The presence of a strong, inherent polar electric field in GaN [1] causes the well-known quantum confined Stark effect [2–4] (QCSE) regarding carrier behavior inside a QW (Fig. 1a). The separation of the center of charge between electron and hole wave functions, caused by the polar E-field, reduces mutual overlap and the related emission probability. The lowering of the confined energy levels, relative to the unperturbed square QW, causes red-shifting of the emitted radiation during electron-hole recombination. This effect has been the subject of extensive perturbative [5] as well as non-perturbative analytic treatments [69], including excitonic effects [1014]. In general earlier analytic theories neglected the modifications to the (intrinsic polar or externally applied) E-field caused by the charge separation and the resulting dielectric shielding, assuming in effect very low carrier densities.

Figure 1
figure 1

a Internal polarization field causes separation in the carrier wave function centers and charge separation. b As carrier density increases the electric field is shielded (reduced) at the center of the well and most of the potential drop occurs near the edges. Wave fucntions are localized at the edges. The energy level separation increases (blue shifts) with increasing wave function confinement (constriction). c At even higher densities the electric field is completely shielded at the center and the voltage drop is localized at nanometer-width charged layers (plasma sheaths). Eventually the energy level is pushed above the edge-well depth Vo and the wave function expands to occupy the entire QW width, for a complete “rectification” of the QCSE

At high carrier densities, charge separation and dipole field formation is sufficient to cause shielding of the intrinsic polarization E-field [15]. The resulting potential gradient across the QW is not uniform, and most of the potential drop is localized across charged layers formed at the edges of the QW (Fig. 1b). The electric gradient scale is of the order of the Debye length. For densities near 1019 cm−3 the Debye length shrinks down to nm-scale (Fig. 1c), and the potential drop is mostly localized at the QW edges while the QW interior is nearly field-free (shielding of the intrinsic E-field). This constitutes the ES-shielded QCSE. It has been anticipated [16] that the shielding of the interior E-field would reduce or even eliminate the QCSE at densities 1019 cm−3. Detailed numerical simulations, employing the self-consistent Poisson–Schrodinger equations [17] have showed that a much higher than expected carrier density, near 1020 cm−3, is required to eliminate the QCSE for QWs wider than 5 nm. This has been attributed to the persistence of carrier confinement in the potential dips at the QW edges, even when the electric field is screened out from the middle. However, an analytic treatment examining the carrier behavior in the ES-shielded QCSE is so far lacking.

This study focuses in finding solutions for the confined carrier wave functions by solving the one-particle Schrodingers’ equation. To gain insight the following simplifying assumptions are used: (a) The shielded potential has exponentially decaying profile on the Debye length ∼λD scale; (b) the peak-to-peak shielded voltage is a given function of the carrier density and the intrinsic polarization strength and (c) excitonic effects are ignored.

The shielded potential results from a self-consistent solution of Poisson’s equation for point-like charges obeying Fermi statistics [15]. Neglecting the charge spreading of the carrier wave function is not too severe when the carrier localization length ∼λD is much smaller than the QW width L. When the Fermi level separation from the lowest occupied levels is much larger than κT, i.e., for nearly Maxwellian distributions, the shielded potential is well approximated by a symmetric profile The exponentially decaying profiles remain a reasonable approximation for Fermi–Dirac distributions in general.

We obtain results based on: (a) a second order perturbative expansion; (b) non-perturbative series expansion; and (c) a numerical solution of Scrodinger’s equation for the carrier envelope wave function. The analytic expressions for the energy levels from (a) are evaluated against numerical the results from (c). The infinite λ D , zero shielding limit reverts to the original (unshielded) QCSE results.

Our analytic models find that increasing the carrier density causes an increase (blue shifting) of the energy levels relative to the unshielded (red-shifted) QCSE values. The confined energy levels asymptote to the values for a flat square QW, and the red shift is effectively eliminated, for densities ≥ 1019cm−3. The perturbative energy levels agree with the numerical values at lowVp, and become inaccurate when the polarization voltage exceeds the energy of the fundamental confined mode in a square QW. Numerical solutions of the Schrodinger equation for high polarization, relevant to GaN parameters, show that at high Vp the perturbation results overestimate the energy level shifts by a factor of 2, but they provide the correct trends over the entire range.

The dependence of the characteristic emission time on the carrier density is computed based on the numerically evaluated eigenfunctions. Despite the adopted simplifications these results reproduce the three order of magnitude increase in the emission rate between densities 1019 and 1021, leading to a complete rectification of the QCSE, as was reported from experimental and detailed computations in Ref. [17].

Interestingly, it is found that elimination of the QCSE-related energy red-shift clearly precedes the recovery of the radiative emission time: the energy red-shifting is gradually eliminated between densities 1017cm−3 and 1019cm−3 while the emission probability is restored at higher densities between 1019cm−3 and 1020cm−3. The first result agrees with the energy recovery behavior obtained in [16] while the emission probability behavior agrees with the results in [17]. The delay in the restoration of the emission probability is explained in terms of carrier trapping at the QW edge.

QW Eigen Modes with ES-shielded Polar Potential

We investigate the wave function profiles and the structure of the energy spectrum inside QWs in the presence of an ES-shielded polarization potential. It can be shown (Appendix1) that the self-consistent charged layer (plasma sheath) potentials can be reasonably approximated by exponentially decaying


where κD = aD scales as the inverse Debye length and a is of order unity. The peak amplitude Vo here is taken equal to half the intrinsic "polarization voltage" The value Φp(0) = 0 at mid-point equals the bottom energy for a polarization-free square well (Fig. 2), and serves as the reference point for electron energy levels. Hole levels are measured from the bottom of the valence well. The above symmetric potential applies for low carrier density and a Fermi level near the mid bandgap. For high doping the reference point xo defined by Φp(xo) = 0 moves closer to the left (right), with unequal edge potentials −Vp(−L/2) > Vp(L/2) (−Vp(−L/2) > Vp(L/2)) for N-doped (P-doped) materials. For analytic simplicity this study will retain the symmetric potential.

Figure 2
figure 2

a Profile of a QW conduction band with a ES-shielded polarization field for characteristic shielding distance (Debye length) λD = 8L, L/2, L/6, L/10, L/20, longer to shorter dash lines. b Energy correction (meV) versus LD, for the lowest five QW modes with Vo = 25 meV and QW width L = 8 nm. c Same versus carrier density N corresponding to λD

Expressing the slowly varying envelope wave function in separable coordinates as casts the 1-D Schrodinger’s equation along x as


where is the net energy contribution from the motion across the well, and k y , k z correspond to the continuous spectrum along the QW. Analytic solutions of (2) are obtained from second order perturbation theory, in terms of an expansion in unperturbed square well modes




A change of variable transforms the integral in the rhs of (4) into


Substituting inside (3) yields


In the zero-shielding, infinite Debye length limit when one recovers the unshielded QCSE levels


The mode energy E n is always measured relative to the middle of the well; the latter always coincides with the bottom energy for the square (un-biased) QW, as shown in Fig. 1.

The shift in energy levels relative to the square QW eigen values, obtained from (7), is plotted in Fig. 2a versus the ratio κDLLD for the lowest three modes. The chosen parameters are peak-to-peak sheath potential 2Vo = 50 meV, QW width L = 8 nm and me*/me = 0.19 for GaN. For λDL/2 the polarization field is nearly unshielded, the potential profile nearly linear, and the red-shifting hovers near the maximum value, characterizing the ordinary QCSE. Red shifting is however reduced rapidly as the screening range becomes equal or shorter than half the QW width, λDL/2, becoming completely negligible at λD < L/4. Beyond this point the energy levels revert to the square QW eigen values and the QCSE is completely "rectified". Using the scaling with the value for the GaN dielectric constant recasts energy shift Fig. 2a in terms of the carrier density Ne, Fig. 2b. Complete shielding of the QCSE occurs at Ne ≥ 1020 cm−3. This value agrees well quantitatively with similar results obtained in [17], based on the observed decrease in the radiative emission time.

As expected, perturbation theory breaks down when the polarization potential exceeds the unperturbed (square QW) energy eigen values e VoE1(0) ∼31 meV. Since the combined inherent and strain-induced polarization fields can reach values up to 5 MeV/cm [18] and up to 2.5 V over a 10 nm QW, numerical solutions of Schrodinger Equation are required for realistic polarization values. For comparison Fig. 3 plots the lowest energy levels obtained from numerical solutions (points) and perturbation theory (curves) versus the ratio L/2λD for Vo = 0.250 V. For unshielded or partially shielded QCSE with λDL/4 the perturbation theory overestimates the red-shift by a factor of 2. Good agreement occurs for λD < L/8 when the charged layer thickness is much smaller than the QW thickness, and thus the size of the perturbation, parameterized by becomes negligible.

Figure 3
figure 3

a Numerical (points) and theoretical energy values (lines) for the lower two eigen modes versus LD for Vo = 0.500 eV. b Numerical energy values for the lower three eigen modes versus LD for Vo = 2.05 eV

It is useful, for the discussion that follows, to obtain an analytic estimate of the carrier energy eigen values for arbitrary Vo and λD. To that end the eigenfunctions of Eq. 2are obtained in terms of an infinite power series expansion a la Frobenius, Appendix1. The fast convergence of the series solutions allows the calculation of the expectation values of the kinetic energy potential energy 〈e Φ(x)〉 and the total energy expectation value, yielding


where K n , W n are functions of eVoT and the quantum number n, and Co is the wave function normalization constant. The kinetic energy increases with decreasing λD, while the potential (“edge-binding”) energy is fixed. For eVo > 5κT the ratio W1/K1 for the fundamental mode is nearly constant and hovers close to 1/2, Appendix 1.

The reduction of the red shift with increasing ES shielding and decreasing shielding distance λD, manifested experimentally as a blue shift relative to the unscreened QCSE, is qualitatively understood as following. For λD < L/2 the sin h(xD) potential behaves like an edge-well inside the square well, instead of a tilted QW floor. If confinement within the edge-well occurs, the lowest energy level must satisfy 〈E1〉 ≤ 0. As long as the confined “kinetic energy” is less than the edge-binding energyeVoW1 then E1 < 0 and the wave function is trapped at the QW edge. Edge-confinement within a range shorter than the well width, λD < L/2, increases the mode energy relative to that for a tilted QW bottom and causes blue shift relative to the unshielded QCSE. The blue-shift increases with increasing carrier density, meaning shorter confinement length λD. Eventually, for large enough density with the kinetic energy exceeds the edge-binding energy and 〈E1〉 > 0, edge confinement ceases, and the wave function shifts to the center to occupy the full QW width. At the same time most of the well bottom becomes nearly as flat as in a square well, since is excluded from most of the interior. Full “rectification” of the QCSE occurs and the eigen values and eigen modes approach that of a square QW.

Transition from edge-confinement to full QW occupation occurs for either Vo < Vth or where is the threshold under given λD, and the threshold under given Vo. This transition is shown in Fig. 4a and b, plotting the fundamental mode profiles Ψ(x) for various values of λD/L, for low and high voltages, respectively Vo= 0.250 V and Vo= 2.05 V. As the screening distance decreases, the center of the wave function moves from the left edge towards the center of the well. The transition to full QW occupancy occurs at shorter screening length λD for higher Vo (Fig. 4b).

Figure 4
figure 4

Normalized wave function profiles (a.u.) for various values λD/L as marked and for: a Vo = 0.25 eVb Vo = 2.05 eV. Transition from edge-trapping to full QW occupation occurs at shorter λD (higher carrier density) for higher polarization voltage

Figure 5a plots the lower two eigen values versus sheath potential, for given λD = L/8. The fundamental E1 becomes positive at about For Vo < Vth the value E1 increases and tends to the square well limit as Vo ≃ 0. Figure 5b shows the fundamental eigen value E1 versus LD for two different voltages Vo. The eigen values asymptote to the square QW limit at shorter screening distance for the case of higher polarization Vo.

Figure 5
figure 5

a Energy levels for the lower two eigen modes versus Vo for fixed λD = L/8 b Fundamental level versus LD for two polarization voltages Vo = 0.250 V and Vo = 2.05 V, corresponding to polarization values MV/cm and MV/cm respectively

Radiative Emission Probability

The changes in the wave function profiles have a profound influence in the eh transition probability during radiative emission, proportional to the dipole moment overlap integral


where are the lattice-periodic parts and the slowly varying envelope functions obtained from (2). Employing, as usual, the space-scale separation between the rapidly varying, on the lattice-constant scale, u c u v , and the slowly varying envelopes, valid for as long as L, λDa, the above is approximated by


Orthogonality among the lattice functions u c u v was used in arriving at (10). The last integral over the unit lattice unit cell volume C is independent of the polarization. For “vertical transitions” with (given that ) the dependence on the polarization voltage Vo and screening distance λD is carried entirely in the overlapping between electron-hole envelopes


with a constant. Here we will assume, due to the symmetry in the sinh potential, that Ψ h (x) = Ψ e (Lx). Taking the transition probability for a flat QW with as reference, and since the emission time one has


The ratio is potted in Fig. 6a versus LD for various peak voltages Vo, using the wave function profiles obtained from numerical solutions. Characteristic emission times tend to increase with increasing applied polarization voltage Vo, and decrease with decreasing screening distance λD. The results of Fig. 6a are plotted verusus the corresponding carrier density N in Fig. 6b, for QW width 8 nm. These results reproduce the three order of magnitude emission increase between densities 1019 and 1021, resulting in complete rectification of the QCSE, that was first obtained using detailed Poisson–Schrodinger simulations in Ref. [17] for a 7 nm QW.

Figure 6
figure 6

a Ratio of radiative emission time for a flat QW to that of the ES-shielded QCSE versus screening distance LD, for low and high polarization voltages b same plotted versus corresponding carrier density N for an 8 nm QW

A careful comparison between the energy blue-shifting with increasing density (screening), Fig. 7a, and the decrease in recombination time, Fig. 7b, shows that the rectification of the QCSE red-shift occurs before the recovery of the radiative emission time: the energy red-shifting is gradually eliminated first, between densities 1017cm−3 and 1019 cm−3, though the radiative emission time remains almost constant there. The emission probability is restored, rather abruptly, at higher densities between 1019 cm−3 and 1020 cm−3. This lagging in restoring the emission probability is explained via edge-carrier trapping, mentioned in the previous discussion. As carrier density increases and the edge-potential range λD narrows down, the increasing edge-confinement of the wave function causes the energy level to increase. As long as the “confinement energy” is smaller than the edge potential depth e Vo electron and hole wave functions remain edge-localized and no significant change in overlap and in recombination time occurs. The abrupt decrease in the radiative emission time (increase in the radiative emission rate) occurs after since at this point the wave function moves from edge-confinement to full QW occupancy. Practically this means that the QCSE-related energy red-shift has already been eliminated before the radiative emission time recovers. This behavior agrees with the results in [17].

Figure 7
figure 7

Comparative evolution of a lowest confined mode energy and b recombination time versus carrier density N, for an 8 nm thickness QW

Shielding of the Peak Polarization Voltage

It has so far been tacitly assumed that the charged layer peak-voltage Vo is independent of the screening carrier density Ne, h and the peak-to-peak voltage 2Vo was taken equal to the “polarization voltage” for an unscreened QW, Fig. 2a. In other words the shielding only modified the potential profile across the QW. However, for given applied and L, the shielded Vo does depend on the carrier density, and in fact Vo is reduced below Vp at high carrier densities. The shielding of the peak voltage is summarized below, based on results from earlier studies [15].

Self-consistent charged layer solutions under Fermi–Dirac thermodynamic equilibrium [15] show that as the QW thickness L increases well beyond λD the peak-to-peak voltage asymptotes rapidly to a maximum saturation value Figure 8a plots 2Vo versus L for various polarization strength values and shows the saturation for LD ≫ 1. Clearly Vs increases with polarization strength The dependence of Vs on density is given in Fig. 8b. The fact that Vs decreases with increasing density stems from Gausses law: it takes a given amount of surface charge to screen a given field. Applying scaling arguments the charge layer thickness is (half of the electric field screened at each QW edge) and the sheath voltage Thus for given polarization the voltage Vs scales roughly as when L > 2λ D .

Figure 8
figure 8

Carrier density effects on the shielded voltage. a peak-to-peak voltage versus QW thickness for doping density ND = 1018 cm−3 and various polarization strengths, as marked b Saturated peak-to-peak voltage versus doping density ND for various polarization strengths c ratio of peak voltage to the polarization potential versus QW thickness for doping density ND = 1018 cm−3

The screened voltage value is always less or equal to the intrinsic “polarization voltage”, This is shown in Fig. 8c, plotting the ratio of the peak-to-peak voltage 2 Vo to Vp, versus sheath length, for given doping density ND= 1018 cm−3. For as long as L≤ 2λD one has unsaturated behavior Once saturation is reached for L > 2λD the peak-to-peak voltage is pinned at Vs, independent of L. This is because when L > 2λD the polarization field is screened-out from the QW interior length L− 2λD that yields a negligible contribution to the voltage difference; Vs comes entirely from two charged layers of width λD. Hence, for wide QWs the peak-to-peak voltage turns out much smaller than the polarization voltage, and the ratio 2Vo/Vp goes as 1/L. Notice that the saturation length Ls where 2Vo dips below Vp depends also on the field strength; letting and yields thus saturation occurs at smaller QW thickness with increasing . According to Fig. 8c, one may apply unsaturated values for QW thicknessL < 10 nm and for up to doping densities 1019 cm−3. This is illustrated in Fig. 9, plotting the ratio 2Vo/Vp versus doping density ND for fixed QWL = 8 nm and for various strengths

Figure 9
figure 9

ratio of peak voltage to the polarization potential versus doping density ND in a QW of thickness L = 8 nm, for various polarization strengths

For given L = 8 nm, the values 2Vo assume their saturation values and the shielded voltage falls significantly below Vp when doping densities exceed ≥1020 cm−3. This is illustrated in Fig. 10, showing the screened potential profiles, 10a, and electric fields, 10b, for various doping levels ND across an 8-nm QW for The peak-to-peak voltage decreases well below Vp with increasing ND. In addition, the electron and hole charged layers become asymmetric: Ve across the negative charged layer is different than Vh across the positive charged layer. In general, reduction of the peak-to-peak voltage, as well as asymmetric electron-hole profiles should be considered for a more accurate description of the ES shielded QCSE. In particular, the drop in Vs < Vp with increasing density could accelerate the cancellation of the QCSE and the blue shifting of the energy levels. For the relevant to our GaN experiments parameters, however, the red-shifting is all but cancelled out at density 1019 cm−3, just before such effects become significant. Thus it appears that energy level blue-shifting caused by the sin h effect in the potential profile cancels to a large degree the QCSE effect, before shielding of the peak amplitude itself becomes important.

Figure 10
figure 10

a Self-consistent shielded potential profiles across an L = 8 nm QW for intrinsic polarization field 0.7 MV/cm, for various carrier densities as marked. b Corresponding shielded electric field profiles


A simplified model employing ES-shielded, exponentially-decaying polarization potentials localized at the QW edges, was employed to study the QCSE at high doping densities. Blue shifting of energy levels relative to the unshielded QCSE occurs with increasing carrier density, due to the wave function constriction within scale length λD < L/2. When the “edge-localization energy” exceeds the peak-voltage of the charged layer eVo the wave function center shifts to the middle of the QW and behavior becomes similar to that of a square (unbiased) QW. In addition, at very high doping the shielded peak voltage is reduced well below the original unshielded “polarization voltage” Vp. Both effects cause gradual elimination of the QCSE red-shifting, an increase in the eh wave function overlap and a decrease of the radiative emission time. A significant reduction of the peak polarization voltage requires higher carrier densities than most practical situations, and screening effects stem mainly from the interior-screening and the localization of the polarization voltage within QW edge-layers. Our theoretical estimates show that the elimination of the QCSE related red-shift in energy precedes the recovery in the radiative emission time, in quantitative agreement with experimental measurements in [17].

Appendix-1: 1-D Edge-confined Modes - Asymptotic Polynomial Expansions

Section "QW Eigen Modes with ES-shielded Polar Potential" derived a perturbative solution for the edge-confined modes in terms of the square well eigen modes. Another approach, involving an infinite series polynomial expansion, will be given here and used to derive the scaling of the edge-confined expectation values for the kinetic and potential energy. First, for λDL/2 one may approximate the sinh potential for x < 0, , as where ζ the distance from the edge The sinh Schrodinger Equation 2 is then approximated by one for an exponential potential which has been analyzed elsewhere.1 A dimensionless scaling measuring length in units of and energy in units of yields


where n labels the energy quantum number A change of variable for ζ > 0 with removes the exponential term and reduces (13) to


The boundary conditions at correspond to w = 1, 0, and are given by A series expansion


inside (14) yields the coefficient recurrence relation or,


where and is found from the normalization condition. Substitution into the series solution and application of the boundary conditions at yields the eigen values from the roots of the following indicial equation


Switching (15) back to the original variables yields the corresponding eigenfunctions as


making use of The leading term goes as and gives the asymptotic behavior at For practical purposes is suffices to keep polynomial terms up to order M equal to twice the integer part inside the infinite sum in (17).

One may now compute expectation values with direct integration of (18). First, orthonormalization yields the normalization constant c o from


The expectation potential energy yields with


and the expectation kinetic energy yields


Thus the energy expectation value 〈E n 〉 is


where the normalization factor from (19). Thus edge detrapping at about 〈E 1 〉 > 0 occurs for Both K and W depend on and on the energy eigen value -ε1 where ε1 = ξ1. The ratio W1/K1 is plotted in Fig. 11 versus the peak voltage (normalized in units of κT) using the lowest mode energy n = 1 inside (20) and (21). Note that for Vo > 5κT the ratio hovers near 1/2 and thus detrapping occurs at

Figure 11
figure 11

Ratio of W1/K1 versus peak-voltage

Appendix 2: Charged Layer Potential

The self-consistent Poisson's equation, including the influence of the charged layer (plasma sheath) potential Φ(x) on the Fermi-Dirac occupation number f in determining the local carrier density is


subject to the boundary conditions This means that equals the unshielded value at each QW edge. Above we have normalized and where No is a reference carrier density and the corresponding Debye length which includes the dielectric shielding ε from core (bound) electrons. The sum of the electron, hole and charged donor charge densities (N-doping is assumed without loss of generality) on the right-hand side follows from the equilibrium Fermi-Dirac occupation numbers,


with EC, EV, F being respectively the conduction, valence, and Fermi levels, G e,h (E) the electron (hole) density of states and ND the dopant density (normalized to No), and The Fermi level F is obtained from the condition ρ[χo|Φ=0] = 0 at the neutral point Φ(xo) = 0. This automatically guarantees total charge neutrality over the QW as follows. The point xo where ρ(xo) = 0 is also the location of the minimum of the screened electric field, since there. Now, from and Gausses law follows and Q-=-Q+. The sheath Eqs. 23 and 24 yield the free carrier dielectric shielding inside a plasma-filled QW capacitor of plate charge under the nonlinear response ρ[Φ].

Analytic solutions of (23) and (24) in terms of the polarization field strength exist for certain degenerate and non-degenerate limits. The simplest treatment illustrating all the salient features is the undoped (intrinsic semiconductor) limit ND = 0. Since the Fermi level in this case lies close to mid-bandgap and the non-degenerate Maxwellian limit applies for the carrier statistics. The carrier density is simply given by where is the zero polarization electron and hold density. Three dimensional density of states is assumed for large enough QW width with small energy spacing Poisson's equation is then simplified to


It has exact analytic solutions, since x = X(Φ) is given in terms of elliptic integrals of complex argument, and hence Φ(x) follows in terms of the elliptic amplitude (Jacobi ) function,


where is the potential drop over half the QW length L and (Different profiles apply for given applied voltages [19] across the sheaths.) The field and voltage profiles have respectively even/odd symmetry about the middle of the QW, reflecting the opposite electron and hole densities for an undoped material. The opposite polarity electron and hole sheath potentials V e = -V h = V o are respectively defined by V e ≡ Φ(0) - Φ(L/2) and V h ≡ Φ(L/2) - Φ(L). The corresponding nominal sheath lengths are L e = L h = L/2. However, when L e, h ≫ λD, the field in each sheath is essentially localized within a few λD while the rest of the length is almost field-free.

Solutions and shielded voltage profiles for both Maxwellian, Eq. 26, as well as Fermi-Dirac distributions in general, Eqs. 23, 24, have been given in [15]. Maxwellian profiles are reasonably well fitted with sinh-profiles employed in the present analysis, such as the bottom of the QW Fig. 2a. The screened profiles remain essentially similar for Fermi-Dirac distributions in general, as shown in Fig. 9a, with one difference: the symmetry between the electron and hole charged-layers is broken, V e ≠ -V h . In addition, F-D statistics yields higher saturation voltages VS under given parameters. The saturation values shown in Fig. 7 correspond to general F-D solutions. Finally, for sufficiently small potentials any sheath profiles, including (26), are reduced to exponential profiles [15] , solutions of the linear differential equation


1 The solutions with are the odd-symmetry eigenfunctions of the general attractive potential