As we show below, such profile provides in-plane parabolic confinement for which the mathematical treatment is significantly easier. For the same reason, we adopt a model with infinite barrier confinement, assuming that dielectric mismatch is much smaller than the mismatches of the conduction and the valence bands; therefore, the probability for the self-tunneling of the particles between QDs is depreciable in comparison with the tunneling provided by the external electric field. In this way, we assume that confinement potential for the electron and for the hole is equal to zero inside the QDs and to infinity otherwise, defined in cylindrical coordinates as:

V e r e = 0 i f 0 < z e < w e ρ e ∞ o t h e r w i s e , V h r h = 0 i f d < z h < d + w h ρ h ∞ o t h e r w i s e ,

Here, three-dimensional position vectors of the electron and the hole in the cylindrical coordinates are r _p = (ρ _p , ϑ _p , z _p ); p = e, h.

The values of the physical parameters pertaining to InAs used in our calculations are dielectric constant ε = 15.2, the effective masses in the InAs material layer for the electron m _e = 0.04m ₀ and for hole m _h = 0.34m ₀, the conduction and the valence bands offsets in junctions are V _0e = 450 meV and V _0h = 316 meV, respectively 19 .

As the quantum dots and the exciton sizes under consideration are much larger than the unit cell of the material, the effective-mass approximation is a suitable approach; therefore the resulting model Hamiltonian of the electron-hole pair in the presence of uniform magnetic and electric fields oriented along the z-axis, perpendicular to the plane of QDs, can be written as:

H = ∑ p = e , h 1 2 m p p p − q p c A p 2 + V p r p ∓ q p F z p − e 2 ε r eh ;

Here, r eh = z e − z h 2 + r e 2 + r h 2 − 2 r e r h cos ϑ e − ϑ h is the electron-hole separation; m _p and p _p are the effective masses and the momentum vectors of the particles, respectively; and the parameter q _p = ± e gives their charges. Choosing the gauge for electron and hole vector potentials as A _e = (B × r _e ) / 2; A _h = (B × r _h )/2, where B is the magnetic field (assumed to be uniform here), the Hamiltonian (3) can be reduced to the following dimensionless form:

H = H 0 e + H 0 h − 2 / r eh ; H 0 p = − η p − Δ p + γ 2 ρ p 2 / 4 − i γ s p ∂ / ∂ ϑ p ∓ α z p + V p r p ; p = e , h ; η h = μ / m h ; η e = μ / m e ; s h = + 1 ; s e = − 1

In these equations H _0e and H _0h represent the Hamiltonians of the unbound electron and hole respectively, confined inside their heterostructures. The following units are used in the dimensionless Hamiltonian (4), the exciton effective Bohr radius a ₀ ^∗ = ℏ ² ε/μ e ² as the unit of length, the effective Rydberg Ry * = e ²/2ε a ₀ ^∗ = ℏ ²/2μ a ₀ ^{∗ 2} as the energy unit, and γ = eℏB/2μ c Ry * and α = ea ₀ ^* F/Ry * as the units of the magnetic and electric field strengths respectively, with μ = m _e m _h/(m _e + m _h) being the reduced mass. The parameters η _e, η _h satisfy the relation η _e + η _h = 1; in our calculations, we assume that η _e > η _h.

Taking into account that typically, in actual self-assembled QDs, the height is much smaller than the lateral dimensions; one can take an advantage of the adiabatic approximation, which allows us to exclude temporarily the rapid particle motions along z-axis from consideration and, in this way, to reduce the dimensionality of the initial three-dimensional problem. Following the adiabatic procedure, described in reference 12 , one can obtain the two-dimensional effective Hamiltonian, which describes only the in-plane particle motion of the form:

H 2 D = H 0 e 2 D + H 0 h 2 D − 2 d 2 + ρ e − ρ h 2 ; H 0 p 2 D = − η p − Δ p 2 D + γ 2 ρ p 2 4 − i γ s p ∂ ∂ ϑ p + π 2 w p 2 ρ p ∓ α z p ; p = e , h

Here, 〈z _p 〉,  p = e, h are the mean values of z-coordinate of the corresponding particle. Finally, for the selected profile (1) the Hamiltonian (5) describes two particles in 2D quantum dot with parabolic confinement:

H 2 D = E 0 − ∑ p = e , h η p − Δ p 2 D + i γ s p ∂ ∂ ϑ p + ω p 2 ρ p 2 4 ∓ α z p + π 2 W p 2 − 2 d 2 + ρ e − ρ h 2 ; ω p 2 = 4 π 2 W p 2 R p 2 + γ 2

Finding the eigenfunctions of the Hamiltonian (6) allows us to introduce relative and center of mass coordinates r = r _e − r _h and R = η _hr _e + η _er _h, respectively. Thus, the Hamiltonian (6) is reduced to:

H = E 0 + H R + H r + U ; E 0 = ∑ p = e , h η e π 2 / W p 2 ; H R = − η e η h Δ R 2 D + η e ω e 2 + η h ω h 2 R 2 / 4 ; H r = − Δ r 2 D + i γ ∂ ∂ ϑ + 1 4 η e 3 ω e 2 + η h 3 ω h 2 r 2 − 2 d 2 + r 2 ; U = 1 2 η e 2 ω e 2 − η h 2 ω h 2 R ⋅ r

Here, E ₀ is the background energy; second and third terms H _R and H _r describe the center of mass and relative motions, respectively, and they correspond to the Hamiltonians for two independent central force problems; while the last term U presents a perturbation. The Hamiltonian H _R coincides with one of the circular oscillator, whose eigenvalues can be found exactly and its eigenfunctions can be expressed in terms of the generalized Laguerre polynomials. It is seen that the Hamiltonian (7) becomes completely separable for a particular case when the external magnetic field is zero (γ = 0), and the geometric parameters of QDs satisfy the following condition:

η e ω e = η h ω h .

In this case, U = 0, and eigenvalues of the Hamiltonian (7) are a sum of the background energy E ₀, the center of mass energy E _R defined by two quantum numbers, radial N and angular M, and the relative energy E _r depending on four quantum numbers, radial n and angular m (N, n = 0, 1, …; M, m = 0, ± 1, …):

E = E 0 + E R N , M + E r n , m ; E R N , M = M + 2 N + 1 η e η h η e ω e 2 + η h ω h 2

The corresponding wave functions are:

Ψ N , M , n , m 0 R , Θ , r , ϑ = C e iMΘ e imϑ R M e − Λ R 2 / 2 L N M Λ R 2 Φ n , m r ; Λ = 1 2 η e ω e 2 + η h ω h 2 / η e η h ,

where L _N ^(M)(x) are generalized Laguerre polynomials and Φ _n,m (r) is the radial part of the wave function describing the relative coordinates evolution, which is a solution of the following ordinary differential equation:

− 1 r d d r r d Φ n , m r d r + m 2 r 2 + γ m + 1 4 η e 3 ω e 2 + η h 3 ω h 2 r 2 − 2 d 2 + r 2 Φ n , m r = E r n , m Φ n , m r .

Once Equation (11) is solved and the set of wave functions (10) is found then it can be used as the basis to calculate the energy corrections due to the presence of the perturbation U in the Hamiltonian (7) in the framework of the so-called exact diagonalization or Galerkin method.