of the eigenbasis either, then the operators would also contain the unknown coefficients C , therefore the problem would be Self-Consistent Field (SCF), since the operators depend on the eigenfunctions, however, our case is not self-consistent). The calculated eigenvalues and eigenvectors are according to: E1  53.6070 ; V1   0.1494, 0, 0.0149,  0.9887  E2  44.6968 ; V2   0.9868, 0, 0.0631, 0.1490  E3  46.0224 ; V3   0.0022, 0,  1, 0.0068,  E4  48.2944 ; V4   0, 1, 0, 0  (69) where we choose the first eigenvalue and eigenvector, since c4 (coefficient of wgmx in Eq. (59)) is near one. It may also be reasonable to investigate five eigenbasis: WGM  c1s  c2 p y  c3 px  c4 wgmx  c5 wgmy in order to observe the difference of energy levels of WGM x and WGM y . (70)

41 The calculated eigenvalues and eigenvectors are as below: E1  44.6968 ; V1   0.9868, 0,  0.0631,  0.1490, 0  E2  46.0224 ; V2   0.0022, 0,  1, 0.0068, 0  E3  53.6070 ; V3   0.1494, 0,  0.0149, 0.9887, 0  E4  53.3365 ; V4   0,  0.0455, 0, 0,  0.999  E5  48.2840 ; V5   0, 0.999, 0, 0,  0.0455  (71) where third and fourth set is associated to the WGM x and WGM y , since c4 and c5 are near one, respectively, which shows a difference of energy about 0.3meV . To evaluate the energy level of Px in the arrangement of the electrons according to the Fig. 18a, we can consider three eigenbasis as: Px  c1s  c2 p y  c3 px (72) which yields E1  47.6672 ; V1   .9948, 0, 0.1019  E2  54.8473 ; V2   0.0775, 0, 0.997  E3  49.3702 ; V3   0,  1, 0  (73) where second set with energy of 54.85 meV is related to the Px which is about 1.5 meV higher than WGM x and 1.3 meV higher than WGM y . This result is corresponding to the previously evaluated energy spectra in Fig. 19(a). 3.5 Magnetic Field Effect Fig. 21 shows eigenergies of QD for 4.1 meV s - p states splitting, where quantum numbers (n, l ) related to the radial and azimuthal number are specified for each of the levels, while Table I shows the corresponding energies for each state.

42 eigenergies (eV) 4,0 3,2 2,4 3,1 2,3 3,0 2,2 2,1 2,0 1,4 0,8 0,7 1,3 0,6 1,2 0,4 1,1 1,0 1,6 1,5 0,1 0,2 0,5 0,3 0,0 azimuthal number (m) Figure 20 - Energy spectra for 4.1meV QD Table 1 - Values corresponding to the Fig. 21 (0,0): 0.019 (0,3): 0.032 (1,3): 0.045 (0,6): 0.049 (4,0): 0.07 (0,1): 0.023 (2,0): 0.039 (0,5): 0.04 (3,1): 0.06 (3,2): 0.068 (1,0): 0.028 (1,2): 0.039 (3,0): 0.052 (2,3): 0.059 (2,4): 0.066 (0,2): 0.028 (0,4): 0.038 (2,2): 0.052 (1,5): 0.058 (1,6): 0.064 (1,1): 0.033 (2,1): 0.045 (1,4): 0.051 (0,7): 0.055 (0,8): 0.061 For 2D harmonic oscillator the energy levels can be determined from: En,l  (2n | l | 1) 0 (74) The energy levels of our system, near to the s & p states can be somehow equivalent to the 2D harmonic oscillator energy levels. To this end, we introduce a detuning parameter n,l as: En,l  (2n | l | 1)( 0   n,l ) (75) The detuning parameter for each energy level is obtaind as following: The energy levels of s - and p- states are considered to be 0 and 2 0 , respectively, with zero detuning parameters, where 0  4.1meV is s & p energy splitting: E0,0  0 ; 0,0  0 E0,1  2 0 ; 0,1  0 (76)

43 For a specific energy level like E0,2 , we can write the energy difference E0,2  E0,0 from Eqs. (75)-(76) as: E0,2  E0,0  3( 0  0,2 )  0  8.537meV (77) Therefore, from Eq. (77), we can calculate: 0,2  0.11meV (78) Similarly, for energy level E0,3 : E0,3  E0,0  4( 0  0,3 )  0  13.31 (79) 0,3  0.25meV (80) we can obtain: On the other hand, energy levels of 2D harmonic oscillator with applying magnetic field can be calculated from: En,l  (2n | l | 1) 1 1 ( c )2  ( 0 )2  l c 4 2 (81) where c is cyclotron frequency which can be evaluated from: c  eB m* ; m*  0.08me (82) For instance for the magnetic field B  1 T we evaluate c  1.4471meV . For our case, we can insert n,l in eq. (81) as: En,l  (2n | l | 1) 1 1 ( c ) 2  ( 0   n,l ) 2  l c 4 2 (83) Therefore, for E0,0 we can write: E0,0  1 1 ( c )2  ( 0 )2  (1.4471)2  (4.1)2  4.1634meV 4 4 (84)

44 while for zero magnetic field, E0,0  4.1 meV which shows that with magnetic field, energy level of s -state would be increased by 0.0634 meV . For p -state, l  1 , E0,1  2 we can write: 1 1 1 1 ( c )2  ( 0 )2  c  2 (1.4471)2  (4.1)2  1.4471  9.0503 4 2 4 2 while for zero magnetic field, E0,1  8.2 , which shows that, p -state (85) would be increased by 0.8503 meV . For E0,2  3 wgm , l  2 we can write: 1 1 ( c )2  ( 0  0,2 )2  c  3 (1.4471)2  (4.1  0.11) 2  1.4471  11.3681 4 4 (86) while without magnetic field, E0,2  12.63 , which shows that it would be decreased by 1.2619 meV For wgm , l  3 , E0,3  4 1 1 3 ( c )2  ( 0  0,3 )2  c  4 (1.4471)2  (4.1  0.25)2  1.4471  15.5684 4 4 2 (87) For zero magnetic field E0,3  17.4 meV which shows that it is decreased by 1.8316 meV . 3.6 Experimental Result Figure below shows experimental result for 4 meV QD. There are two close picks between s- and p-states which are related to the WGMs. Therefore, we can predict that these two close picks are corresponding to 2 electrons in WGM orbital, where each pick related to one of these 2 electrons. This result is corresponding to our numerically calculated energy spectra for 4 meV QD, Fig. 19 (a), in which 2 electrons are supposed to be in the excited state and 4 electrons in the ground state. In Fig. 19 (a), we similarly calculate the energy splitting between s- and p-states equal to 4 meV, and attain a WGM between these two states. Although, these two electrons have degenerate energy levels

45 in numerical result, but in experiment, this degeneracy can be split due to the defect in the dot. Besides, the size of the wavefunction distribution which is obtained by NearField Scanning Optical Microscopy (NSOM) is almost identical with single particle (numerical) calculation. Figure 21 - Experiment result of energy spectra and wavefunction distribution for 4 meV QD 3.7 Surface Modes in Lateral QD We could achieve surface states in lateral QD, in two orthogonal directions, φand z-directions. 3.7.1 WGMs The resonance in the φ-direction which is referred to WGMs shows that WGM with higher angular momentum has larger effective length of the QD. Figure 22 shows WGMs with different angular momentums, in which the radius size of the resonance is increased for higher angular momentums.

46 Figure 22 – WGMs from left to right, top: m=3, m=5, bottom: m=7, m=9. 3.7.2 Resonance across the Height For each of the angular momentums (m), it is possible to achieve different modes of distribution along the height of the QD. Figure 23 shows these modes for m=7, which are related to 5th, 7th, 9th and 11th eigenvalues.

47 Figure 23 - Wavefunctions of QD for m=7 ; from left to right, top: 5th, 7th, bottom: 9th, 11th eigenfunctions For 6 meV QD, for m=0, quantum number associated to the height, starts to change from 7th eigenvalue, while for 4 meV QD, it is 6th eigenvalue. Generally speaking, with increaing the s-p splitting, resonances across the height occur in a higher number of eigenstates. Figure 24 shows surface states of 10 meV QD which can exist only for m=0 (7th and 9th eigenvalues), m=1 (7th and 9th eigenvalues), m=2 (7th eigenvalue) and m=3 (7th eigenvalue). The interesting feature in this figure is that the confinement of the distribution in the QD is reduced which can be attributed to the both approaching the eigenvalues to the applied voltage, and also decreasing the height of QD.

48 Figure 24 - Surface states of 6 meV QD; from left to right, top: 7th (m=0), 9th (m=0), 7th (m=1), bottom: 9th (m=1), 7th (m=2) , 7th (m=3) eigenfunctions 3.8 Perturbation 3.8.1 Lateral QD Figure 25 shows that the perturbation which interfere a state can change it, otherwise it would be unaffected. This figure shows the comparison of three modes of surface states between unperturbed and perturbed QD, for 5th eigenvalue of m=0,1, 2 (unperturbed dot) and 6th eigenvalue of m=0, 5th eigenvalue of m=1,2 (perturbed dot).

49 Figure 25 - Wavefunctions of 4 meV QD; from left to right, top (unperturbd): 5th (m=0), 5th (m=1), 5th (m=2), bottom (perturbed): 6th (m=0), 5th (m=1), 5th (m=2) eigenfunctions It shows that the first mode which is mostly affected by perturbation is changed considerably, however, the other two modes are changed a little. The perturbation at the center of the dot leads the height of QD be decreased which cause the surface state for m=0 occur at a higher eigenvalue. 3.8.2 Spherical QD Figure 26 shows that smooth deformation along the surface keep the surface states unchanged. This figure is related to the 1st, 2nd and 3rd eigenvalues of angular momentum m=10, for both spherical and deformed spherical dot, which shows that distribution and also the eigenvalues are almost identical in these dots. However, for higher order modes, due to the rapidly changing of the distribution along the surface, the deformation would change the states.

50 Figure 26 - Wavefunctions of spherical QD for first three eigenmodes of m=10, top: unperturbed, bottom: perturbed 3.8.3. Triangular QD Figure 27 shows that by changing the geometry of the QD, we can handle the distribution of the electrons in the dot. This figure is related to the first three eigenmodes of angular momentum m=10 for the dots with the vertex at symmetrical and also unsymmetrical positions. It shows that electrons tend to concentrate at the vertex of these dot. Besides, in the unsymmetrical one, the distribution of electrons are more likely along the surface.

51 Figure 27 - Wavefunctions of triangular QD for first three eigenmodes of m=10, top: symmetrical vertex, bottom: unsymmetrical vertex

52 Conclusion In this thesis, we study WGMs in Quantum Dots (QDs). In lateral QD the Coulomb interaction between the electrons plays a significant role in its energy spectra. We reveal that Coulomb interaction can be comparable to the interband transition energy, which cause some higher energy levels fall between the two lowest energy levels s- and pstates. Furthermore, some resonances along the height of QD are detected which are unusual in these dots. We reveal the perturbation and topological effect on the surface states in these dots as well as in spherical and triangular dots. In these dots WGMs with higher angular momentums shows larger effective lengths in the QD. Resonances along the height of these dots show less confinement in the dot, with decreasing the height of the dot. It is shown that perturbation can affect the local distributions, while the non-local states would be unaffected. Smooth perturbation in spherical QD keep the surface states unaffected. Furthermore, surface states can be manipulated by changing the location of the vertex, in triangular dots.

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