Electronic spectra and chemical potential of 2-D multi-electron quantum dots in magnetic field: exact multi-pole expansion of coulomb correlation

Two-dimensional superlattices, TMDCs and graphene family exhibit strong coulomb correlations among e–e, e–h, electron-phonon and etc. As the number of electron N increases, confining gate voltage and transverse magnetic field are superseded by increasing coulomb interactions (N(N−1)/2 factors) that composes non-trivial Schrödinger equations. Representing such equations in Whittaker-M functions yields a modest alternate formalism, that accommodates integrals of coulomb (exchange) correlations in single-summed, finite and exact Lauricella functions via Chu-Vandermonde identity. For higher carrier density (N = 3, 4, 5, 6, ..), the multipole expansion is incurred as exact and finite-summed coulomb, coulomb-type and dipole-type integrals. Although, fermionic exchange symmetry of many electron systems could be included in terms of various two-electron integrals, for the sake of brevity we have aimed to reproduce experimental results without spin. Signature of interplay among gate voltage, magnetic field, dielectric constant, mass and density of carriers is examined in electronic spectra, magnetization, chemical potential for the systems spanning over wide range of materials (He, BN, GaAs and etc.). Interestingly, chemical potential and addition energy as a function of magnetic field and number of carriers monitor the statistics between strongly degenerate to weakly degenerate composite fermions, coulomb blockade and shell structure of 2-D superlattices. At the most, quadrapole and octapole suffice the convergence.


Introduction
Confined ballistic systems such as quantum dots (0-D), quantum wires (1-D) and quantum wells (2-D) are engineered to study structural, optical, transport and magnetic properties of N-e systems by monitoring gate voltage, transverse magnetic field, dielectric constant and mass of carriers [1][2][3][4]. In addition to certain limitations on the issues like inclusion of Pauli exclusion principle, fermionic exchange symmetry, arbitrary magnetic and electric fields, as the number of correlation terms increases by a factor N(N−1)/2 an impenetrable barricade emerges in studying of such quantum systems. For 2-D quantum dots, Laughlin and Jain have set landmarks by introducing Laughlin's wavefunction (LW) for 3-e and composite-fermion (CF) wavefunction for N-e systems respectively [5][6][7]. Other magnificent contributions in solving 2-D multi-electron systems have been improvised by Chakraborty et al, Hamaguchi et al and Taut [8][9][10][11][12][13][14][15][16]. Recently, our group proposed a modest alternate to many-body approximation methods for strong e-e and e-h correlations of 2-D quantum dots in magnetic field [17][18][19]. As many-body physics holds great importance, we have designed a prototype to solve exactly generalized N-e systems using multi-pole expansion (section 2) [20]. In this paper, synergistic coulomb-and dipole-type integrals evaluated by us are attributed to finitely single-summed terminated Lauricella function via Chu-Vandermonde identity [21]. In section 3, we have studied electronic spectra of 2-e, 3-e, 4-e and 5-e systems with respective chemical potential (m N ) and addition energy (Δμ) to explain extent of fermionic character, shell structure and coulomb blockade in detail. Chemical potential and addition energy as a function of gate voltage, magnetic field and number of electrons may reveal underlying physics related to phase transition and existence of both strongly and weakly degenerate character of unusually stabilized composite fermions [22]. In a dot structure, electrons are bound by a confining potential which gives rise to discretized energy states or shell structure of quantum dots with magic number N=2, 6,12,20 representing the stable configurations [4].

Theoretical development
The spin free hamiltonian of 2-D N-e harmonically confined system (ω 0 ) in transverse magnetic field having a vector potential (A(r i )) of the form where m e and ò are effective mass of electron and static dielectric constant of the material respectively. Here, H i 0 represents the unperturbed hamiltonian of the ith particle and ¢ H represent coulomb interaction term among charge carriers. For N=2, the hamiltonion can be segregated into relative and center-of-mass coordinates [17]. But for N>2, the interaction term ( ¢ H ) replicates by a factor of N(N−1)/2. Moreover, surface integrals of both Dirichlet and Newman forms of Coulomb interaction in Green's function expansion exhibit sharp falls in the values for harmonic oscillator and reach multipole expansion of generic coordinates (r i ). Thus, each interaction term ( ¢ H ) is expanded via well known multi-pole expansion corroborating to [20]: Although, Schrödinger equations for both generic coordinates and relative/center-of-mass coordinates resemble to each other but their scaling factors differ [17]. Thus, it is very necessary to examine the nonrelativistic equation for generic coordinates. Evaluating, , 0 for each ith particle of 2-D systems in magnetic field (w c =eB/c) where, n and m are the principal and magnetic quantum numbers respectively.

Result and discussion
The onset of inter-electronic repulsions constitutes mixing among states of atoms, molecules, superlattices and quantum dots due to massive interplay of ω 0 , ω c , m e , N and ò. These level crossings/anti-crossings and stabilization in lower bound states of 2-D N-e systems are reflected in magnetization (M) (T∼1K ), chemical potential (μ) and addition energy with varying magnetic field. Consequently, to understand its physical origin, N-e eigen values are realized using similarity transformation (Jacobi Rotation). In the present work, we have considered systems of two, three, four and five electrons (m=0,±1,±2, ..,±5 for N=3 and m=0,±1, ±2, K,±4 for N=4). A sound convergence at =2 (quadrapole) is attained with the condition  r r i j (for arbitrary r i and r j ). Monopole and dipole factors have significant contribution to coulomb (exchange) correlations in comparison to quadrapole factor. Contribution of monopole factor to Coulomb (exchange) integral can roughly be formulated as (quantum number of all other coordinates are conserved) [17]: for bound states, nearly equating to Laughlin's prediction of | | m 1 for states having higher angular moment (m) and high ω c . A key point to be noted is that states of conserved angular momentum (m 1 =m 2 =m 3 =.....=m) mutually exclude dipole contribution. Our formalism of absolutely terminating coulomb-and dipole type integrals does not meet any secular divergence problem at all [11]. Level spacing statistics and specific heat measurements (C v ) are also investigated in detail. It is evident that switching over of statistics between Poisson and Wigner distribution is very much prominent for N=2 in comparision to N=3, 4, 5, 6, K and consequently, C v shows maximum fluctuations for N=2 rather than the cases of higher carrier density [17]. Therefore, for the brevity and compactness of the paper we have refrained to discuss it.

Energy level diagram
The electronic structures of different materials [He, Li, Be, GaAs and BN] scrutinized with N=2, 3, 4 and 5 at ω 0 =0.004 a.u. and 0.000 04 a.u. for different m e and ò (He, Li and Be (m e =1.0 a.u. and ò=1.0), GaAs (m e =0.07 a.u. and ò=12.35) and BN (m e =0.26 a.u. and ò=7.1)) register an unusual pattern on energy spectra ( figures (1-2)). In our previous work we have discussed energy level diagram of SiO 2 (m e =0.42 a.u. and ò=3.9). But here, we have excluded SiO 2 (m e =0.42 a.u. and ò=3.9) system. Although, it also required a detailed discussion for SiO 2 with varying carrire density, size and confinements, for the sake of brevity we have refrained from the corresponding results [17]. An enhancement in energy levels with increasing carrier density (N=2, 3, 4 and 5) is noteworthy for atomic systems (He, Li, Be) in comparison to other systems (GaAs and BN) because of low dielectric constant. In general, lowering of the confinement frequency by 10 times, a relative decrease in coulomb interaction nearly by three times is observed in comparison to non-interacting energy (equation (34)) [17]. In the next section we will discuss the role of confinement frequency (ω 0 ) on energy spectrum.
3.1.1. Strong confinement systems (w 0 =0.004 a.u.) In strongly confined systems, apparently weak correlations marks a feeble blow on energy spectrum. But for systems with higher mass (m e ) and lower dielectric constant (ò), it appears little disordered. The early crossing of ground state shifts monotonically to lower field (ω c 0.000 9 a.u., 0.000 4 a.u. and 0.000 3 a.u.) with increasing carrier density (figures 1(g)-(i) GaAs system). In BN quantum dots (figures 1(d)-(f)), multiple transitions with stabilization of ground state is observed at weak field (ω c 0.0 a.u. to 0.001 5 a.u.) for N=3 on account of interplay between enhanced exchange correlation and magnetic field ( figure 1(e)). The stabilization becomes more pronounced for N=4 ( figure 1(d) BN system). Atom-like quantum dots (figures 1(a)-(c)) imitate exceptional features of crossings/anti-crossings, stabilizations and enhanced degeneracies in lower bound states as a function of N. For N=2 (He), the nodeless ground state at ω c =0.0 comprises of huge crossing/anticrossing at moderate ω c (=0.001 a.u., 0.002 a.u., 0.004 a.u.) (figure 1(c)). However, an immense stabilization of lower bound states for N=3 and doubly degenerate ground state with quick evolution is recognized for N=4 (figures 1(a), (b)). 3.1.2. Weakly confined systems (w 0 =0.000 04 a.u.) Inter-electronic repulsions greatly suppress magnetic field at weaker confinement (ω 0 =0.000 04 a.u.) depending upon the number of carriers and static dielectric constant. As a result of it, symmetry of lower bound states is destroyed and the ground state become more involved in crossings/anti-crossings as the survey spans over ò=1.0 (atomic) to ò=7.1 (BN) and ò=12.35 (GaAs). Exchange interactions cause massive stabilization, enhanced degeneracy of ground state and numerous transitions among lower bound states for N=3 and N=4 electrons (figures 2(b), (e), (h) (a), (d), (g)). Ground state remains doubly degenerate in case of atom-like quantum dots and criss-crossing among states is diminished due to acute upsurge of coulomb interactions for high carrier density ( figure 2(a)). The similar impression of excess stabilization and/or enhanced degeneracy among lower bound states is recognized for either low dielectric systems with low carrier density (N=3, ò=1.0) or high dielectric systems with high carrier density (N=4, ò=7.1) (figures 2(g), (e) (b), (c), (d) and (1(d) 2(g)), (1(b) 2(e))). Particularly, in case of BN (Figures 2(d)-(f)) and GaAs (figures 2(g)-(i)) quantum dots, enhanced exchange interactions cause unusual stabilization and shuffling of few lowest bound states in weak field regime for N=3 (figures 2(e), (h)). But as N increases doubly degeneracy of ground state or excess stabilization or both appears due to profound coulomb (exchange) interaction (figures 2(d), (g)).

Magnetization (M)
Magnetization is another aspect that highlights the proliferating effect of coulomb interaction with increasing N through sharp discontinuities as a function of magnetic field [29,30]. It can be expressed as (within the low temperature limit T∼1K ): [31]  where y= + e -( ) e 1 1 1 , ε 1 =ΔE 1 /k B T, ΔE 1 =(E 1 -E 0 ) and E is the energy at specific ω c . E 0 and E 1 are the eigen value of ground and 1st excited states respectively. In the following sections, magnetization spectra of 2-D N-e dot of GaAs (m e =0.07 a.u. and ò=12.35), BN (m e =0.26 a.u. and ò=7.1) and atom-like quantum dot systems (m e =1.0 a.u. and ò=1.0) with ω 0 =0.004 a.u. and 0.000 04 a.u. are investigated.
3.2.1. Weakly confined system (ω 0 =0.000 04 a.u.) Smooth decay in magnetization curves for low dielectric system (ò=1.0) are registered in weakly confined system due to relatively pronounced coulomb interaction for N=2-4 ( figure 3(d)). Further, increase in dielectric constant for BN and GaAs quantum dots introduces discontinuities in curves that show occurrence of ground state transitions near crossing points (figures 3(e), (f)). These transitions primarily address change in angular momentum of ground state. Even at ω c =0.0 a.u., a quantitative difference in magnetization is noticed with increasing dielectric constant for systems of N=2 and N=3 (figures 3(d)-(f)).

3.2.2.
Highly confined system (ω 0 =0.004 a.u.) A sharp jump in magnetization for N=4 and little step wise decrease for N=2 and 3 is observed in low dielectric system (ò=1.0) ( figure 3(a)). In BN system two sharp jumps in the curve (M − ω c ) at two different magnetic field (ω c ∼0.008 a.u. and 0.001 6 a.u.) for N=4 and single jump for N=2 (ω c ∼0.001 6 a.u.) are observed ( figure 3(b)). It is seen that number of jumps (discontinuities) have reduced in comparison to weakly confined systems because of increasing confinement frequency scales down the strength of coulomb interaction.
In GaAs system, coulomb interaction diminishes further and results in smooth curve for N=2 ( figure 3(c)).

Chemical potential (m N ) and addition energy ( m D )
Level crossings/anti-crossings in energy level diagram appear as cusp or transition in chemical potential with changing angular momentum of lowest bound states [4]. The number of transitions or cusps is either enhanced or results in fluctuation with number of electrons, dielectric constant (ò), mass of carrier (m e ), confinement frequency (ω 0 ) and magnetic field (ω c ). Thus, in this section a variation of chemical potential • Gallium Arsenide (GaAs) quantum dots (ò=12.35 and m e =0.07 a.u.): Magnetic-field dependence of μ 1 is reflected in the smooth curve due to absence of crossings/anti-crossings of the ground state ( figure 4(a)). But with the onset of coulomb interaction a very small cusp emerges at ω c =0.000 9 a.u. in μ 2 whereas μ 3 and μ 4 exhibit prominent cusps or transitions at ω c =0.000 45 a.u. and ω c =0.000 320 a.u. respectively in contrast to μ 1 and μ 2 ( figure 4(a)). The transitions at ω c =0.000 2 a.u., 0.000 3 a.u. and 0.000 45 a.u. are observed for μ 5 . These discontinuities in the form of maxima and minima in chemical potential may give an impression of fractional Landau level filling factors [32,33]. The gap between μ 1 and μ 2 appears to be constant which indicates equal synergistic effect of coulomb interaction on lowest bound states for zero and non-zero magnetic field (figures 4(a)). Moreover, addition energies are also plotted against the number of electrons N at different magnetic fields which map to the 'shell structure' already reported by Kouwenhoven and his group [4] at B=0 for strongly confined system ( figure 4(c)). The peak structure fades away gradually into a smooth curve at high magnetic field (ω c =0.001 a.u.). High magnetic field compacts electrons and increases interactions to such an extent that shuffling among states is reduced. In case of GaAs, chemical potential and addition energy plots partially resemble to the non-interacting system due to reduced coulomb interaction. • Boron Nitride (BN) quantum dots (ò=7.1 and m e =0.26 a.u.): A smooth curve of m 1 against magnetic field expectedly appears owing to absence of cross over points among lowest bound states ( figure 5(a)). The appearance of cusp or transition of μ 2 at ω c =0.001 54 a.u. becomes more prominent than that of GaAs quantum dot due to enhanced e-e interaction (figures 4(a)-5(a)). μ 3 decreases upto a threshhold magnetic field (ω c =0.001 4 a.u.) and then increases with increasing magnetic field ( figure 5(a)). This peculiar behavior is ascribed to the enhanced exchange interactions in low field which may emerge as a remnant of switch over from strongly degenerate to weakly degenerate fermionic character. There are two cusps in μ 4 at ω c =0.000 6 a.u. and at ω c =0.002 a.u. due to crossing of ground state for N=3 and N=4 dots ( figure 5(a)). However, μ 5 remains nearly constant upto certain magnetic field (ω c =0.000 8 a.u.) and then a steep increase with increasing magnetic field on account of upsurging coulomb interaction due to enhanced carrier density. These oscillations, i.e, rise and fall in chemical potential reflect acute response of dots to magnetic field and number of particles (N=2, 3, 4, 5, ..) ( figure 5(c)). At low magnetic field (=0.0 a.u. and 0.000 8 a.u.) an experimentally observed 'shell structure' is observed which inverts at high magnetic field (=0.001 6 a.u., 0.002 4 a.u., 0.003 2 a.u. and 0.004 a.u.) due to augmentation of coulomb interaction ( figure 5(c)). and GaAs (ò=12.35, m e =0.07 a.u.) systems. Therefore, Chemical potential and addition energy respond differently. Absence of cusps in μ 1 , μ 2 and μ 5 signify no level crossing/anticrossing of the ground states. Anamolously, μ 3 has a huge fall and lies far below μ 2 with the onset of magnetic field ( figure 6(a)). This is due to enhanced exchange interactions which cause excess stabilization to the N=3 system. μ 4 and μ 5 show similar trend but a little cusp near ω c =0.005 a.u. is observed in μ 4 . Interestingly, the spacing between μ 1 and μ 2 is extremely small in comparison to μ 3 and μ 4 and μ 4 and μ 5 which exhibit deviation from the behavior reported by Tarucha et al [34] for moderate and high dielectric quantum dots (figure 6(a)). Addition energy plot also displays a complete opposite trend in comparion to the usual shell structure (figure 6(c)). It is clearly evident from the studies of chemical potential and addition energy that formation of tightly bound composite-fermions is highly facilitated by magnetic field for N=2 and 3 than that of N=4 and 5 which is a reminiscence of Cooper pair formation.
3.3.2. Weakly confined system (w 0 =0.000 04 a.u.) Coulomb repulsion profoundly dominates in larger dots which brings about a noticeable effect in chemical potential and addition energies. Thus, study of larger dots displays different signature in chemical potential and addition energy plots.
• Gallium Arsenide (GaAs) system (ò=12.35 and m e =0.07 a.u.): Similar impression of nearly smooth curve appears for μ 1 and μ 2 as there is negligible e-e coulomb correlation ( figure 4(b)). Alike strongly confined atom-like quantum dots a monotonic decrease in μ 3 is found due to unusual stabilization induced by exchange interactions. A sharp increase with magnetic field is observed for μ 4 and μ 5 . The energy gap between μ 3 and μ 4 widend up with increase in magnetic field ( figure 4(b)). Subsequently, addition energy follows opposite trend to that of highly confined GaAs system and the corresponding peaks become sharp with increase in magnetic field (figure 4(d)) • Boron Nitride (BN) system (ò=7.1 and m e =0.26 a.u.): m 1 shows an increasing behaviour whereas μ 2 and μ 3 decreases with the arrival of cyclotron frequency. μ 2 and μ 3 coinside with μ 1 near ω c =0.000 025 a.u.
( figure 5(b)). However, μ 4 and μ 5 increases as usual with magnetic field ( figure 5(b)). The widening of energy gap between μ 2 or μ 3 and μ 4 is also noticeable in BN system. Addition energy plot against number of electrons