Materials and Device Modeling and Simulation

Enhanced efficiency of Schottky-barrier solar cell with periodically nonhomogeneous indium gallium nitride layer

[+] Author Affiliations
Tom H. Anderson, Tom G. Mackay

University of Edinburgh, School of Mathematics and Maxwell Institute for Mathematical Sciences, Edinburgh, Scotland, United Kingdom

Pennsylvania State University, Department of Engineering Science and Mechanics, NanoMM—Nanoengineered Metamaterials Group, University Park, Pennsylvania, United States

Akhlesh Lakhtakia

Pennsylvania State University, Department of Engineering Science and Mechanics, NanoMM—Nanoengineered Metamaterials Group, University Park, Pennsylvania, United States

J. Photon. Energy. 7(1), 014502 (Feb 03, 2017). doi:10.1117/1.JPE.7.014502
History: Received December 8, 2016; Accepted January 18, 2017
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Open Access Open Access

Abstract.  A two-dimensional finite-element model was developed to simulate the optoelectronic performance of a Schottky-barrier solar cell. The heart of this solar cell is a junction between a metal and a layer of n-doped indium gallium nitride (InξGa1ξN) alloy sandwiched between a reflection-reducing front window and a periodically corrugated metallic back reflector. The bandgap of the InξGa1ξN layer was varied periodically in the thickness direction by varying the parameter ξ(0,1). First, the frequency-domain Maxwell postulates were solved to determine the spatial profile of photon absorption and, thus, the generation of electron–hole pairs. The AM1.5G solar spectrum was taken to represent the incident solar flux. Next, the drift-diffusion equations were solved for the steady-state electron and hole densities. Numerical results indicate that a corrugated back reflector of a period of 600 nm is optimal for photon absorption when the InξGa1ξN layer is homogeneous. The efficiency of a solar cell with a periodically nonhomogeneous InξGa1ξN layer may be higher by as much as 26.8% compared to the analogous solar cell with a homogeneous InξGa1ξN layer.

Figures in this Article

A variety of light-trapping strategies capitalizing on structures engineered on the order of the wavelength of solar light have been examined both experimentally and theoretically to enhance the efficiencies of solar cells.1 These strategies include texturing the front surface of the solar cell,2,3 incorporating a periodically corrugated metallic back reflector,46 coating the front surface with an “antireflection” (AR) layer,7,8 embedding nanoparticles inside the light-absorbing layer(s),911 and using light concentrators.1215

An attractive approach for boosting the solar-cell efficiency requires the use of photon-absorbing component materials whose electromagnetic properties are periodically nonhomogeneous along the thickness direction. Recently, it was demonstrated that the incorporation of a periodically nonhomogeneous intrinsic layer (i.e., i layer), along with a periodically corrugated back reflector, in amorphous silicon p-i-n junction solar cells can improve overall efficiency by up to 17%.16 This improvement is likely due to

  • the periodic corrugation of the metallic back reflector46,17 facilitating the excitation of surface-plasmon-polariton waves1820 and waveguide modes21 to intensify the electric field inside the semiconductor region, leading to an increase in the electron–hole pair (EHP) generation rate;
  • the periodic nonhomogeneity of the i-layer that may facilitate the excitation of multiple surface-plasmon-polariton waves22 and waveguide modes,17 thereby further boosting the EHP generation rate; and
  • the accompanying spatial gradient in the bandgap that may also aid charge separation and reduce the EHP recombination rate.23,24

The study described herein concerns an especially simple type of solar cell that has received scant attention from researchers to date: a Schottky-barrier solar cell.25 The Schottky barrier is provided by a metal–semiconductor junction.26,27 Recent theoretical studies,28,29 along with an earlier experimental study,30 have suggested that Schottky-barrier solar cells may be a particularly promising proposition if the semiconductor layer (i.e., the absorbing layer) was made from alloys of indium gallium nitride (InξGa1ξN), since the bandgap for these alloys can closely match the range of energies of photons across the entire solar spectrum (i.e., 0.70 to 3.42 eV) by varying the relative proportions of indium and gallium through the parameter ξ(0,1).31 Specifically, indium nitride (i.e., ξ=1) has a bandgap of 0.7 eV32,33 and absorbs efficiently across the infrared regime in the solar spectrum, while gallium nitride (i.e., ξ=0) has a bandgap of 3.42 eV and absorbs efficiently across the near-ultraviolet portion of the solar spectrum.

With current technologies, there are significant challenges to overcome in the production of InξGa1ξN alloys for all values of ξ, particularly for smaller-bandgap alloys. When the proportion of indium is large (i.e., ξ0.3), poor crystal growth plagues the realization of solar-cell applications. Such poor growth results in decreased carrier transport, background n doping due to Fermi pinning above the conduction band edge,34 and a bandgap that is greater than expected.35,36 By using a Schottky-barrier junction with n-doped InξGa1ξN, as opposed to a p-i-n junction, the difficulty of p doping the material is avoided.

In the following sections, a two-dimensional (2-D) numerical simulation is described for a Schottky-barrier solar cell made with InξGa1ξN. It is essential that the coupled processes of optical absorption and electrical-current generation are simultaneously accommodated in the simulation. This is because a focus on the computation of only the optical (i.e., at full quantum efficiency) short-circuit current density, but not of the open-circuit voltage, overplays the importance of the EHP generation rate by not taking the EHP recombination rate into account.16 In our simulation, particular attention is paid to the role of periodic nonhomogeneity of the absorbing material. Thus, the effect of periodically varying the fractional composition parameter ξ of InξGa1ξN in the direction perpendicular to the mean plane of the back reflector is explored, the back reflector being periodically corrugated in one direction. The model is briefly described in Secs. 2.1 and 2.2, with further details being available elsewhere.16,37 Numerical results are presented in Secs. 3.1 and 3.2. Closing remarks are presented in Sec. 4.

Physical Model

A schematic diagram of the simulated Schottky-barrier solar cell is shown in Fig. 1. The back reflector is corrugated along the x-axis, and the z axis is normal to the plane of the Schottky contact and the mean plane of the back reflector. In the remainder of this paper, the term “width” refers to extent in the x-direction, while “thickness” refers to extent in the z-direction. Insolation is provided via an excitation port at z=LAirLwLc. A planar AR window made of an insulating material occupies the region LwLc<z<Lc. To align with our earlier work, the optical permittivity of this layer was taken to be identical to that of aluminum-doped zinc oxide.38

Graphic Jump Location
Fig. 1
F1 :

Schematic illustration of a Schottky-barrier solar cell. Only one back-reflector period is shown [i.e., (Lx/2)<x<(Lx/2)].

The region 0<z<Lz is occupied by n-doped InξGa1ξN, forming a Schottky junction and two ohmic junctions with a metal in the region 0<z<Lc. For calculations, the metal is assumed to be silver.39 The Schottky contact of width Ls is centered at x=0. The two ohmic contacts, each of width Lo/2, are centered about x=±(LxLo/2)/2. Note that Lo+Ls<Lx. The two regions between the metal contacts for 0<z<Lc are occupied with the same material as the AR window that occupies LwLc<z<Lc.

A silver back reflector, covered with an insulating window (made of the same material as the window that occupies LwLc<z<Lc), occupies the region Lz<z<Lz+Lr. This back reflector is periodically corrugated in the x-direction with period Lx. The region Lz<z<g(x) is filled with the window material, while the region g(x)<z<Lz+Lr is filled with silver. The corrugation in the unit cell is specified by the function Display Formula

g(x)={Lz+daLgcos(πxζLx),2x/Lx(ζ,ζ)Lz+da,2x/Lx(ζ,ζ),(1)
where Lx is the corrugation period, Lgda is the corrugation height, and ζ(0,1] is the duty cycle. As the corrugation is invariant along the y-axis, our model is 2-D.

Absorption of the normally incident solar flux with AM1.5G spectrum40 is calculated by solving the frequency-domain Maxwell postulates. The semiconductor charge-carrier drift-diffusion equations model the electron and hole density spatial distributions.41,42 Because of the nonhomogeneity of the semiconductor (i.e., InξGa1ξN), the effective dc electric field acting on (a) electrons includes a contribution from gradients in the electron affinity and (b) holes includes contributions from gradients in both the electron affinity and the bandgap. Direct, mid-gap Shockley–Read–Hall, and Auger recombination are all included in our simulation. The current density J, which is averaged over the Schottky contact (or, identically, both of the ohmic contacts), is calculated for a range of external biasing voltages Vext.

By varying the proportion of indium relative to that of gallium, the bandgap of InξGa1ξN can be engineered to take any value from EgInN=0.7  eV (i.e., for InN when ξ=1) continuously through to EgGaN=3.42  eV (i.e., for GaN when ξ=0). Thus, allowing for nonhomogeneity in the z-direction, the InξGa1ξN alloy has bandgap given by Display Formula

Eg0(z)=ξ(z)EgInN+[1ξ(z)]EgGaNbξ(z)[1ξ(z)],z(0,Lz),(2)
where the bowing parameter b=1.43  eV.43 To estimate ξ for a material with a specific bandgap Eg0, Eq. (2) may be solved as Display Formula
ξ(z)=b+(EgGaNEgInN)4b[Eg0(z)EgGaN]+(b+EgGaNEgInN)22b.(3)

The electron affinity χ0 for InξGa1ξN is modeled in an analogous manner to the bandgap. Hence Display Formula

χ0(z)=ξ(z)χInN+[1ξ(z)]χGaNbξ(z)[1ξ(z)],z(0,Lz),(4)
where χInN and χGaN are the electron affinities of InN and GaN, respectively. For all other material parameters of InξGa1ξN, as described in the first column of Table 1, Vegard’s law of linear interpolation44 is assumed to apply, i.e., Display Formula
τInξGa1ξN(z)=ξ(z)τInN+[1ξ(z)]τGaN,z(0,Lz),(5)
where τ{NC,NV,Cn,Cp,Crad,αoc}. Data for τInN and τGaN are provided in Table 1.

Table Grahic Jump Location
Table 1Electronic data used for GaN and InN. The composition of InξGa1ξN was estimated using Eq. (3), with linear interpolation used to estimate data for the semiconductor-filled region 0<z<Lz with bandgaps not presented here in all cases, except for the electron affinity χ0 which uses Eq. (4).

The narrowing of the bandgap associated with doping was incorporated through the Slotboom model.45 While this model was developed for silicon, similar narrowing behavior under heavy-doping conditions has been observed in GaN.46 When doped, the bandgap of the semiconductor narrows to Eg=Eg0ΔEg, while the electron affinity reduces to χ=χ0αocΔEg, with the conduction-band fraction αoc being a material-specific parameter. Here, the bandgap narrowing is estimated as Display Formula

ΔEg=Eref{ln(Nd+NaNref)+[ln(Nd+NaNref)]2+12},(6)
where Nd and Na are the doping densities of the donor and the acceptor, respectively, and Nref and Eref are the empirically determined reference doping density and reference energy, respectively.

The real part of the optical refractive index of InξGa1ξN is Display Formula

Re{n}=Re(AA(ξ){[Eg(ξ)Eγ]2[21+EγEg(ξ)1EγEg(ξ)]+BA(ξ)}),(7)
per the Adachi model,29 where AA and BA are parameters given in Table 1. The photon energy is Eγ=hc0/λ0, where h=6.62607004×1034  m2kgs1 is the Planck constant, c0=299792485  ms1 is the speed of light in free space, and λ0 is the free-space wavelength. The imaginary part of the refractive index is Display Formula
Im{n}=αoptλ04π.(8)
The absorption coefficient αopt is modeled in Ref. 29Display Formula
αopt(ξ)=105C(ξ)(EγEg)+D(ξ)(EγEg)2nm1,(9)
where the constants Display Formula
C(ξ)=(3.52518.28ξ+40.22ξ237.52ξ3+12.77ξ4)eV1D(ξ)=(0.6651+3.616ξ2.460ξ2)eV2},(10)
come from interpolation of parameters given by Brown et al.47

An empirical low-field mobility model—called either the Caughey–Thomas 29 or the Arora 48 mobility model—describes the variations of the electron mobility μn and the hole mobility μp with temperature and doping. Thus Display Formula

μ=μ(1)(TTref)αct,+μ(2)(TTref)βμ(1)(TTref)αct,1+[(TTref)γ(Nd+NaNcrit)]δ,{n,p},(11)
where =n for electrons and =p for holes. In the foregoing equation, μ(2) is the maximum value of the carrier mobility μ when the material is undoped, μ(1) is the minimum value of μ when the material is heavily doped, Ncrit is the critical doping density of the InξGa1ξN alloy, T is the lattice temperature and Tref=300  K is the reference temperature, and αct,, β, γ, and δ are empirically determined parameters. The solar cell was taken to be operating at T=Tref=300  K; therefore, it should be noted that the choices of αct,, β, and γ have no effect on the results. Following Hamady et al.,29 in lieu of experimental data, we assumed that αct,=β=γ=1 for all simulations reported here. With these assumptions, the carrier mobilities simplify to Display Formula
μ=μ(1)+μ(2)μ(1)1+N˜δ,{n,p},(12)
where N˜=(Nd+Na)/Nref is the ratio of doping density to the critical doping density. The electronic data used for InξGa1ξN presented in Table 1 were provided by Hamady et al.,29 who also demonstrated that a relatively large metal work-function Φ for the Schottky-barrier contact improves efficiency. Accordingly, the relatively large value of Φ=6  eV was chosen here.

The bandgap profile of InξGa1ξN for the solar cells simulated in this study is given by Display Formula

Eg0(z)=Eg*A(1{12[sin(2πzLp2πφ)+1]}α),(13)
where Eg* is the baseline (maximum) bandgap, A is the amplitude, Lp=Lz/κ is the period with κ>0, φ is a phase shift, and α is a shaping parameter, which governs the profile’s gradient. Three example bandgap profiles are shown in Fig. 2 for α{0.2,1,10}, when κ=3 and φ=0.75.

Graphic Jump Location
Fig. 2
F2 :

Bandgap profiles given by Eq. (13) for α{0.2,1,10}, when κ=3 and φ=0.75.

Computational Model

A 2-D finite-element optoelectronic model was implemented in the COMSOL Multiphysics (V5.1) software package48 in two major steps, as described now. Let it be noted that terms in block capitals are COMSOL Multiphysics48 terms.

In the first major step, the Electromagnetic Waves, Frequency Domain module was used to calculate the 2-D EHP generation rate as a function of x(Lx/2,Lx/2] and z(0,Lz). Normally incident monochromatic light sampled at 5-nm intervals on the λ0-scale across the AM1.5G spectrum, 50% s polarized and 50% p polarized, was activated by the Periodic ports option at z=LcLwLAir. Diffraction Order ports for diffraction orders {3,2,1,1,2,3} were added. The inclusion of diffraction ports for even higher orders has little impact on the resulting EHP generation rate, predominantly due to strong absorption of shorter wavelength photons by InξGa1ξN, the majority of which are absorbed close to the surface of the device before they can be scattered. The boundaries parallel to the z-axis have Floquet periodicity with the wavevector provided by the periodic ports. The region z>Lz+L+f behind the back reflector was taken to be a perfectly matched layer. The semiconductor region was first meshed with a Mapped mesh with a 10-nm Distribution in both the z- and x-directions and then split into to a triangular distribution using the insert center points conversion. The back-reflector region was covered with an Extra fine, Delaunay, and Free Triangular mesh. Data for a spatial map of the computed EHP generation rate were stored in an external file.

The EHP generation rate can be used to compute the optical short-circuit current density JSCOpt, assuming that every absorbed photon creates an EHP in the InξGa1ξN layer and that no recombination takes place. Neglect of recombination implies that JSCOpt is necessarily larger than the short-circuit current density JSC, which is the electronically simulated current density that flows when the solar cell is illuminated and no external bias is applied (i.e., when Vext=0).

In the second major step, the Semiconductor module was used to calculate the electron and hole densities in the semiconductor region, and thereby the current densities. Due to the symmetry of the unit cell, only its right half (i.e., 0xLx/2) needs to be electrically simulated. Fermi-Dirac carrier statistics, along with continuous quasi-Fermi levels at any internal boundary, were employed. Finite volume (constant shape function) discretization was employed as this inherently conserves current throughout the solar cell.48 COMSOL uses a Scharfetter–Gummel upwinding scheme for solving the charge carrier transport equations. The Free triangular, Delaunay mesh has a maximum element size of 15 nm.

A potential difference of Vext was applied between the ohmic and Ideal Schottky contacts. Thermionic currents, with standard Richardson coefficients of An*=110  AK2cm2, and Ap*=90  AK2cm2 were applied at the Schottky barrier.29,48Insulator Interfaces were placed at the remaining external electrical boundaries. External Mathematica™ or MATLAB™ codes were used to calculate the User-Defined Generation from the output of the first major step. Recombination was incorporated via Auger, Direct and Trap-Assisted (Midgap Shockley-Read-Hall) pathways, with parameters as provided in Table 1. To facilitate convergence, the nonhomogeneity, EHP generation, and EHP recombination physics were slowly activated as the solver progressed by use of a continuation parameter. An analytic doping model was used to set the donor concentration to Nd. The Free triangular, Delaunay mesh used has a maximum element size of 15 nm.

Design of Periodically Corrugated Back Reflector

The numerical results presented here are for devices with parameters listed in Table 2. A preliminary study was carried out to ascertain reasonable values for the dimensions of the device. Figure 3 presents JSCOpt as a function of the back-reflector period Lx(300,1000)nm for a Schottky-barrier solar cell with a homogeneous InξGa1ξN layer of thickness Lz{200,400,600,860,1000}nm. For this set of simulations, Lg=130  nm, da=135  nm, and ζ=0.5 were determined as optimal for a solar cell with a homogeneous InξGa1ξN layer characterized by Eg0=1.6  eV, Lx=600  nm, and Lz=860  nm. While this optimization is far from exhaustive, significant optical variation was seen to occur for thinner InξGa1ξN solar cells: a 6% variation in JSCOpt was calculated for a solar cell with Lz=200  nm, with a maximum JSCOpt of 24.9  mAcm2 when Lx=480  nm. Importantly, for solar cells with InξGa1ξN layers thicker than about 600 nm, JSCOpt saturates as Lx increases beyond a threshold value. Therefore, Lx=600  nm was chosen to be optimal. While the peak value of JSC of a Schottky-barrier solar cell with a 600-nm-thick InξGa1ξN layer is obtained for larger periods than this (Lx680  nm), JSC falls more rapidly for devices with back reflectors of overly large periods than for those with periods that are too small.

Table Grahic Jump Location
Table 2Summary of parameters used for the simulations.
Graphic Jump Location
Fig. 3
F3 :

Optical short-circuit current density JSCOpt as a function of the back-reflector period Lx for a Schottky-barrier solar cell with InξGa1ξN layer of thickness Lz{200,400,600,860,1000}nm. The absorbing layer of InξGa1ξN is homogeneous with ξ=0.6, i.e., the bandgap Eg0=1.45  eV. Thin solid lines show JSCOpt for the equivalent solar cells with a flat back-reflector.

Effect of Periodic Nonhomogeneity of InξGa1−ξN Layer

The essential effects of including periodic nonhomogeneity in the thickness direction in the InξGa1ξN layer may be inferred from Fig. 4. For a fixed bandgap-nonhomogeneity amplitude A{0,0.2,0.4,0.6,0.8,1}, the baseline bandgap Eg* is shown to dramatically affect the simulated efficiency Display Formula

η=maxVextVextJ(Vext)1000  W/m2,(14)
which equals the maximum power produced by the solar cell as a function of Vext, divided by the incident AM1.5G solar power flux of 1000  W/m2. The solar spectrum was discretized at 5-nm λ0-intervals; with the minimum λ0 chosen to be 300 nm; the maximum λ0 chosen as min[2400,1240/(Eg*A)] nm; the other bandgap-nonhomogeneity parameters fixed at κ=3, α=2, and φ=0.75. At the minimum bandgap for an InξGa1ξN alloy, i.e., Eg=0.7  V which arises at ξ=1, there is a minimum permissible baseline bandgap Eg* for each value of A. Irrespective of the choice of value of A(0,0.8)eV, a baseline bandgap of Eg*=1.5  eV turns out to be optimal. From Fig. 4, it may be inferred that the difference in the efficiencies with a homogeneous InξGa1ξN layer and a periodically nonhomogeneous InξGa1ξN layer is greatest when

  • the bandgap-nonhomogeneity amplitude A is the greatest and
  • when the maximum bandgap Eg*1.5  eV.

Graphic Jump Location
Fig. 4
F4 :

Efficiency η as a function of baseline bandgap Eg* for bandgap-nonhomogeneity amplitude A{0,0.2,0.4,0.6,0.8,1}eV, when κ=3, α=2, and φ=0.75.

Indeed, at Eg*=1.5  eV, the efficiency is 13.26% for A=0 and 16.81% for A=0.8  eV, i.e., the relative increase in efficiency attributable to the periodic nonhomogeneity of the InξGa1ξN layer is 26.8%, which is this paper’s most significant result. For A=1  eV, Eg*=1.7  eV is optimal, yielding an efficiency of 15.8%.

The importance of the periodicity in the bandgap variation when seeking maximal efficiency may be inferred from Figs. 5 and 6. In Fig. 5, the thickness of the InξGa1ξN layer is fixed at Lz=600  nm while the ratio κ varies from 0.5 to 4. The efficiency of the solar cell is seen to reach a maximum value of about 17% at κ=1.4 and κ=3. Between these two values of κ, only a slight decrease in η was found. Significantly, the efficiency for all values of κ>0 is greater than it is when the InξGa1ξN layer is homogeneous.

Graphic Jump Location
Fig. 5
F5 :

Efficiency η as a function of the ratio κ=Lz/Lp for the InξGa1ξN layer when Lz=600  nm, Eg*=1.5  eV, A=0.8  eV, α=2, and φ=0.75. The horizontal red line indicates the efficiency when the InξGa1ξN layer is homogeneous.

Graphic Jump Location
Fig. 6
F6 :

Efficiency η as a function of thickness Lz of the InξGa1ξN layer, which is either homogeneous (solid black curve) or periodically nonhomogeneous with Lp=200  nm (dashed red curve) or Lp=400  nm (dotted blue curve), when Eg*=1.54  eV, A=0.8  eV, α=5, and φ=0.75. Integer values of the ratio κ=Lz/Lp are identified.

Figure 6 illustrates the effect of varying the thickness Lz of the InξGa1ξN layer when the nonhomogeneity period Lp is fixed. Irrespective of the period, the solar cell with the nonhomogeneous InξGa1ξN layer performs better than the one with a homogeneous InξGa1ξN layer. Furthermore, the solar cell with the smaller period (Lp=200  nm) is found to perform more efficiently than the one with larger period (Lp=400  nm). Distinct maximums can be seen at κ=1.5 and κ=3, which correspond to Lz=300  nm and Lz=600  nm, respectively. The efficiencies at these points, respectively, are 17.28% and 16.88%, which correspond to relative increases in efficiency of 16.7% and 28.5% as compared to the analogous solar cell with a homogeneous InξGa1ξN layer.

By varying the bandgap-nonhomogeneity shaping parameter α, the spatial profile of the InξGa1ξN nonhomogeneity can be varied. Thus, α=1 holds for a sinusoidal bandgap profile, while smaller and larger values of α yield bandgap profiles with steeper gradients, as illustrated in Fig. 2. For all three values of α in Fig. 7, the efficiency of the solar cell is substantially greater than it is for the corresponding solar cell with a homogeneous InξGa1ξN layer. The maximum increase in efficiency is found for a nearly sinusoidal bandgap profile; indeed, the efficiency increases from 13.2% for a Schottky-barrier solar cell with a homogeneous InξGa1ξN layer to 16.88% for its analog with a nonhomogeneous InξGa1ξN layer with α=2, which is a relative increase in efficiency of 28.5%.

Graphic Jump Location
Fig. 7
F7 :

Efficiency η plotted against bandgap-nonhomogeneity shaping parameter α, when Eg*=1.54  eV, A=0.8  eV, Lz=600  nm, κ=3, and φ=0.75.

Let us note here that as α becomes much larger or smaller than unity, the spatial gradients of the bandgap increase in magnitude and the peaks in Fig. 2 become narrower. Accordingly, at extreme values of α, the semiclassical carrier-transport equations implemented in our model become less appropriate as quantum processes become significant. Then, it may be necessary to take into account the quantization of allowed energy states between the peaks of Eg0(z) and tunneling through those peaks, especially for large amplitudes A.

A 2-D finite-element model was devised to simulate the combined optical and electrical performances of InξGa1ξN Schottky-barrier solar cells. First, it was found that a periodically corrugated back reflector of a period of 600 nm is optimal for photon absorption in a solar cell containing a homogeneous layer of InξGa1ξN. Second, the effects of periodic nonhomogeneity of the InξGa1ξN layer were elucidated. The nonhomogeneity was directed perpendicular to the mean plane of the periodically corrugated back reflector. For the particular model investigated here, the efficiency of a solar cell with a 600-nm-thick layer of InξGa1ξN was found to increase by 26.8% when suitable periodic nonhomogeneity was incorporated. Thus, the incorporation of a periodically nonhomogeneous InξGa1ξN layer in a Schottky-barrier solar cell can substantially increase the efficiency as compared to the analogous solar cell with a homogeneous InξGa1ξN layer. A comprehensive optimization of study of material and design parameters may well yield even greater nonhomogeneity-induced increases in efficiency. However, such an optimization study—which is justified by the substantial efficiency boosts reported herein—represents a major undertaking that lies beyond the scope of this paper. Similarly, it would be of some value to delineate the photon-absorption mechanisms that underpin the boost in the light-to-electricity conversion efficiency that arises following the introduction of periodic nonhomogeneity, but this too is matter for future study.

The feasibility of producing InξGa1ξN alloys with the prescribed spatial variation in ξ to achieve efficiency boosts in Schottky-barrier solar cells is a matter for our experimentalist colleagues to shed light on. On the basis of our numerical study and the relatively huge boosts in efficiency that may be attained, it would be worthwhile for major efforts to be directed toward the production of appropriate InξGa1ξN alloys.

Last, the principal finding of our study is that the introduction of periodic nonhomogeneity can, in principle, substantially boost the efficiency of a Schottky-barrier solar cell. To focus on efficiency, our attention was restricted to only normally incident solar radiation for all cell configurations considered. The influence of the angle of incidence is planned for a future study in which the mechanisms for efficiency boosting will be more fully explored.

T.H.A. thanks the Charles Godfrey Binder Endowment for partial financial support during a six-month stay at Pennsylvania State University. T.G.M. acknowledges the support of EPSRC Grant No. EP/M018075/1. A.L. thanks the National Science Foundation for partial financial support under Grant No. DMS-1619901, and he is grateful to the Charles Godfrey Binder Endowment for ongoing support of his research.

Singh  R., , Alapatt  G. F., and Lakhtakia  A., “Making solar cells a reality in every home: opportunities and challenges for photovoltaic device design,” IEEE J. Electron Devices Soc.. 1, , 129 –144 (2013).CrossRef
Verlinden  P.  et al., “The surface texturization of solar cells: a new method with V-grooves with controllable sidewall angles,” Sol. Energy Mater. Sol. Cells. 26, , 71 –78 (1992). 0927-0248 CrossRef
Yamada  N.  et al., “Characterization of antireflection moth-eye film on crystalline silicon photovoltaic module,” Opt. Express. 19, , A118 –A125 (2011). 1094-4087 CrossRef
Sheng  P., , Bloch  A. N., and Stepleman  R. S., “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett.. 43, , 579 –581 (1983). 0003-6951 CrossRef
Heine  C., and Morf  R. F., “Submicrometer gratings for solar energy applications,” Appl. Opt.. 34, , 2476 –2482 (1995). 0003-6935 CrossRef
Solano  M.  et al., “Optimization of the absorption efficiency of an amorphous-silicon thin-film tandem solar cell backed by a metallic surface-relief grating,” Appl. Opt.. 52, , 966 –979 (2013) [errata: 54, 398–399 (2015)]. 0003-6935 CrossRef
Dhungel  S. K.  et al., “Double-layer antireflection coating of MgF2/SiNx for crystalline silicon solar cells,” J. Korean Phys. Soc.. 49, , 885 –889 (2006). 0374-4884 
Boden  S. A., and Bagnall  D. M., “Sunrise to sunset optimization of thin film antireflective coatings for encapsulated, planar silicon solar cells,” Prog. Photovoltaics Res. Appl.. 17, , 241 –252 (2009). 1062-7995 CrossRef
Schaadt  D. M., , Feng  B., and Yu  E. T., “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett.. 86, , 063106  (2005). 0003-6951 CrossRef
Lee  J.-Y., and Peumans  P., “The origin of enhanced optical absorption in solar cells with metal nanoparticles embedded in the active layer,” Opt. Express. 18, , 10078 –10087 (2010). 1094-4087 CrossRef
Shokeen  P., , Jain  A., and Kapoor  A., “Silicon nanospheres for directional scattering in thin-film solar cells,” J. Nanophotonics. 10, , 036013  (2016). 1934-2608 CrossRef
Sinton  R. A.  et al., “27.5-percent silicon concentrator solar cells,” IEEE Electron Device Lett.. 7, , 567 –569 (1986). 0741-3106 CrossRef
Nishioka  K.  et al., “Evaluation of InGaP/InGaAs/Ge triple-junction solar cell and optimization of solar cells structure focusing on series resistance for high-efficiency concentrator photovoltaic systems,” Sol. Energy Mater. Sol. Cells. 90, , 1308 –1321 (2006). 0927-0248 CrossRef
Solano  M. E.  et al., “Buffer layer between a planar optical concentrator and a solar cell,” AIP Adv.. 5, , 097150  (2015). 2158-3226 CrossRef
Liu  L.  et al., “Planar light concentration in micro-Si solar cells enabled by a metallic grating-photonic crystal architecture,” ACS Photonics. 3, , 604 –610 (2016).CrossRef
Anderson  T. H.  et al., “Combined optical-electrical finite-element simulations of thin-film solar cells with homogeneous and nonhomogeneous intrinsic layers,” J. Photonics Energy. 6, , 025502  (2016).CrossRef
Liu  L.  et al., “Experimental excitation of multiple surface-plasmon-polariton waves and waveguide modes in a one-dimensional photonic crystal atop a two-dimensional metal grating,” J. Nanophotonics. 9, , 093593  (2015). 1934-2608 CrossRef
Anderson  L. M., “Parallel-processing with surface plasmons, a new strategy for converting the broad solar spectrum,” in  Proc. 16th IEEE Photovoltaic Specialists Conf. , Vol. 1, , pp. 371 –377 (1982).CrossRef
Anderson  L. M., “Harnessing surface plasmons for solar energy conversion,” Proc. SPIE. 408, , 172 –178 (1983). 0277-786X CrossRef
Polo  J. A.  Jr., , Mackay  T. G., and Lakhtakia  A., Electromagnetic Surface Waves: A Modern Perspective. ,  Elsevier ,  Waltham, Massachusetts  (2013).
Khaleque  T., and Magnusson  R., “Light management through guided-mode resonances in thin-film silicon solar cells,” J. Nanophotonics. 8, , 083995  (2014). 1934-2608 CrossRef
Faryad  M., and Lakhtakia  A., “Enhancement of light absorption efficiency of amorphous-silicon thin-film tandem solar cell due to multiple surface-plasmon-polariton waves in the near-infrared spectral regime,” Opt. Eng.. 52, , 087106  (2013) [errata: 53, 129801 (2014)].CrossRef
Kabir  M. I.  et al., “Amorphous silicon single-junction thin-film solar cell exceeding 10% efficiency by design optimization,” Int. J. Photoenergy. 2012, , 460919  (2012).CrossRef
Iftiquar  S. M.  et al., “Single- and multiple-junction p-i-n type amorphous silicon solar cells with p−a−Si1−xCx:H and nc-Si:H films,” in and Yun  I., Ed., Photodiodes—From Fundamentals to Applications. ,  InTech ,  Rijeka, Croatia  (2012).
Landsberg  P. T., and Klimpke  C., “Theory of the Schottky barrier solar cell,” Proc. R. Soc. A: Math. Phys. Eng. Sci.. 354, , 101 –118 (1977). 1364-5021 CrossRef
Swami  N. K., , Srivastava  S., and Ghule  H. M., “The role of the interfacial layer in Schottky barrier solar cells,” J. Phys. D: Appl. Phys.. 12, , 765 –771 (1979). 0022-3727 CrossRef
Colinge  J.-P., and Colinge  C. A., Physics of Semiconductor Devices. ,  Kluwer Academic ,  New York, New York  (2002).
Mahala  P.  et al., “Metal/InGaN Schottky junction solar cells: an analytical approach,” Appl. Phys. A. 118, , 1459 –1468 (2015).CrossRef
Hamady  S. O. S., , Adaine  A., and Fressengeas  N., “Numerical simulation of InGaN Schottky solar cell,” Mater. Sci. Semicond. Process.. 41, , 219 –225 (2016). 1369-8001 CrossRef
Xue  J.-J.  et al., “Au/Pt/InGaN/GaN heterostructure Schottky prototype solar cell,” Chin. Phys. Lett.. 26, , 098102  (2009). 0256-307X CrossRef
Wu  J.  et al., “Superior radiation resistance of InxGa1-xN alloys: full-solar-spectrum photovoltaic material system,” J. Appl. Phys.. 94, , 6477 –6482 (2003). 0021-8979 CrossRef
Wu  J.  et al., “Unusual properties of the fundamental band gap of InN,” Appl. Phys. Lett.. 80, , 3967 –3969 (2002). 0003-6951 CrossRef
Saito  Y.  et al., “Growth temperature dependence of Indium Nitride crystalline quality grown by RF-MBE,” Physica Status Solidi B. 234, , 796 –800 (2002).CrossRef
Li  S. X.  et al., “Fermi-level stabilization energy in group III nitrides,” Phys. Rev. B. 71, , 161201  (2005).CrossRef
Osamura  K.  et al., “Fundamental absorption edge in GaN, InN and their alloys,” Solid State Commun.. 11, , 617 –621 (1972). 0038-1098 CrossRef
Yodo  T.  et al., “Strong band edge luminescence from InN films grown on Si substrates by electron cyclotron resonance-assisted molecular beam epitaxy,” Appl. Phys. Lett.. 806, , 968 –970 (2002). 0003-6951 CrossRef
Anderson  T. H.  et al., “Towards numerical simulation of nonhomogeneous thin-film silicon solar cells,” Proc. SPIE. 8981, , 898115  (2014). 0277-786X CrossRef
Gao  X. Y., , Liang  Y., and Lin  Q. G., “Analysis of the optical constants of aluminum-doped zinc-oxide films by using the single-oscillator model,” J. Korean Phys. Soc.. 57, , 710 –714 (2010). 0374-4884 CrossRef
Palik  E. D., Ed., Handbook of Optical Constants of Solids. ,  Academic Press ,  Boston, Massachusetts  (1985).
“Reference solar spectral irradiance: air mass 1.5,”  NREL , http://rredc.nrel.gov/solar/spectra/am1.5/ (28  July , 2015).
Nelson  J., The Physics of Solar Cells. ,  Imperial College Press ,  London, United Kingdom  (2003).
Fonash  S. J., Solar Cell Device Physics. , 2nd ed.,  Academic Press ,  Burlington, Massachusetts  (2010).
Wu  J., and Walukiewicz  W., “Band gaps of InN and group III nitride alloys,” Superlattices Microstruct.. 34, , 63 –75 (2003). 0749-6036 CrossRef
Vegard  L., “Die Röntgenstrahlen im Dienste der Erforschung der Materie,” Z. Kristallogr.. 67, , 239 –259 (1928).CrossRef
Slotboom  J. W., and de Graaff  H. C., “Measurements of bandgap narrowing in Si bipolar transistors,” Solid-State Electron.. 19, , 857 –862 (1976).CrossRef
Lee  I.-H.  et al., “Band-gap narrowing and potential fluctuation in Si-doped GaN,” Appl. Phys. Lett.. 74, , 102 –104 (1999). 0003-6951 CrossRef
Brown  G.F.  et al., “Finite element simulations of compositionally graded InGaN solar cells,” Sol. Energy Mater. Sol. Cells. 94, , 478 –483 (2010). 0927-0248 CrossRef
http://www.comsol.com/ (19  January  2015).

Tom H. Anderson received his BSc degree in mathematics and physics from the University of Edinburgh in 2011. He received his MSc degree in fusion energy from the University of York in 2012 and his PhD in applied mathematics from the University of Edinburgh in 2016. His research interests include the optical and electrical modeling of thin-film solar cells.

Tom G. Mackay is a reader at the School of Mathematics, the University of Edinburgh, and an adjunct professor in the Department of Engineering Science and Mechanics at Pennsylvania State University. He is a graduate of the Universities of Edinburgh, Glasgow, and Strathclyde and a fellow of the Institute of Physics (UK) and SPIE. His research interests include the electromagnetic theory of complex materials, including homogenized composite materials.

Akhlesh Lakhtakia received degrees from the Banaras Hindu University and the University of Utah. He is the Charles Godfrey Binder professor of engineering at the Pennsylvania State University. His research interests include surface multiplasmonics, bioreplication, forensic science, solar cells, sculptured thin films, and mimumes. He is a fellow of Optical Society of America, SPIE, Institute of Physics, American Association for the Advancement of Science, American Physical Society, Institute of Electrical and Electronics Engineers, and Royal Society of Chemistry. He received the 2010 SPIE Technical Achievement Award and the 2016 Walston Chubb Award for Innovation.

© The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Citation

Tom H. Anderson ; Tom G. Mackay and Akhlesh Lakhtakia
"Enhanced efficiency of Schottky-barrier solar cell with periodically nonhomogeneous indium gallium nitride layer", J. Photon. Energy. 7(1), 014502 (Feb 03, 2017). ; http://dx.doi.org/10.1117/1.JPE.7.014502


Figures

Graphic Jump Location
Fig. 1
F1 :

Schematic illustration of a Schottky-barrier solar cell. Only one back-reflector period is shown [i.e., (Lx/2)<x<(Lx/2)].

Graphic Jump Location
Fig. 2
F2 :

Bandgap profiles given by Eq. (13) for α{0.2,1,10}, when κ=3 and φ=0.75.

Graphic Jump Location
Fig. 3
F3 :

Optical short-circuit current density JSCOpt as a function of the back-reflector period Lx for a Schottky-barrier solar cell with InξGa1ξN layer of thickness Lz{200,400,600,860,1000}nm. The absorbing layer of InξGa1ξN is homogeneous with ξ=0.6, i.e., the bandgap Eg0=1.45  eV. Thin solid lines show JSCOpt for the equivalent solar cells with a flat back-reflector.

Graphic Jump Location
Fig. 4
F4 :

Efficiency η as a function of baseline bandgap Eg* for bandgap-nonhomogeneity amplitude A{0,0.2,0.4,0.6,0.8,1}eV, when κ=3, α=2, and φ=0.75.

Graphic Jump Location
Fig. 5
F5 :

Efficiency η as a function of the ratio κ=Lz/Lp for the InξGa1ξN layer when Lz=600  nm, Eg*=1.5  eV, A=0.8  eV, α=2, and φ=0.75. The horizontal red line indicates the efficiency when the InξGa1ξN layer is homogeneous.

Graphic Jump Location
Fig. 6
F6 :

Efficiency η as a function of thickness Lz of the InξGa1ξN layer, which is either homogeneous (solid black curve) or periodically nonhomogeneous with Lp=200  nm (dashed red curve) or Lp=400  nm (dotted blue curve), when Eg*=1.54  eV, A=0.8  eV, α=5, and φ=0.75. Integer values of the ratio κ=Lz/Lp are identified.

Graphic Jump Location
Fig. 7
F7 :

Efficiency η plotted against bandgap-nonhomogeneity shaping parameter α, when Eg*=1.54  eV, A=0.8  eV, Lz=600  nm, κ=3, and φ=0.75.

Tables

Table Grahic Jump Location
Table 1Electronic data used for GaN and InN. The composition of InξGa1ξN was estimated using Eq. (3), with linear interpolation used to estimate data for the semiconductor-filled region 0<z<Lz with bandgaps not presented here in all cases, except for the electron affinity χ0 which uses Eq. (4).
Table Grahic Jump Location
Table 2Summary of parameters used for the simulations.

References

Singh  R., , Alapatt  G. F., and Lakhtakia  A., “Making solar cells a reality in every home: opportunities and challenges for photovoltaic device design,” IEEE J. Electron Devices Soc.. 1, , 129 –144 (2013).CrossRef
Verlinden  P.  et al., “The surface texturization of solar cells: a new method with V-grooves with controllable sidewall angles,” Sol. Energy Mater. Sol. Cells. 26, , 71 –78 (1992). 0927-0248 CrossRef
Yamada  N.  et al., “Characterization of antireflection moth-eye film on crystalline silicon photovoltaic module,” Opt. Express. 19, , A118 –A125 (2011). 1094-4087 CrossRef
Sheng  P., , Bloch  A. N., and Stepleman  R. S., “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett.. 43, , 579 –581 (1983). 0003-6951 CrossRef
Heine  C., and Morf  R. F., “Submicrometer gratings for solar energy applications,” Appl. Opt.. 34, , 2476 –2482 (1995). 0003-6935 CrossRef
Solano  M.  et al., “Optimization of the absorption efficiency of an amorphous-silicon thin-film tandem solar cell backed by a metallic surface-relief grating,” Appl. Opt.. 52, , 966 –979 (2013) [errata: 54, 398–399 (2015)]. 0003-6935 CrossRef
Dhungel  S. K.  et al., “Double-layer antireflection coating of MgF2/SiNx for crystalline silicon solar cells,” J. Korean Phys. Soc.. 49, , 885 –889 (2006). 0374-4884 
Boden  S. A., and Bagnall  D. M., “Sunrise to sunset optimization of thin film antireflective coatings for encapsulated, planar silicon solar cells,” Prog. Photovoltaics Res. Appl.. 17, , 241 –252 (2009). 1062-7995 CrossRef
Schaadt  D. M., , Feng  B., and Yu  E. T., “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett.. 86, , 063106  (2005). 0003-6951 CrossRef
Lee  J.-Y., and Peumans  P., “The origin of enhanced optical absorption in solar cells with metal nanoparticles embedded in the active layer,” Opt. Express. 18, , 10078 –10087 (2010). 1094-4087 CrossRef
Shokeen  P., , Jain  A., and Kapoor  A., “Silicon nanospheres for directional scattering in thin-film solar cells,” J. Nanophotonics. 10, , 036013  (2016). 1934-2608 CrossRef
Sinton  R. A.  et al., “27.5-percent silicon concentrator solar cells,” IEEE Electron Device Lett.. 7, , 567 –569 (1986). 0741-3106 CrossRef
Nishioka  K.  et al., “Evaluation of InGaP/InGaAs/Ge triple-junction solar cell and optimization of solar cells structure focusing on series resistance for high-efficiency concentrator photovoltaic systems,” Sol. Energy Mater. Sol. Cells. 90, , 1308 –1321 (2006). 0927-0248 CrossRef
Solano  M. E.  et al., “Buffer layer between a planar optical concentrator and a solar cell,” AIP Adv.. 5, , 097150  (2015). 2158-3226 CrossRef
Liu  L.  et al., “Planar light concentration in micro-Si solar cells enabled by a metallic grating-photonic crystal architecture,” ACS Photonics. 3, , 604 –610 (2016).CrossRef
Anderson  T. H.  et al., “Combined optical-electrical finite-element simulations of thin-film solar cells with homogeneous and nonhomogeneous intrinsic layers,” J. Photonics Energy. 6, , 025502  (2016).CrossRef
Liu  L.  et al., “Experimental excitation of multiple surface-plasmon-polariton waves and waveguide modes in a one-dimensional photonic crystal atop a two-dimensional metal grating,” J. Nanophotonics. 9, , 093593  (2015). 1934-2608 CrossRef
Anderson  L. M., “Parallel-processing with surface plasmons, a new strategy for converting the broad solar spectrum,” in  Proc. 16th IEEE Photovoltaic Specialists Conf. , Vol. 1, , pp. 371 –377 (1982).CrossRef
Anderson  L. M., “Harnessing surface plasmons for solar energy conversion,” Proc. SPIE. 408, , 172 –178 (1983). 0277-786X CrossRef
Polo  J. A.  Jr., , Mackay  T. G., and Lakhtakia  A., Electromagnetic Surface Waves: A Modern Perspective. ,  Elsevier ,  Waltham, Massachusetts  (2013).
Khaleque  T., and Magnusson  R., “Light management through guided-mode resonances in thin-film silicon solar cells,” J. Nanophotonics. 8, , 083995  (2014). 1934-2608 CrossRef
Faryad  M., and Lakhtakia  A., “Enhancement of light absorption efficiency of amorphous-silicon thin-film tandem solar cell due to multiple surface-plasmon-polariton waves in the near-infrared spectral regime,” Opt. Eng.. 52, , 087106  (2013) [errata: 53, 129801 (2014)].CrossRef
Kabir  M. I.  et al., “Amorphous silicon single-junction thin-film solar cell exceeding 10% efficiency by design optimization,” Int. J. Photoenergy. 2012, , 460919  (2012).CrossRef
Iftiquar  S. M.  et al., “Single- and multiple-junction p-i-n type amorphous silicon solar cells with p−a−Si1−xCx:H and nc-Si:H films,” in and Yun  I., Ed., Photodiodes—From Fundamentals to Applications. ,  InTech ,  Rijeka, Croatia  (2012).
Landsberg  P. T., and Klimpke  C., “Theory of the Schottky barrier solar cell,” Proc. R. Soc. A: Math. Phys. Eng. Sci.. 354, , 101 –118 (1977). 1364-5021 CrossRef
Swami  N. K., , Srivastava  S., and Ghule  H. M., “The role of the interfacial layer in Schottky barrier solar cells,” J. Phys. D: Appl. Phys.. 12, , 765 –771 (1979). 0022-3727 CrossRef
Colinge  J.-P., and Colinge  C. A., Physics of Semiconductor Devices. ,  Kluwer Academic ,  New York, New York  (2002).
Mahala  P.  et al., “Metal/InGaN Schottky junction solar cells: an analytical approach,” Appl. Phys. A. 118, , 1459 –1468 (2015).CrossRef
Hamady  S. O. S., , Adaine  A., and Fressengeas  N., “Numerical simulation of InGaN Schottky solar cell,” Mater. Sci. Semicond. Process.. 41, , 219 –225 (2016). 1369-8001 CrossRef
Xue  J.-J.  et al., “Au/Pt/InGaN/GaN heterostructure Schottky prototype solar cell,” Chin. Phys. Lett.. 26, , 098102  (2009). 0256-307X CrossRef
Wu  J.  et al., “Superior radiation resistance of InxGa1-xN alloys: full-solar-spectrum photovoltaic material system,” J. Appl. Phys.. 94, , 6477 –6482 (2003). 0021-8979 CrossRef
Wu  J.  et al., “Unusual properties of the fundamental band gap of InN,” Appl. Phys. Lett.. 80, , 3967 –3969 (2002). 0003-6951 CrossRef
Saito  Y.  et al., “Growth temperature dependence of Indium Nitride crystalline quality grown by RF-MBE,” Physica Status Solidi B. 234, , 796 –800 (2002).CrossRef
Li  S. X.  et al., “Fermi-level stabilization energy in group III nitrides,” Phys. Rev. B. 71, , 161201  (2005).CrossRef
Osamura  K.  et al., “Fundamental absorption edge in GaN, InN and their alloys,” Solid State Commun.. 11, , 617 –621 (1972). 0038-1098 CrossRef
Yodo  T.  et al., “Strong band edge luminescence from InN films grown on Si substrates by electron cyclotron resonance-assisted molecular beam epitaxy,” Appl. Phys. Lett.. 806, , 968 –970 (2002). 0003-6951 CrossRef
Anderson  T. H.  et al., “Towards numerical simulation of nonhomogeneous thin-film silicon solar cells,” Proc. SPIE. 8981, , 898115  (2014). 0277-786X CrossRef
Gao  X. Y., , Liang  Y., and Lin  Q. G., “Analysis of the optical constants of aluminum-doped zinc-oxide films by using the single-oscillator model,” J. Korean Phys. Soc.. 57, , 710 –714 (2010). 0374-4884 CrossRef
Palik  E. D., Ed., Handbook of Optical Constants of Solids. ,  Academic Press ,  Boston, Massachusetts  (1985).
“Reference solar spectral irradiance: air mass 1.5,”  NREL , http://rredc.nrel.gov/solar/spectra/am1.5/ (28  July , 2015).
Nelson  J., The Physics of Solar Cells. ,  Imperial College Press ,  London, United Kingdom  (2003).
Fonash  S. J., Solar Cell Device Physics. , 2nd ed.,  Academic Press ,  Burlington, Massachusetts  (2010).
Wu  J., and Walukiewicz  W., “Band gaps of InN and group III nitride alloys,” Superlattices Microstruct.. 34, , 63 –75 (2003). 0749-6036 CrossRef
Vegard  L., “Die Röntgenstrahlen im Dienste der Erforschung der Materie,” Z. Kristallogr.. 67, , 239 –259 (1928).CrossRef
Slotboom  J. W., and de Graaff  H. C., “Measurements of bandgap narrowing in Si bipolar transistors,” Solid-State Electron.. 19, , 857 –862 (1976).CrossRef
Lee  I.-H.  et al., “Band-gap narrowing and potential fluctuation in Si-doped GaN,” Appl. Phys. Lett.. 74, , 102 –104 (1999). 0003-6951 CrossRef
Brown  G.F.  et al., “Finite element simulations of compositionally graded InGaN solar cells,” Sol. Energy Mater. Sol. Cells. 94, , 478 –483 (2010). 0927-0248 CrossRef
http://www.comsol.com/ (19  January  2015).

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