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The inability of a single-gap solar cell to absorb energies less than the band-gap energy is one of the intrinsic loss mechanisms which limit the conversion efficiency in photovoltaic devices. New approaches to “ultra-high” efficiency solar cells include devices such as multiple quantum wells (QW) and superlattices (SL) systems in the intrinsic region of a p-i-n cell of wider band-gap energy (barrier or host) semiconductor. These configurations are intended to extend the absorption band beyond the single gap host cell semiconductor. A theoretical model has been developed to study the performance of the strain-balanced GaAsP/InGaAs/GaAs MQWSC, and GaAs/GaInNAs MQWSC or SLSC. Our results show that conversion efficiencies can be reached which have never been obtained before for a single-junction solar cell.

Quantum well and superlattice solar cells approaches consist on the insertion of a multiple quantum well or superlattice system in the intrinsic region of a p-i-n cell of wider bandgap (barrier or host) semiconductor in order to improve the spectral response of the cell in the energy region below the absorption edge of host material. In simple terms, the multiple quantum wells solar cell (QWSC) and superlattice solar cell (SLSC) are p-i-n structures with sandwiched very thin layers (quantum wells) in the intrinsic layer between the two n and p doped bulk semiconductors. This idea was pioneered by Barnham et al. in 1990 [

As mentioned, the purpose of these designs is to improve the spectral response of the cell in the energy region below the absorption edge of host material, in order to gain an extra photocurrent and therefore an increment in the short-circuit current. Under solar radiation, a drop in open circuit voltage (V_{OC}) has been observed, due to the inclusion of lower bandgap material and interfaces, which could be overcompensated by the increased shortircuit current (I_{SC}) from the quantum wells [

Other more successful approach has been to include strain-balanced GaAsP/InGaAs quantum wells and barriers in the intrinsic region [

The dilute nitride (GaIn) (NAs) is a novel material system grown lattice matched to GaAs. This compound is gaining increasing interest in recent years due to its very unique physical properties and a wide range of possible device applications. The band gap of GaAs decreases rapidly with the addition of small atomic fractions of N [^{2} in absence of anti-reflection coating (ARC) and output voltages around 0.6V for a 1.1 eV MQW cell. These short circuit current and open circuit voltage values are much smaller than those reported for conventional GaAs solar cells, surely due to high interface recombination as consequence of the lattice mismatch between GaAs and GaNAs layers.

We report a theoretical model that shows that the insertion of GaInNAs multiple quantum wells into depletion region or a p-i(MQW)-n solar cell can significantly enhance the conversion efficiencies. Open circuit voltage, current densities, I-V curves and conversion efficiencies (h) are calculated as functions of the well and barrier band gaps, the width and depth of the wells and, the number of the wells in the intrinsic region. These results supply a broad representation of quantum well solar cells operation, and should provide a profitable guide for designing and interpreting the performance characteristics of quantum well solar cells fabricated from a wide variety of materials. In this paper, an accurate modeling of a GaAsP/InGaAs/GaAs SB-QWSC is also presented, showing that high performance devices are achievable. Our model takes into account the effect of the strain over the energy bands of the material by the definition of the strain Hamiltonian of the structure. The effect of tensile and compressive strain on the band structure for both conduction band and valence band state are rigorously calculated in order to compute the electron and hole dispersion relation E(k). Similarly, the optical transitions in quantum well and barriers as a function of tensile and compressive strain were evaluated without fitting parameters to calculate the quantum efficiency, dark current and the photocurrent and, to compare them with experimental data. The GaAsP/InGaAs/GaAs solar cell is optimized to reach the maximum performance by evaluating the current-voltage curves under illumination. Our model was used to determine the highest efficiencies for cells containing quantum wells under varying degrees of strain, but it could also allow the optimization of the photocurrent or the open circuit voltage in a triplejunction concentrator cell based on a SB-QWSC middle cell.

Another approach, where GaAs/GaInNAs multiplequantum wells and superlattice are added within the intrinsic region of conventional GaAs p–i–n solar cells is also presented. First, a GaAs/GaInNAs MQWSC is described in order to research the conversion efficiency as a function of wells width and depth. A discussion about N fraction and quantum well width to reach the maximum of the solar cell performance is addressed. Second, a variably spaced MQW or SL was optimized to enhance the resonant tunneling between adjacent wells following the method reported in references [4,11]. A theoretical study of the GaAs/GaInNAs SLSC viability is considered where the effective density of states and the absorption coefficient for the SL structure are calculated in order to determinate the J-V characteristic. The influence of the superlattice width and number on the conversion efficiency is discussed. The SLSC conversion efficiency is compared with the maximum conversion efficiency obtained for the MQWSC. Finally, we present GaAs/ GaInNAs SLSC conversion efficiency as a function of solar concentration, showing an amazing increment in its performance.

In this paper, we made the common assumptions of homogeneous composition in doped and intrinsic layers, the depletion approximation in the space-charge region, and total photogenerated carrier collection.

The first cell characteristic simulated was the spectral response, yielding the cell quantum efficiency (QE) and short circuit current for a given spectrum. Transport and Poisson equations were used to compute the quantum efficiency in the charge-neutral layers. The fit to the QE determines the recombination characteristics independently in charge neutral and space-charge regions. This determines the radiative and non-radiative recombination currents in these regions as a function of applied bias. The overall photocurrent is simply expressed in terms of superposition, adding photocurrent to the dark current in order to ascertain the light current characteristic. At the same time, we assumed an equal carrier temperature in all regions.

A MQW solar cell with N_{W} wells each of width L_{W} in the intrinsic region of length W with barrier band gap Eg_{B} and well band gap Eg_{W} was studied in previous works [

where q is the electron charge, V is the terminal voltage, kT is the thermal energy, and

J_{0} is the reverse saturation current density, A_{B} is the non-radiative coefficient for barriers in the depletion region, which is related to barrier non-radiative lifetime τ_{B} by A_{B} = 1/τ_{B}, B_{B} is the radiative barrier recombination coefficient, which includes the charge-neutral Shockley injection current, and the space-charge region radiative current, n_{iB} is the equilibrium intrinsic carrier concentration for the barrier material, r_{R} and r_{NR} are the radiative enhancement ratio and non-radiative enhancement ratio respectively, and represent the fractional increase in radiative and non-radiative recombination in the intrinsic region, due to the insertion of the quantum wells. These parameters are given by the following expressions:

where DE = Eg_{B} – Eg_{W}, f_{W }is the fraction of the intrinsic region volume replaced by quantum well material, g_{DOS} = g_{W}/g_{B} is the density of states enhancement factor, with g_{W} and g_{B} as the effective density of states for the wells and barriers, and γ_{B}, g_{A} are “oscillator enhancement factor” and “lifetime reduction factor”, respectively. The built in field is denoted by F and x is the position in the wells, so r_{R} and r_{NR} are position dependent. The photocurrent J_{PH} is calculated from the quantum efficiency of the cell. The p-region and n-region contribution to QE was classically evaluated solving the carrier transport equations at room temperature within the minority carrier and depletion approximations. The contribution of photo-generated carriers in the intrinsic region to QE values is calculated by the expression:

where R(l) is the surface reflectivity spectrum of the antireflection layer. The first exponential factor is due to the attenuation of light in the layers between the surface of the cell and the depletion layer (see _{i} and z_{i} are the absorption coefficient and the width of these precedent layers, respectively, the a_{B} is the absorption coefficient of the bulk barrier material, and is the dimensionless quantum well absorption coefficient, used for energies below the barrier band gap.

When mixing between light and heavy valence subands is neglected, the absorption coefficient can be calculated as follows:

where and are sums over well states n and m, which numbers depend on the quantum wells width and depth, and are the absorption coefficients due to electron-heavy hole and electron-light hole transitions to conduction band, respectively; α_{W} is the well absorption coefficient and L is the ‘‘quantum thickness of the heterostructure’’. To match accurately with experimental data in the long wavelength region, the exciton absorption is considered in the theoretical calculation. The exciton binding energies are analytically evaluated in the framework of fractional-dimensional space.

Once the total QE is calculated, by means of the AM1.5 solar spectrum F(l), the photocurrent is determined by integration:

where l_{1} and l_{2} are limits of the taken solar spectrum.

Then, Equation (1) is completely determined and conversion efficiency (h) can be evaluated. The present model was previously corroborated comparing experimental quantum efficiency and open circuit voltage with the corresponding calculated values using AlGaAs/GaAs MQWSC sample from the research group at Imperial College London [

In a QWSC, the different quantum wells are independent and there is no coupling between neighboring quantum wells. However, for the superlattice solar cell, there is an interaction between neighboring wells and the wave function becomes extended over the whole superlattice, and the discrete levels in isolated quantum wells spread into a miniband. In presence of electric field across the intrinsic region, perpendicular to the layers, the superlattice minibands split into a band energy ladder and as the electric field is increased, considering wells of equal width with a single energy level; an alignment between the energy subbands is getting worse.

On the other hand, when wells have more than a single quantum energy level, it is expected that the bands move in and out of alignment as the field is varied. This effect reduces tunneling current causing a decrease of the solar cell performance. This way, the electric field causes a localization of the electron in the quantum well that produces vanishing of the miniband, and then the superlattice advantages disappear. Hence, the major challenge is to achieve the conditions for resonant tunneling for a specific value of the electric field which allows the escape of the carrier from the superlattices. This difficulty is overcome fabricating variable spaced superlattice where a well width variation is chosen so that the electron levels are resonant at the operating bias.

From the theoretical point of view, the advantages of an SLSC over an MQWSC are the following: 1) provides quantum levels for electrons and holes within specific eigenenergies (minibands); 2) enhances the miniband absorption; 3) eliminates deep-level recombination between single and double heterojunction; 4) the carriers are capable of tunneling along the growth direction through thin barriers while they are essentially free along the transverse direction; and 5) allows an efficient escape rate of carriers out of quantum wells, which are collected in the emitter and base regions.

In a SLSC, the optical transitions take place between minibands and not between quantum levels, so optical transitions between electron, heavy, and light holes minibands must be included in the J-V characteristic. Equation (1) can be extended to describe a SLSC taking into account that the coefficients (Equations (2) and (3)) and the photocurrent J_{PH} should be related to superlattice structure. Now, the f_{W} factor is the intrinsic region fraction replaced by superlattices, with g_{SL} and g_{B} as the effective density of states for superlattices and barriers, γ_{B} = B_{SL}/B_{B} and g_{A} = A_{SL}/A_{B} are the radiative and non radiative recombination coefficients which are referred to superlattices and barriers respectively. The photocurrent is evaluated including the miniband absorption coefficients. We calculated the effective density of states, the absorption coefficient, the radiative recombination coefficient and the photocurrent in a superlattice system in order to determine the J-V characteristic for a SLSC.

The density of states N(E), for a superlattice was estimated considering that each state of energy in a single quantum well becomes the bottom of a two dimensional subband in the density of states. In the same way, each Bloch state in a superlattice behaves as the bottom of a subband. The cosine approximation for a singe miniband of width G was considered. Once N(E) is known, assuming an instantaneous transition at the barrier band edge to the usual three dimensional density of states, considering that the miniband energy extends only the length of the cluster L_{SL} and making the Boltzmann approximation to the Fermi-Dirac distribution, the effective density of states for electrons was found [

where E_{e} is the electron miniband bottom, erfc is the complementary error function, m_{e} is the electron effective mass, ΔEc = Qc(Eg_{B} – Eg_{W}) is the conduction band well depth, Qc is the band offset, d_{SL}_{ }is the superlattice period and e is the conduction miniband width. Analogous expressions are held for the heavy hole and light hole effective density of states. Then the total superlattice effective density of states was calculated as:

The absorption coefficient for the transitions between light hole and electron minibands was also computed as a function of their widths, G_{lh} and G_{e}:

where is the matrix element between the initial i and the final f states, a is a unit vector in the direction of propagation, the momentum p is a differential operator, n_{r} is the heterostructure refraction index, ε_{0} is the vacuum dielectric constant, ω is the radiation frequency, m_{lh} is the light hole effective mass, and E_{lh} is the light hole miniband top. A similar expression of the absorption coefficient is found for the transitions between heavy hole and electron minibands. Hence, the superlattice absorption coefficient can be written as:

The absorption allows us to evaluate the radiative recombination loss in Equation (1). According to detailed balance theory, assuming a quasi-Fermi level separation constant and equal to the applied voltage V, the radiative recombination coefficient is given by:

The Equation (4) should be modified to consider the absorption of photons through a miniband and not by means of the quantum well discreet levels. The contribution of photo-generated carriers in the intrinsic region to QE values is calculated by the expression:

where L_{SL} is the superlattice width or cluster width and N is the number of clusters.

Once the expressions for the effective density of states, the absorption coefficient, the radiative recombination coefficient, and photocurrent were found for SLSC, it is possible to evaluate the J-V characteristic (Equation (1)). It is then possible to calculate the conversion efficiency.

The SB-QWSC is a GaAs p-i-n solar cell with quantum well layers incorporated into the i-region with InGaAs as well material and GaAsP as barrier material.

The compressive strain in the InGaAs QW is matched by tensile strain in GaAsP barriers, overcoming the lattice-mismatch limitation. The GaAsP and InGaAs layer widths were chosen to ensure the average lattice parameter across the i-region was equal to that of GaAs. Elastic constants were considered to evaluate the tensile and compressive strain in GaAsP and InGaAs layers. Thus, if L_{B} is the barrier width, L_{W} is the well width, and are the respective well and barrier lattice constants; we define:

The p and n regions were designed with 200 and 500 nm widths respectively, and a 40 nm Al_{0.8}Ga_{0.2} As window layer was incorporated into the p-region to reduce front surface recombination. The hole and electron concentrations are p = 10^{18} cm^{−3} and n = 10^{18} cm^{−3}, respectively. The ARC is a 70 nm MgF:SiN layer. A passivation layer in the solar cell rear with 200 cm/s surface recombination velocity was assumed.

Biaxial strain can only be achieved on the nanoscale, giving nanostructured solar cells a fundamental advantage over bulk semi-conductor solar cells. The changes of band structure of the layer under strain have significant effects on the SB-QWSC. The growth of strained GaAsP and InGaAs layers allows a wider choice of P and In compositions to fit the energy levels in the quantum wells. Thus, the balanced strain between GaAsP and InGaAs layers is also designed as an extra parameter to tailor the layer materials and the SB-QWSC performance.

The total strain in the layer may be resolved into a hydrostatic component and an axial component. For unstrained bulk material, the heavy hole (hh) and light hole (lh) bands at the top of the valence band are degenerate at the Brillouin zone centre. The hydrostatic component of strain acts on the band edges, thereby changing the band gaps. However, it is the axial strain component, acting on degenerate bands such as in the valence bands. In the valence band, the axial component lifts the degeneracy that exists for the heavy-hole and light-hole bands at the band edge (point). Under compressive strain, bottom energy of the conduction band is displaced to higher energy and the valence band splits, with the lh band moving further from the conduction band suppressing the lh transition [

The lattice constant of the epitaxial layer in the growth interface is forced to be equal to the lattice constant of the substrate. Therefore, there is an in-plane strain_{, }where i, j = x, y, z. For biaxial strain in (001) plane in the strained layers, the values along [

where, and are the new energy level values under strain for heavy and light holes and electrons, respectively, E_{g} is the band gap, a_{n} and a_{c} are so-called hydrostatic deformation potentials, b is the shear deformation potential and D_{SO} is the spin-orbit splitting of the VB at G point. The separation of the total hydrostatic deformation potential in conduction (a_{c}) and valence band (a_{n}) contributions is important at heterointerfaces.

Varying the wave vector values, the E versus k diagram is obtained for both materials, InGaAs and GaAsP, which are shown in

strains, the In_{0.2}Ga_{0.8} As layer experiences a 121 meV band gap increment, while for GaAs_{0.7}P_{0.3 }layer a decrease of 176 meV is obtained. When the In and P compositions as well as their layer widths are varied, such that the condition given by Equation (14) is satisfied, the strain is changed in the well and barrier layers, causing in both films a deviation in the absorption threshold.

The envelope function approximation is here assumed to compute QW energy levels in conduction band. The electron energy E_{e} and wave function y_{e} can be calculated with the effective mass approximation. Then, the motion of the conduction band electron in the QW is described by Schrödinger equation. In order to determine the QW energy levels in the hh and lh bands under varying compressive strain a 4 × 4 k∙p Kohn-Luttinger Hamiltonian was used:

where H_{KL} is the Kohn-Luttinger Hamiltonian without strain and H^{} is the strain Hamiltonian that for epilayers grown in [

In a QW system, some splitting of the confined valence band levels takes place due to the differences in effective mass. This splitting can be greatly enhanced if the QW is strained. When the well material is strained we have the confinement effects, but in addition we have the consequences of the non-degenerate bulk band edges. For example under (001) biaxial compressive strain the lh band is shifted further energetically from the conduction band. In a sense, compressive strain acts in the same manner as the effect of confinement.

Then, biaxial strain, in well and barriers layers, lifts the degeneracy in the valence band such that it is possible to considerer independently the hh and lh bands. Under above approximations, the QW energy levels for hh and lh bands were found.

Schrödinger equation corresponding to Hamiltonian is not separated so it is assumed that the “off-diagonal” terms, which lead to valence band mixing, are small enough that they can be neglected. With this assumption the Schrödinger equation becomes separable:

where and are the envelope functions with spin direction. The Equations (18) and (19) are solved in barrier and well regions with the corresponding potential and Konh-Luttinger parameter (and) values in each layer. Computed the E_{e}, E_{h} and E_{lh} values the optical transitions are calculated by Fermi’s golden rule and the Equation (6) is evaluated to determine the absorption in the quantum wells. Then, the procedure to calculate the photocurrent is analogous to that described in 2.1.

The QE as a function of wavelength was computed by means of Equation (4) and compared with qt1897b sample from Quantum Photovoltaic Group at Imperial College of London. The cell is a p-i-n diode with an i-region containing five QWs that are 9.6 nm wide of compressively strained In_{0.16}Ga_{0.84} As inserted into tensilestrained GaAs_{0.91}P_{0.09} barriers at strain-balance condition.

_{e} and E_{h} or E_{lh} quantum well levels is smaller.

The strain and electric field effects that do not explicitly

show in Equation (1) were considered to evaluate the J-V characteristic. When the In and P compositions are varied or their width layer are changed, the generated strain modifies the absorption threshold. In a QW system, some splitting of the confined valence band levels takes place due to the differences in effective mass. This splitting can be greatly enhanced if the QW is strained. Similarly, when W is changed, the electric field variation also modifies the absorption threshold.

The dependence of conversion efficiency on In and P composition is examined in _{W} = 15 nm. In and P composition are varied in the range from 0.5% to 20% during the development process of the contour plot.

The L_{B} values are varied in order to satisfy strain-balanced condition (Equation (14)). The white region corresponds to In and P compositions for which the optical transitions between the ground heavy hole and electron QW levels are larger than GaAs bandgap. These optical transitions were not considered in the calculations.

The dependence of conversion efficiency on In composition and quantum well width is shown in _{W} are varied in the range from 0.5% to 20% and 4 nm to 20 nm, respectively, during the development process of the contour plot. In this case, L_{B} and N_{W} were necessarily varied to satisfy strain-balanced condition. The broken lines are due to the emergence of QW energy levels when the well width and barrier height are increased.

For an In composition given, as it can be observed in _{ }values with a maximum at x = 0.02 and L_{W} = 18 nm.

A GaAs p-i-n solar cell is modeled where the i-region is filled with GaAs/GaInNAs multiple quantum wells. The electron and hole concentrations, in base and emitter regions are 1.8 × 10^{17} cm^{−3} and 2.3 × 10^{18} cm^{−3}, respectively, while their widths are 0.15 mm (p-region), 0.60 mm (i-region), and 0.46 mm (n-region). A 40 nm Al_{0.8}Ga_{0.2}As window layer was incorporated into the p-region to reduce front surface recombination and a 70 nm MgF:SiN layer as ARC was used.

In order to investigate the GaAs/Ga_{1-x}In_{x}N_{y}As_{1-y} MQWSC conversion efficiency, small nitrogen concentrations were considered to modify the quantum well depth. Also, the x = 2.85y condition was satisfied so that Ga_{1-x}In_{x}N_{y}As_{1-y} lattice matches to GaAs. GaInNAs parameters were taken from reference [

GaAs/GaInNAs solar cells were studied inserting now superlattices in the i-region of a GaAs p-i-n solar cell, keeping the other device parameters identical.

The conditions for resonant tunneling were computed by the well-known transfer matrix method (TMM) without back-scattered wave. A variably spaced multiple quantum well or superlattice was considered to enhance the resonant tunneling between adjacent wells [

resonance takes place at 12 kV/cm, which was evaluated taking into account the uniform doping levels, p and n-regions, and intrinsic region width. We have used a fixed field (a parameter in our model), which is not such a difficult thing to do experimentally, once that doping levels and intrinsic thickness are as those reported here.

To study the GaAs/GaInNAs SLSC performance, some different arrangements with ten variably spaced multiple quantum wells (clusters) were optimized to enhance the resonant tunneling between adjacent wells. These clusters are composed by quantum wells of 1% nitrogen composition and barriers of 1 nm wide. A series of clusters are inserted in the i-region, independent from each other, in such a way that there is no coupling between neighboring clusters.