# New Results in Optical Modelling of Quantum Well Solar Cells

Authors: Silvian Fara; Paul Sterian; Laurentiu Fara; Mihai Iancu; Andreea Sterian
Journal: International Journal of Photoenergy
 (2012)

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## Abstract

This project brought further advancements to the quantum well solar cell concept proposed by Keith Barnham. In this paper, the optical modelling of MQW solar cells was analyzed and we focussed on the following topics: (i) simulation of the refraction index and the reflectance, (ii) simulation of the absorption coefficient, (iii) simulation of the quantum efficiency for the absorption process, (iv) discussion and modelling of the quantum confinement effect, and (v) evaluation of datasheet parameters of the MQW cell.


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## Body

## 1. Background

Quantum well (QW) photovoltaic cells, third-generation solar cells, were proposed in 1990 by the research group of Professor Barnham [1] aiming at extending the spectral response and at increasing the photocurrent.The quantum well solar cell (QW) is a p-i-n structure having quantum wells built in the intrinsic. Doped regions on each side will produce an electric field perpendicular to the layers of quantum wells. It is also possible to have induced electric fields, because of the piezoelectric effects. These fields are induced by the intensities from the semiconductor layers, because of the disparity between network constants [2].The multiple quantum well (MQW) solar cell is a system that contains a layer in which the carrier assemblies comply quantum rules in connection with layers in which the carrier assemblies are performing classical, and its modelling is conducted on two-dimensional levels [3]:(1)
quantum level in which one computes the energy spectrum of the electron and the absorption coefficient of the MQW;(2)
macroscopic level in which one studies the transport of charge carriers in a similar manner to the conventional p-i-n diode [4]. The reflection of light is brought considering MQW cell a pseudohomogeneous medium. This combination of quantum and classical elements used in modelling of MQW solar cell justifies the name of hybrid given to the model of MQW solar cell. An example of application of this hybrid model (HM) [5] is the case of solar cell based on a ternary alloy semiconductor AxB1-xC (AlxGa1-xAs) which has the lowest BC band gap equal with zero and achieve the greatest value (x=1) for the AC semiconductor band gap.In Figure1 are graphically represented the transmittances and energetic levels in the MWQ system as a result of the transport matrix algorithm [6]. The proper values from Figure 1(a) and resonances from Figure 1(b) are identical and mutually confirmed. The levels represented in Figure 1(a) are obtained by applying Kramer’s rule, while the transmittances represented in Figure 1(b) are calculated via the transfer matrix. It is noted that a large variation of alloy composition x does not determine significant changes in position of the first level from the quantum wells, which means that there is a weak dependence between short-circuit current and the ternary alloy composition. Each curve in Figure 1(b) is calculated in 104 points in less than a minute.(a) Schematic representation of levels in the valence bands and conduction for the MQW system Al0.3Ga0.7As/GaAs containing 30 quantum wells. The corresponding level for the heavy gaps is shown with dotted line. (b) Transmission coefficient based on the electron energy for the same system. The curves parameter is the number of quantum wells.
(a)(b)Figure2 shows the geometry of an MQW solar cell in the HM hybrid model. This model allows determining the configuration that leads to maximum conversion efficiency. This layer configuration can further be designed, by using the transfer matrix method and the properties of the real materials. In other words, starting from a hypothetic layer system, the HM model allows optimization of the cell configuration in order to obtain maximum output electrical power, for a certain state of illumination [4].Figure 2
Geometry of a solar cell in the HM hybrid model.The HM model may have the capability to evaluate the performances of the MQW cell. These are represented bythe shape factor of cell(1)Ff=VmImVCDJSCand conversion efficiencyη=VmIm∫G0(λ)dλ,
where Vm and Im are the operation point coordinates when the power supplied in the load is maximum.


## 2. Modeling and Simulation of Refraction Index and Reflectance

Investigation of electric field effects on therefraction indexof MQW solar cells can be achieved by calculating the absorption coefficient [7]. In the case of solar cells with quantum wells was introduced the hypothesis of an electrostatic field generated by the load [8]. For this reason, it is expected a nonvarying change in the refraction index. Based on the results obtained by Barnham, it should be noted that the quantum wells from the solar cells of this type are designed to use the spectrum region, with the possibility of generating particles in those quantum wells. For the refraction index of the solar cell with quantum wells it was used the following expression:(2)n2=n=1+EdEo+EdEo3E2+ηπE4ln⁡(2Eo2-Eg2-E2Er2-E2).
The parameters are Eg=1.6729;Er=1.4235;Eo=2.6+0.7Er; Ed=F/Eo and η=πEd/(2Eo3(Eo2-Er2)).The simulation results for the refraction index depending on the photon energy are represented in Figure3.Figure 3
The refraction index of the cells with quantum wells depending on the photon energy.Theindex of refraction model can be used to explore the relation between the index of refraction and reflectance. To achieve this, the refraction index model must be combined with the Fresnel relations,thus allowing the finding of the losses caused by reflection of the quantum wells structure. The R reflectance of the solar cell can be calculated using a Fresnel-type relation [9](3)R=r12+r22+2r1r2cos⁡2θ1+r12r22+2r1r2cos⁡2θ,
where
(4)r1=n0-n1n0+n1;r2=n1-n2n1+n2;θ=2ππ1d1λ.The following notations were introduced:(i)
n0 the refraction index of the incident environment; for air n0=1;(ii)
n1 the refraction index of the antireflecting coating;(iii)
n2 the refraction index of the under layer (the solar cell with quantum wells).Thicknessd1 of the antireflecting coating can be determined by minimizing the last relation (5). It was noted that this value is approximately 600–650 nm, for the minimum reflection.It can be evaluated theeffect of the quantum well numberon the index of refraction and on the reflection losses so the optimal number of the quantum well for the structure could be calculated [10].In Figure4 are the results of the optical simulation of solar cells with quantum well (based on relations (3)-(4)), in the from of dependence on reflectance R, depending on the wavelength for different thicknesses d for the antireflecting coating; were considered two cases of antireflecting coating (n1 = 2.4—TiO2 and n1 = 1.4—SiO). Results from model are consistent with experimental results obtained. The reflectance model can be used to determine the variation effects of quantum well number on the index of refraction.Figure 4
Simulation of the reflectance for a solar cell with QW structure, for a SiO antireflecting coating.The simulation of refraction index and reflectance of the solar cells with quantum wells have been made with the Octave software, version 3.02.The cell reflectance can be calculated using the refraction indices of GaAs semiconductor and of the Al0.3Ga0.7As alloy (obtained by filtering the experimental data [11].(5)nGaAs={3.24123-(4.8304085/λ)+(2.82482/λ2)-(0.769037/λ3)+(0.08198/λ4)1-(1.5308/λ)+(0.9123972/λ2)-(0.2508648/λ3)+(0.026769/λ4),if0.325<λ≤4,2.8434068-(1.8916996/λ)+(0.4189801/λ2)-(0.0308637/λ3)1-(0.682372/λ)+(0.154593/λ2)-(0.01159/λ3),if0.2≤λ≤0.325.In (5) λ is expressed in μm. The refraction index of ternary alloy AlxGa1-xAs is calculated using the same translation procedure of the axes as in the case of the absorption coefficient:(6)nAlxGa1-xAs(λ)=[1.05-0.53x+0.09x2]nGaAs(λx)withλinμm.To minimize the reflection losses the solar cells are frequently coated withantireflecting coating (ARC).In Figure5(a) are the values calculated for the four refraction indices listed according to the wavelength. The MQW layer was considered to consist of 30 quantum wells of GaAs of 20 nm width separated by barriers Al0.3Ga0.7As with width of 10 nm.(a) The refraction index for the materials involved in constructing a MQW cell. (b) Spectral reflectance of the MQW cell for different width (nm) of the antireflecting coating. The HM model was run for GaAs/Al0.3Ga0.7As with Nw=30, lw=20 nm and lb=10 nm.
(a)(b)In Figure5(b) are summarized the results of reflection losses for the various thicknesses of the antireflecting coating. In the absence of ARC, reflection is an important mechanism of losses exceeding 30% on the spectral range in which the cell absorbs. The other curves show that it is possible to reduce losses through reflection by covering with an antireflecting coating of appropriate thickness.


## 3. Modelling and Simulation of Absorption  Coefficient

In applications related to calculating the conversion efficiency of solar cells, but also in other applications, the coefficient of absorption (the absorbance) is practically described by continuous functions. For GaAs, dropping from the experimental data [12] was determined the following function that approximates the acceptable rate of absorption:(3)αGaAs(λ)={eF1(λ),0.7<λ≤0.88,eF2(λ),0.24<λ≤0.7,eF2(0.24),0<λ≤0.24,0,otherwise,F1(λ)=-0.7863+5.3115[1+erf⁡(-λ-0.842910.038)],F2(λ)=-447.432+4201.2λ2+6835.128λ2ln⁡λ-3781.193λ3+3.9049λ2.In Figure6(a) is graphically represented the absorption coefficient for GaAs calculated with the algorithm from [13] for different compositions of the alloy.(a) Absorption coefficient ofAlxGa1-xAs depending on wavelength; x is the curves parameter. (b) Absorption coefficient in GaAs crystal; the ternary compound Al0.5Ga0.5As is calculated using axes translation in MQW system.
(a)(b)In Figure6(b) are graphically compared the absorption coefficients for GaAs, Al0.5Ga0.5As, and MQW system. It is noted that with increasing energy the absorption coefficient from MQW system increases in steps; this is because of the quantifying of the density of states in confinement of the carriers’ direction.Absorption coefficientαx(λ) of the AlxGa1-xAs alloy is generated from the absorption coefficient of the GaAs: αx(λ)=α0(λx) (h is Planck constant, c is the speed of light, a=0.62 and b=0.5 [13])(8)λx=hc(hc/λ)-Eg(x)+Eg(0)+α[(hc/λ)-Eg(0)]bx.


## 4. Increasing Quantum Efficiency of the QW  Solar Cells

### 4.1. Simulation of the Quantum Efficiency of QW Solar Cells by Using a Semiquantum Model

The analyzed model uses the transport equation ofAlxGa1-xAs quantum well solar cells, where x represents the Aluminium concentration. We considered the results obtaining by Hutchby and Fudurich [14], Connolly in his doctoral thesis [15], and Barnham’s research group [16].The model used the transport equation ofAlxGa1-xAs solar cells. The expression of the quantum efficiency is given by(9)QE≅αk(α-S)(k2-α2)(Ssinh⁡(kz)-kcosh⁡(kz))≅α-Sα2(S-1).It is introduced the parameterS of the boundary condition on the surface. It is resulted S≈2εf,  α  = spectral absorption coefficient, εf = effective electric field due to the minoritary carriers in the band.The simulated results are presented in the Figures7(a) and 7(b).(a) Variation of the quantum efficiency with the wavelength, for different fractions of Al. (b) Variation of the quantum efficiency with the wavelength, for different solar cell thicknesses. (c) Variation of the quantum efficiency with the wavelength, for different effective field.
(a)(b)(c)From the analysis of the data obtained, one can draw the following comments:(i)
quantum efficiency increases with any increase inλ, reaching significant values of maximum 90%, in the case of large diffusion wavelength;(ii)
quantum efficiency increases with any increase ofz and εf, respectively.


### 4.2. Quantum Confinement Effect

In order to find ways for increasing efficiency through the simulation of nanostructured solar cells (QW), we have used a mathematical model for the quantum efficiency of the transitions between the resonant levels (in GaAs) and another one for the quantum efficiency of the transitions between the confining levels (inAlxGa1-xAs) [17].In the case of resonant levels, quantum efficiency of the absorption process is given as follows [18]:(10)QEλ=8π2e2cℏε0εrw[(1-sin2i4εrw)+tw2L2sin2i4εrw],
where α is prism angle, i is incidence angle, εr is relative permittivity, and n is confinement level of quantum efficiency.For transitions between the confining levels, the quantum efficiency is given by(11)QEλn=4096e2π2cℏε0εrw,b2tw,b2L2n2sin2i(4n2-1)4.
In expressions (10) and (11), index w refers to GaAs, and index b refers to AlxGa1-xAs.For GaAs one obtains the curves of quantum efficiency of the transitions between the resonant levels (see Figures8(a) and 8(b)) and between the confining levels, depending on the prism refraction index and on the incidence angle of the ray that falls on the prism, for α=π/6 and α=2π/3. Similarly one obtains the quantum efficiency, for AlxGa1-xAs, Np=1.5 (Np=2 resp.), α=π/6 and α=2π/3, depending on the quantity of Aluminium.(a) Quantum efficiency of the transitions between resonant levels for GaAs as a function of the prism refraction index. It is considered the following two main cases: (1) prism angleα=π/6 and incidence angle: i=0 rad (red curve), i=0.8 rad (green curve), i=1 rad (black curve). (2) prism angle α=2π/3 incidence angle i=0 rad (blue curve), i=0.8 rad (violet curve), i=1 rad (brown curve). (b) Quantum efficiency of the transitions between resonant levels for GaAs as a function of the incidence angle for the ray falling on the prism. (c) Quantum efficiency of the transitions between confinement levels for AlxGa1-xAs as a function of  the prism refraction index. (d) Quantum efficiency of the transitions between confinement levels for AlGaAs   as a function of the incidence  angle of the ray falling on the prism. (e) Quantum efficiency of the transitions between confinement levels for AlxGa1-xAs as a function of the aluminium quantity; prism refraction index is Np=1.5. (f) Quantum efficiency of the transitions between confinement levels for AlxGa1-xAs as a function of the aluminium quantity; prism refraction index is Np=2.
(a)(b)(c)(d)(e)(f)From the analysis of the results, one can conclude the following.(i)
The quantum efficiency of the absorption process for nanostructured (quantum) solar cells is higher for a larger range of wavelengths, in comparison to the semiquantum solar cells.(ii)
Although the quantum efficiency of the absorption on the confining levels is very low, the appearance of electrons in the gap greatly increases the photocurrent through photoassisted tunnelling.(iii)
Monitoring the operating parameters (processor loading, memory used, network traffic, etc.) of the cluster (which allowed a large volume of computations) was done with the help of the Ganglia application.


## 4.1. Simulation of the Quantum Efficiency of QW Solar Cells by Using a Semiquantum Model

The analyzed model uses the transport equation ofAlxGa1-xAs quantum well solar cells, where x represents the Aluminium concentration. We considered the results obtaining by Hutchby and Fudurich [14], Connolly in his doctoral thesis [15], and Barnham’s research group [16].The model used the transport equation ofAlxGa1-xAs solar cells. The expression of the quantum efficiency is given by(9)QE≅αk(α-S)(k2-α2)(Ssinh⁡(kz)-kcosh⁡(kz))≅α-Sα2(S-1).It is introduced the parameterS of the boundary condition on the surface. It is resulted S≈2εf,  α  = spectral absorption coefficient, εf = effective electric field due to the minoritary carriers in the band.The simulated results are presented in the Figures7(a) and 7(b).(a) Variation of the quantum efficiency with the wavelength, for different fractions of Al. (b) Variation of the quantum efficiency with the wavelength, for different solar cell thicknesses. (c) Variation of the quantum efficiency with the wavelength, for different effective field.
(a)(b)(c)From the analysis of the data obtained, one can draw the following comments:(i)
quantum efficiency increases with any increase inλ, reaching significant values of maximum 90%, in the case of large diffusion wavelength;(ii)
quantum efficiency increases with any increase ofz and εf, respectively.


## 4.2. Quantum Confinement Effect

In order to find ways for increasing efficiency through the simulation of nanostructured solar cells (QW), we have used a mathematical model for the quantum efficiency of the transitions between the resonant levels (in GaAs) and another one for the quantum efficiency of the transitions between the confining levels (inAlxGa1-xAs) [17].In the case of resonant levels, quantum efficiency of the absorption process is given as follows [18]:(10)QEλ=8π2e2cℏε0εrw[(1-sin2i4εrw)+tw2L2sin2i4εrw],
where α is prism angle, i is incidence angle, εr is relative permittivity, and n is confinement level of quantum efficiency.For transitions between the confining levels, the quantum efficiency is given by(11)QEλn=4096e2π2cℏε0εrw,b2tw,b2L2n2sin2i(4n2-1)4.
In expressions (10) and (11), index w refers to GaAs, and index b refers to AlxGa1-xAs.For GaAs one obtains the curves of quantum efficiency of the transitions between the resonant levels (see Figures8(a) and 8(b)) and between the confining levels, depending on the prism refraction index and on the incidence angle of the ray that falls on the prism, for α=π/6 and α=2π/3. Similarly one obtains the quantum efficiency, for AlxGa1-xAs, Np=1.5 (Np=2 resp.), α=π/6 and α=2π/3, depending on the quantity of Aluminium.(a) Quantum efficiency of the transitions between resonant levels for GaAs as a function of the prism refraction index. It is considered the following two main cases: (1) prism angleα=π/6 and incidence angle: i=0 rad (red curve), i=0.8 rad (green curve), i=1 rad (black curve). (2) prism angle α=2π/3 incidence angle i=0 rad (blue curve), i=0.8 rad (violet curve), i=1 rad (brown curve). (b) Quantum efficiency of the transitions between resonant levels for GaAs as a function of the incidence angle for the ray falling on the prism. (c) Quantum efficiency of the transitions between confinement levels for AlxGa1-xAs as a function of  the prism refraction index. (d) Quantum efficiency of the transitions between confinement levels for AlGaAs   as a function of the incidence  angle of the ray falling on the prism. (e) Quantum efficiency of the transitions between confinement levels for AlxGa1-xAs as a function of the aluminium quantity; prism refraction index is Np=1.5. (f) Quantum efficiency of the transitions between confinement levels for AlxGa1-xAs as a function of the aluminium quantity; prism refraction index is Np=2.
(a)(b)(c)(d)(e)(f)From the analysis of the results, one can conclude the following.(i)
The quantum efficiency of the absorption process for nanostructured (quantum) solar cells is higher for a larger range of wavelengths, in comparison to the semiquantum solar cells.(ii)
Although the quantum efficiency of the absorption on the confining levels is very low, the appearance of electrons in the gap greatly increases the photocurrent through photoassisted tunnelling.(iii)
Monitoring the operating parameters (processor loading, memory used, network traffic, etc.) of the cluster (which allowed a large volume of computations) was done with the help of the Ganglia application.


## 5. The Results Obtained during Simulation of  the Cell Parameters

In Figure9, for the cell parameters Nw=30, lw=20 nm, and lb=10 nm are summarized  the calculation results of the conversion efficiency. In Figure 9(a) are represented the current-intensity characteristic with the power dissipated by the cell in the external circuit. Hence the main parameters of the cell are JSC=48.5 mA/cm2, VCD = 0.886 V, Ff = 0.86, and η=0.371. The numerical values are closed to those reported in the literature [19, 20]. As for the conventional cell with a p-n junction the maximum possible efficiency is estimated at 34% [3]. The result nor confirms nor invalidates that the insertion of MQW in the space-load region improves the conversion efficiency.(a) Voltage-current characteristic (line) and the output power (points) of the cell with the following MQW system: GaAs/Al0.3Ga0.7As, Nw=30, lw=20 nm, lb=10 nm and the antireflecting coating depth d=64 nm. (b) The MQW cell efficiency depending on the antireflecting coating depth. The dotted line indicates the ideal cell efficiency with zero reflectance. (c) The efficiency of MQW cell based on the number of quantum wells.
(a)(b)(c)Figure9(b) shows the conversion efficiency dependence on the antireflecting coating depth. It can be noted that in the ideal case, when the cell reflectance is neglected, the MQW cell efficiency based on GaAs/AlxGa1-xAs does not exceed 40%.Unlike the reflectance, in Figure9(c) it is observed that the conversion efficiency is strongly correlated with the number of quantum wells up to Nw=30; over this value the efficiency is saturated in proportion to Nw. Inside the MQW system the saturation is installed when the road length travelled by light is comparable with the absorption length.This is a one way to use the HM model. Knowing the materials componence of MQW cell and its optoelectronic properties, for a given geometric configuration the conversion efficiency can be calculated. Obviously the calculation can be repeated varying different geometrical and material parameters in order to determine the optimal configuration, that is, one that maximizes efficiency.Recently good experimental results were reported in MQW solar cells [21]. The fabricated solar cells based on In0.3Ga0.7N/GaN MQWs exhibit an open-circuit voltage of about 2 V, fill factor of about 60%, and an external efficiency of 40%–10% at 420 nm–50 nm.


## 6. Conclusions

The results were based on the HM hybrid model of the MQW solar cells; after simulation there were highlighted the following conclusions.(i)
It could be evaluated the quantum well number effect over the refraction index and reflection losses by evaluation of the quantum well optimal number of the structure.(ii)
The reflectance model proposed by authors could be used to determine variation effect of the quantum well number over the refraction index.(iii)
It was developed an absorption coefficient model which was used for studying the simulation of the MQW solar cells in special weather conditions.(iv)
It was obtained the internal quantum efficiency for the absorption process between the confining levels.(v)
The results generated by the model simulator agree with the experimental ones [22].There were studied several MQW solar cells configurations for optimized values of the optical parameters (reflectance, refraction index, and absorption), so the conversion efficiency could be improved. Although in our study, we considered only rectangular quantum wells, the HM hybrid model allows computing the levels whatsoever the form of the quantum wells is.New solar devices could be conceived based on optical properties of nanostructured materials [23, 24].QuantaSol [25] is a company form UK which developed firstly quantum wells and quantum dots for solar photovoltaics as well as concentrating solar cells for individual requirements.



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