Exciton effective mass enhancement in coupled quantum wells in electric and magnetic fields

We present a calculation of exciton states in semiconductor coupled quantum wells (CQWs) in the presence of electric and magnetic fields applied perpendicular to the QW plane. The exciton Schr\"odinger equation is solved in real space in three dimensions to obtain the Landau levels of both direct and indirect excitons. Calculation of the exciton energy levels and oscillator strengths enables mapping of the electric and magnetic field dependence of the exciton absorption spectrum. For the ground state of the system, we evaluate the Bohr radius, optical lifetime, binding energy and dipole moment. The exciton mass renormalization due to the magnetic field is calculated using a perturbative approach. We predict a non-monotonous dependence of the exciton ground state effective mass on magnetic field. Such a trend is explained in a classical picture, in terms of the ground state tending from an indirect to a direct exciton with increasing magnetic field.


I. INTRODUCTION
Indirect excitons in coupled quantum wells (CQWs) present a model system for the study of a statistically degenerate Bose gas in solid-state materials. This is due to the long optical lifetime of indirect excitons in comparison to their thermalization time which permits the creation of a cold and dense exciton gas. [1][2][3] Numerous studies have focused on Bose-Einstein condensation of excitons. [4][5][6][7] The extension of the indirect exciton lifetime over that of direct excitons owes to the reduced overlap of electron and hole wave functions when confined to spatially separated QW layers. The extent of this effect is controlled via an electric field applied perpendicular to the growth direction and may be varied over several orders of magnitude. 8,9 The static dipole moment of indirect excitons gives rise to a number of interesting and useful effects. Firstly, transport of indirect excitons can be controlled via patterned electrodes positioned adjacent to the CQWs. [10][11][12] This utilizes the quantum confined Stark effect to create an indirect exciton electrostatic potential landscape in the plane of the QWs. The shape of the electrodes and applied voltage can be chosen to create quite arbitrary static and dynamic potentials. Numerous investigations have studied excitons in electrostatic traps, [13][14][15][16][17] linear potential gradients 10,12 and stationary 18 and moving 19 lattices. Secondly, the dipolar interactions provides indirect excitons with an ability to screen the QW disorder potential. This potential is intrinsic to the QW structures due to interface roughness and defects. 20 At relatively low densities (≈ 10 9 cm −2 ), screening of the QW disorder facilitates transport of indirect excitons over tens of micrometers. 21 At higher densities, indirect excitons may screen externally applied potentials up to 10 meV deep leading to localization-delocalization transitions. 18 In addition, the dipolar interactions cause a blue shift in the exciton emission which provides a probe of the exciton density. [22][23][24] A number of remarkable effects have been observed in the exciton photoluminescence including a rich variety of spatial pattern formations [25][26][27] and spin transport phenomena. 28 In the strong coupling regime, an exciton and a photon inside a planar Bragg-mirror microcavity form a microcavity polariton. 29 This system offers new tools to study fundamental physics with greater control. Bose-Einstein condensation is accessible at a higher temperature due to the small polariton effective mass. 30 Recently, CQWs were embedded in a microcavity 31 to create a new state known as a dipolariton. The cavity is designed such that a photon mode is resonant with the direct exciton, which itself is electronically coupled to the indirect exciton at the electron resonant tunneling condition. 32 The dipolariton is thus a superposition of a cavity photon, a direct and an indirect exciton and thus acquires a static dipole moment. 33,34 Dipolaritons have been suggested for observation of super radiant THz emission, 35 continuous THz lasing 36 and polariton bistability. 37 The development of photonic devices that exploit the diverse properties of excitons and polaritons has also attracted much attention recently. The high degree of flexibility and control that is readily achieved in such systems has led to demonstrations of excitonic 38 and polaritonic 39 optical transistors.
Another tool in the arsenal of techniques with which to study and control excitons and polaritons is the application of magnetic fields to the CQW structure. It has been demonstrated that the exciton effective mass increases in the presence of a magnetic field. 40 This effect can be understood by considering a simple classical analogy. When two oppositely charged particles travel through a uniform magnetic field that is perpendicular to their motion, they experience forces that act to increase their relative separation. The separation comes at an energy cost as it is opposed by the mutual Coulomb attraction between the charges. The manifestation of this effect is a resistance to change in momentum and thus an effective mass increase that depends on the strength of the applied magnetic field.
The focus of this paper is a numerical solution of the exciton Schrödinger equation in QW heterostructures with electric and magnetic fields applied perpendicular to the QW plane. We also present a precise calculation of the electric field dependence of the exciton effective mass arXiv:1502.07094v1 [cond-mat.mes-hall] 25 Feb 2015 enhancement in a magnetic field. A theoretical study of excitons in CQWs with finite barrier and well width was done in Ref. 41, using two-dimensional (2D) free electronhole pair states in magnetic field as a basis for expansion of the exciton wave function. This approach however is only well suited for treating high magnetic fields. Also, a calculation of the exciton mass renormalization due to the magnetic field was absent. The mass renormalization in the 2D limit was calculated for excitons in single QWs 42 and indirect excitons in double QWs. 43 In those works, the electron and hole wave functions in the QW growth direction were approximated by delta functions located at the QW positions. This corresponds to the limit of infinitely deep QWs of zero width. Such a treatment is applicable only for the analysis of either purely direct or purely indirect excitons in the case of high electric field and strong QW confinement. It cannot describe in any detail the smooth crossover between direct and indirect exciton states for intermediate electric fields. This is in contrast to the recently developed rigorous multisub-level approach 9 (MSLA). Previously, this approach was used to calculate the CQW exciton states in perpendicular electric fields. The outcomes, which included a calculation of the dependence of the absorption spectrum and optical lifetime on electric field, were found to be in good agreement with available experimental data. Here, we use the MSLA to solve the exciton Schrödinger equation in CQWs with both electric and magnetic fields applied perpendicular to the QW plane. This method solves the exciton in-plane problem by a numerical discretization in real space and is thus equally well suited for both low and high magnetic fields. Here, we have refined the MSLA to a highly accurate scheme by using Numerov's algorithm, 44 a fourth-order linear multistep method, for obtaining the energies and wave functions of the exciton states. It is found that the direct-to-indirect exciton crossover with varying electric and magnetic fields is an important ingredient in describing the magnetic field induced exciton effective mass renormalization, which was not captured in earlier works. We note that the MSLA can also be used to solve the exciton Schrödinger equation coupled to Maxwell's equations in order to model microcavity polaritons. 34 However, modeling polaritons in magnetic fields is beyond the scope of this paper and forms the subject of future study.
In Section II, we describe the Schrödinger equation for a CQW exciton in the presence of electric and magnetic fields pointing in the growth direction and provide details of the MSLA. The perturbative approach used to obtain the electric and magnetic field dependence of the exciton effective mass is outlined in Section III. In Section IV, we present calculations of exciton states and their associated properties for 8-4-8 nm GaAs/AlGaAs CQWs. Comparisons with the 2D methods are made in Section V where we show that our method converges to these descriptions in the limit of zero QW thickness. Conclusions are made in Section VI.

II. EXCITONS IN MULTIPLE QWs IN ELECTRIC AND MAGNETIC FIELDS
To calculate the bound exciton states in a semiconductor nanostructure with external bias and magnetic field oriented normal to the QW plane, we consider the following Hamiltonian: Here, r e,h are the electron and hole coordinates. For a multi-layered heterostructure, we adopt a cylindrical coordinate system (ρ, φ, z) with the z-axis along the QW growth direction. The electron and hole effective mass tensors,m e,h (z), are composed of in-plane and perpendicular components, m || e,h and m ⊥ e,h (z), respectively. As in Ref. 9, we neglect here any z-dependence in the inplane electron and hole effective masses, which is justified by relatively low mass contrast in the heterostructures treated here and a minor contribution to the exciton problem of the electron and hole wave functions outside the well regions.m e,h (z) are layer-dependent step functions, as are the QW confinement potentials V QW e,h (z). The confinement potential and the external bias are included in the potentials U e,h (z) = V QW e,h (z) ± eF z where F is the electric field. For the band gap E g and in-plane masses m || e,h , values for the QW layers are used. ε is the average permittivity of the structure. For a perpendicular magnetic field (along the growth direction), one takes the vector potential, using the symmetric gauge, in the form A(r) = 1 2 B × r. It can be shown 45,46 that the solution to the equation H 0 Ψ = EΨ has the following variable-separable form: with the center of mass motion of an exciton carrying 2D momentum P described solely by where are the 2D in-plane center of mass and relative coordinates, respectively, M x = m || e + m || h is the in-plane exciton mass, and m is the exciton magnetic quantum number. Such a substitution enables removal of the center of mass coordinate from the problem. The exciton relative motion is then described by the HamiltonianĤ P where the Hamiltonians of the electron and hole perpendicular motion arê These describe the Coulomb-uncorrelated single particle electron and hole states in the absence of a magnetic field, with the wave functions ψ e,h q (z), where the index q labels the electron (hole) states quantized in the CQW heterostructure potentials. The Schrödinger equationŝ H ⊥ e,h ψ e,h q = E e,h q ψ e,h q are solved numerically using the shooting method with Numerov's algorithm. Figure 1 illustrates these solutions showing the ground and first excited electron and hole states for finite electric fields in a GaAs/AlGaAs CQW.
The kinetic term in Eq. (7),K(ρ), is given bŷ where µ is the in-plane reduced exciton mass, 1/µ = 1/m e + 1/m h . The potential due to the magnetic field is where 1/κ = 1/m e − 1/m h is the inverse mass difference of the electron and hole, called the exciton magnetic dipole mass. 43 The Schrödinger equationĤ 0 x |k, m = E k,m |k, m , in which |k, m is an exciton state in a CQW in finite electric and magnetic fields, is solved using the MSLA developed in Ref. 9. The principle of the method is that we expand the exciton wave function ϕ(ρ, z e , z h ) in Eq. (5) into the set of Coulomb uncorrelated electron-hole pair states leading to: where Φ n (z e , z h ) = ψ e pn (z e )ψ h qn (z h ). The mapping n → (p n , q n ) leads to N e N h pair states formed from N e electron and N h hole states. We then calculate the radial components of the wave function φ k,m n (ρ), using a matrix generalization of the shooting method with Numerov's method incorporated in the finite difference scheme. This solved the problem of the electron and hole in-plane and perpendicular motion mixed by the Coulomb interaction, leading to the exciton quantization here labeled by the index k. The numerical solution is generated on a logarithmic grid in the range ρ ∈ [ρ 0 , R]. A sufficiently small inner limit ρ 0 is chosen so that the wave functions and extracted data have converged with respect to decreasing ρ 0 . The outer limit R of the wave functions is sufficiently large that all radial components asymptotically approach zero. The existence of such a limit is guaranteed for nonzero B as the potential V B grows with ρ 2 . This is in contrast to the case of B = 0 where truncation of the solution range leads to a discretization of the unbound e-h continuum states. For the range of electric fields considered (F < 25 kV/cm), it was previously found for B = 0 that high accuracy is achieved using N = 2 electron and hole states. 9 The state with zero angular momentum has radiative linewidth Γ The oscillator strength, f (k) , is calculated from the overlap of the electron and hole wave functions, where d cv is the dipole matrix element between the conduction and valence bands.

III. EXCITON MASS RENORMALIZATION IN A MAGNETIC FIELD
For an exciton with a finite center of mass momentum P = 0, the Hamiltonian Eq. (7) is modified tô where the perturbationV P is given by (see Appendix A) and P = |P|. To calculate the exciton mass renormalization in a magnetic field, we treat P as a small parameter and use the perturbation theory up to 2nd order. The 1st term of Eq. (14) contributes to the exciton energy in 1st order only leading the bare mass, while the 2nd term, vanishing in 1st order, gives rise to a quadratic in P correction in 2nd order and is thus responsible for the mass renormalization. Neglecting non-parabolicity of the exciton band, which would be accounted for by treating the 2nd term in higher perturbation orders, we find the correction to the exciton energy proportional to P 2 and renormalized effective mass M * k,m of exciton state |k, m in magnetic field: (15) Here the index j counts over the eigenstates with the angular momenta m ± 1 which only contribute in 2nd order and which are calculated at given values of electric and magnetic field. Details of calculation of the matrix elements in Eq. (15) are given in Appendix B.
The advantage of our approach compared to some previous calculations of the exciton mass renormalization is that the perturbation is in P only allowing the B-field to be arbitrarily large. In contrast, the approach developed in Ref. 42 used the magnetic field as a small parameter of the perturbation theory. Consequently, the applicability of the latter is restricted to low magnetic field (up to ≈ 2 T in the structure considered in this paper, see also comparison in Sec. V). Another significant benefit of our approach is that the full 3D solution of the exciton Schrödinger Eq. (7) describes the inter-well coupling that is neglected by any 2D approximation. 42,43 IV. APPLICATION TO GaAs/AlGaAs CQWs All results in this section refer to the structures studied in Refs. 1,2,8,12,18,19,21 which consist of symmetric GaAs/Al 0.33 Ga 0.67 As QWs with barrier and well thickness of 4 nm and 8 nm, respectively. A complete list of parameters used in the calculations are given in Table I. Although we restrict the present analysis to two QWs, the method is applicable to any number of QWs since one can always obtain, either analytically or numerically, eigenstates of the single particle electron and hole Hamiltonians in Eq. (8). Fig. 2(a) shows the optical transition energy for different exciton states as a function of electric field at B = 10 T. For each state, the circle area is proportional to the oscillator strength. One can identify four different exciton states with each state being duplicated in each of the first three Landau levels. Firstly, we see a pair of bright states at E x − E g ≈ 51 meV whose energy is almost independent of electric field. These are the first Landau level (L 1 ) direct exciton states occupying adja-  Fig. 2(a). Secondly, we see a pair of darker states, one whose energy decreases and one whose energy increases with electric field from E x − E g ≈ 60 meV at F = 0. These are the L 1 indirect exciton states. Their transition energy dependence on electric field is almost linear with the gradient corresponding to the nominal distance between the QWs. The state with energy that decreases linearly with F is composed predominantly of the ground electron and hole states e1h1 shown in Fig. 1. The other is composed predominantly of the excited electron and hole states e2h2. The corresponding L 2 and L 3 replicas of these four states appear at higher energies. The spacing between the Landau levels for each type of exciton is approximately equal.
In the previous work using the MSLA, 9 which did not include magnetic field, it was necessary to calculate a large number of unbound e-h pair states in order to determine the exciton spectrum. These states were discretized due to a finite domain of the solution which imposed an infinite potential barrier at ρ = R. It was found that with increasing R, the density of such discretized continuum states increased while their oscillator strengths decreased. The absorption spectrum converged for sufficiently large R. In the present case, however, the magnetic field removes any need to consider the continuum states. The potential due to the magnetic field, Eq. (10), contains a term in ρ 2 which naturally closes the domain of the solutions. Therefore, a finite value of R can always be found such that the wave functions are sufficiently small there. This point explains the apparent cleanliness of Fig. 2(a) which does not contain any unbound e-h pair states. Fig. 2(b) shows a magnification in the region of anticrossing between the L 1 direct and indirect exciton states, labeled in Fig. 2(a). We identify the electric field at the anti-crossing F a-c as the value of F where the minimum separation occurs between the upper and lower black curves. This is shown in Fig. 2(c) as a function of B for the first three Landau levels. The magnetic field provides a means to tune F a-c over several kV/cm. The corresponding energy splitting between the upper and lower black curves in Fig. 2(b) is about 2 meV and varies by less than 10% in the investigated magnetic field range.
Using the Lorentzian model of absorbing oscillators, the absorption spectrum is calculated from the transition energy and linewidth of each exciton state. The magnetic field dependence of the absorption spectrum is plotted in Fig. 3. For clarity, the spectrum has been broadened by convolution with a Gaussian of full width at half maximum of 1 meV. Most clearly visible is the Landau fan of the direct exciton states. These bright spectral lines are due to the two direct exciton states overlapped in en- ergy. Fainter fans are seen for the indirect exciton statesone starting from about 20 meV and another from about 80 meV in Fig. 3(b). The lower (upper) fan corresponds to indirect excitons composed mainly of an electron and a hole in the ground (excited) state, e1h1 (e2h2).
A number of properties of each eigenstate can be calculated from the exciton wave functions Eq. (11). In Fig. 4, we focus on the characteristics of the ground state of the system. At zero magnetic field, the results presented here agree 47 with those of Ref. 9. Fig. 4(a) shows the electric and magnetic field dependence of the in-plane Bohr radius, r B = ρ 2 . Increasing the electric field increases r B due to a reduction in the e-h Coulomb interaction which is caused by their increased separation in the z direction. Increasing the magnetic field has the reverse effect and shrinks the wave function in the QW plane. This shrinkage enhances the e-h interaction which, in turn, leads to a more likely localization of the electron within the same QW as the hole and thus the ground state becomes a direct exciton. This is seen in Fig. 4(b) where a sharp reduction in the dipole moment | z e − z h | takes place for increasing B. The value of B at which this transition occurs increases with F as a stronger confinement of the in-plane wave function is needed to sufficiently enhance the e-h interaction and to induce the indirect-todirect transition. In the indirect exciton regime at high electric field (F = 24 kV/cm), the dipole moment corre- sponds to approximately the center-to-center distance of the QWs which is 12 nm.
From the radiative linewidth, the radiative lifetime of the ground state exciton, τ R = /(2Γ (0) R ), is found. Fig. 4(c) shows τ R as a function of electric and magnetic fields. τ R is inversely proportional to the overlap of the electron and hole wave functions (see Fig.1). Therefore, we find a decrease in τ R with decreasing F or increasing B, consistent with the shrinking of the exciton radius shown in Figs. 4(a) and (b). We note that in experiments, 48 contrary to our results, the radiative lifetime of QW excitons is found to increase with increasing magnetic field. This, however, can be explained by considering the intrinsic QW disorder potential. Formally, Eq. (12) describes to the case of perfect QWs with no defects. Compared to this result, weak localization of the exciton to minima in the disorder potential acts to reduce the oscillator strength, as the exciton center-of-mass motions spreads in momentum space leaving some part of the wave function outside the radiative light cone. As the exciton effective mass increases (see Fig. 5) the localization increases, also increasing the non-radiating part of the wave function. This reduces the oscillator strength and extends the radiative lifetime. Quantifying this effect is beyond the scope of the present study as it is highly dependent on the form of the disorder potential which is specific to the structure's growth conditions. = E e 1 + E h 1 is the energy of the unbound e-h pair state in zero magnetic field, and the last term takes into account the Landau quantization energy of the free pair in magnetic field. The decrease in the size of the exciton, caused by either increasing B or decreasing F , leads to an enhancement of the e-h interaction. This causes a tighter binding of the exciton and, therefore, an increase in E b .
The effective exciton mass, M * k,m , is calculated using second order perturbation theory (see Section III). Fig. 5 shows the electric and magnetic field dependence of the effective mass, M * 0,0 , for the exciton ground state. We identify two limiting cases: Firstly, for small electric fields (F < 3 kV/cm) where the exciton ground state is predominantly direct, there is a monotonic increase of the effective mass with increasing magnetic field. The rate of this increase is weakly dependent on the electric field. Secondly, for electric fields greater than 3 kV/cm, there is an initial increase in the effective mass with increasing magnetic field, followed by an eventual decrease. At higher magnetic fields, the effective mass asymptotically approaches the direct exciton effective mass.
To interpret the data presented in Fig. 5, we use the classical analogy introduced in Section I. The exciton is approximated as two oppositely charged masses at an equilibrium separation a B = r 2 B + z e − z h 2 with center of mass motion perpendicular to the magnetic field. The Lorentz force F L = eBv/c acts on each particle to separate the charges but is balanced by the Coulomb restoring force F C that is approximately linear for small changes in separation (corresponding to small velocity and small Lorentz force): F L = F C ≈ k R ∆ρ. In this picture, one derives from the energy dependence on the exciton velocity v, M x v 2 /2 + k R (∆ρ) 2 /2, an estimate of the effective mass, M * cl = M x +e 2 B 2 /(c 2 k R ). The 'spring constant' k R is the coefficient of the restoring force. It can be approximated via linearization of the slope in the Coulomb potential at ρ = r B which gives The inset in Fig. 5 shows this classical calculation using the parameters r B and z e − z h 2 characterizing the exciton taken from the full calculation, see Figs. 4(a) and (b). The curves for F = 3 kV/cm and 6 kV fully describe the main trends seen in the full calculation and even demonstrate to some extent a quantitative agreement. For larger electric and magnetic fields this classical model fails because the spring constant k R becomes too small and the linearization of the restoring force used in the model no longer works.
The classical model allows us to understand the exciton mass dependence in Fig. 5. Indeed, from the calculation presented in Fig. 4, we find a greater k R for direct excitons than indirect excitons. Qualitatively, this is because the indirect exciton has a weaker binding due to the increased spatial separation of the electron and hole. The difference in k R leads to a greater rate of effective mass increase with magnetic field for indirect excitons than for direct excitons. This explains the effective mass dependence in the low and high electric field limits in Fig. 5. At intermediate fields (6−12 kV/cm), the initial increase and then decrease of the effective mass is a consequence of the ground state tending from an indirect to a direct exciton. This happens due to a shrinkage of the exciton wave function caused by the magnetic field. In turn, the e-h Coulomb attraction is enhanced, making the direct state lower in energy than the indirect state.

V. COMPARISON WITH THE 2D LIMIT
In order to verify our model, we make comparisons with previous results in which the limiting case of 2D excitons were treated. To make this comparison, we include just one electron and one hole state [N e = N h = 1 in Eq. (11)] taking the wave functions in the form |ψ e (z)| 2 = δ(z) and |ψ h (z)| 2 = δ(z − d z ), where δ(z) is the Dirac delta function and d z is the nominal distance between the QWs. Technically, this is done using normalized Gaussian functions for ψ e,h (z e,h ), for the electron and hole centered on 0 and d z , respectively, and taking the limit of their widths to zero.
First, we consider the mass renormalization of 2D direct excitons in single QWs (d z = 0). This was calculated by Arseev and Dzyubenko,42 and the analytical expression for the exciton mass, which was derived by treating the magnetic field as a perturbation, was given as 49 This is relevant in the low magnetic field limit where the magnetic length is much greater than the exciton Bohr radius, l B = c/eB a x = ε 2 /µe 2 . For the considered system, this corresponds to B 2.6 T. Fig. 6(a) shows the inverse of the effective mass renormalization as a function of B 2 calculated numerically via Eq. (15) and using the analytic Eq. (17). There is a strong agreement between the different approaches for small B. For increasing B above 1T, the results depart as expected, since the magnetic length approaches the Bohr radius.
We also compare our results with the work of Lozovik 43 which treated the case of 2D indirect excitons in CQWs. In that approach, the exciton Schrödinger equation was solved with a finite center of mass momentum that enabled calculation of the dispersion relation. The effective mass was then extracted from a parabolic fit to the bottom of the dispersion band. The mass calculated by this method is shown in Fig. 6(b). To make a suitable comparison, we calculate the 2D effective mass using the same parameters as in Refs. 43 and 40: d z = 11.5 nm, µ = 0.049 m 0 , κ = 0.11 m 0 , and ε = 12.1. We also show the effective mass dependence on magnetic field as was determined experimentally 40,43 using an in-plane component in the magnetic field to map the indirect exciton dispersion by shifting it with respect to the photon cone. The discrepancy between the full calculation (Fig. 5) and the experimental data could be due to the limited accuracy of the effective mass measurement which entails fitting a parabola to an almost flat dispersion curve. On that basis, our data is in good agreement with the experimentally measured exciton effective mass as well as the calculation done in Ref. 43.

VI. CONCLUSIONS
We have studied the combined effect of perpendicular electric and magnetic fields on exciton states in coupled quantum wells. This was done using the rigorous multi-sub-level approach (MSLA) to solve the full exciton Schrödinger equation in the effective mass approximation. In this approach, the exciton wave function is expanded into Coulomb uncorrelated electron-hole pair states and found using a matrix generalization of the shooting method with high stability and accuracy provided by Numerov's algorithm. We calculated the optical transition energy and linewidths for different exciton states and obtained the electric and magnetic field dependence of the exciton absorption spectrum. For the exciton ground state, we evaluated the in-plane Bohr radius, static dipole moment, radiative lifetime and binding energy. Furthermore, we have calculated the magnetic field-induced exciton mass renormalization in arbitrary electric and magnetic field, by treating the exciton momentum as a small perturbation. In particular, our calculations demonstrate a non-monotonous behavior of the exciton effective mass in magnetic field which is under-stood in terms of the direct-to-indirect exciton transition and also illustrated in a simple classical model. Our results for the quantum well width tending to zero are in good agreement with different 2D approaches available in the literature. However, in contrast to the 2D approaches, the full MSLA captures the direct-to-indirect transitions of the exciton ground state for varying electric and magnetic fields which are necessary for a full and detailed description of the system.
Using the factorizable form of the wave function Eqs. (5-6) we make a transformation of the full Hamiltonian Eq. (1) removing the in-plane center of mass coordinate from the problem. The exciton Hamiltonian then takes the form H P x (ρ, z e , z h ) =Ĥ ⊥ e (z e ) +Ĥ ⊥ h (z h ) +Ŵ P B (ρ) + E g +V C ρ 2 + (z e − z h ) 2 , whereĤ ⊥ e,h and V C are given by Eq.
For brevity of notation, we have introduced in Eq. (A4) γ = e/ c, A e,h = A(ρ e,h ), and ∇ e,h = ∇ ρ e,h , where ∇ ρ is a 2D nabla-operator. For the same reason, we also drop below the index P in χ P , x in M x , and in m e,h . Taking an arbitrary function f (ρ) we obtain from Eq. (A4): −2iγA e ·(χ∇ e f + f ∇ e χ) + γ 2 A 2 e χf Noting that ∇ e f = ∇f and ∇ h f = −∇f , Eq. (A5) can be written aŝ Finally, noting that obtain ∇ e,h χ = i m e,h M P ∓ γA h,e χ, and after some algebraic simplifications using A(ρ) = A e −A h , find the required termŴ P B of the exciton Hamiltonian Eq. (A2): This term is the only one in Eq. (A2) which carries in the dependence on the magnetic field B and the exciton in-plane momentum P. For the exciton wave function in the form of Eq. (5), with the split-off angular motion, the exciton Hamiltonian is further reduced to Eq. (7) for P = 0 and to Eq. (14) for P = 0.