Electrostatically Shielded Quantum Confined Stark Effect Inside Polar Nanostructures
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The effect of electrostatic shielding of the polarization fields in nanostructures at high carrier densities is studied. A simplified analytical model, employing screened, exponentially decaying polarization potentials, localized at the edges of a QW, is introduced for the ES-shielded quantum confined Stark effect (QCSE). Wave function trapping within the Debye-length edge-potential causes blue shifting of energy levels and gradual elimination of the QCSE red-shifting with increasing carrier density. The increase in the e−h wave function overlap and the decrease of the radiative emission time are, however, delayed until the “edge-localization” energy exceeds the peak-voltage of the charged layer. Then the wave function center shifts to the middle of the QW, and behavior becomes similar to that of an unbiased square QW. Our theoretical estimates of the radiative emission time show a complete elimination of the QCSE at doping densities ≥1020 cm−3, in quantitative agreement with experimental measurements.
At high carrier densities, charge separation and dipole field formation is sufficient to cause shielding of the intrinsic polarization E-field . 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  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  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 . 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. .
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  while the emission probability behavior agrees with the results in . 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
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 κDL ≡ L/λD 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 λD ≫ L/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, λD ≤ L/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 , based on the observed decrease in the radiative emission time.
where K n , W n are functions of eVo/κT 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(x/λD) 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.
Radiative Emission Probability
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 .
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 e−h 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 .
Appendix-1: 1-D Edge-confined Modes - Asymptotic Polynomial Expansions
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).
Appendix 2: Charged Layer Potential
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].
where is the potential drop over half the QW length L and (Different profiles apply for given applied voltages  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 . 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  , solutions of the linear differential equation
1 The solutions with are the odd-symmetry eigenfunctions of the general attractive potential