# Carrier-induced ferromagnetism in 2D magnetically-doped semiconductor
structures

V. A. Stephanovich<EMAIL_ADDRESS>E. V. Kirichenko  G. Engel  Institute of
Physics, Opole University, Opole, 45-052, Poland  Yu. G. Semenov
<EMAIL_ADDRESS>Department of Electrical and Computer Engineering, North
Carolina State University, Raleigh, North Carolina 27695, USA  V. Lashkaryov
Institute of Semiconductor Physics of National Academy of Sciences of Ukraine,
41 Nauky prospekt, Kyiv 03680, Ukraine  K. W. Kim<EMAIL_ADDRESS>Department of
Electrical and Computer Engineering, North Carolina State University, Raleigh,
North Carolina 27695, USA  Department of Physics, North Carolina State
University, Raleigh, North Carolina 27695, USA  Institute of Physics, Opole
University, Opole, 45-052, Poland  Department of Electrical and Computer
Engineering, North Carolina State University, Raleigh, North Carolina 27695,
USA  V. Lashkaryov Institute of Semiconductor Physics of National Academy of
Sciences of Ukraine, Kyiv 03680, Ukraine  Department of Electrical and
Computer Engineering, North Carolina State University, Raleigh, North Carolina
27695, USA  Department of Physics, North Carolina State University, Raleigh,
North Carolina 27695, USA

###### Abstract

We show theoretically that the magnetic ions, randomly distributed in a two-
dimensional (2D) semiconductor system, can generate a ferromagnetic long-range
order via the RKKY interaction. The main physical reason is the discrete
(rather than continuous) symmetry of the 2D Ising model of the spin-spin
interaction mediated by the spin-orbit coupling of 2D free carriers, which
precludes the validity of the Mermin-Wagner theorem. Further, the analysis
clearly illustrates the crucial role of the molecular field fluctuations as
opposed to the mean field. The developed theoretical model describes the
desired magnetization and phase-transition temperature $T_{c}$ in terms of a
single parameter; namely, the chemical potential $\mu$. Our results highlight
a path way to reach the highest possible $T_{c}$ in a given material as well
as an opportunity to control the magnetic properties externally (e.g., via a
gate bias). Numerical estimations show that magnetic impurities such as Mn2+
with spins $S=5/2$ can realize ferromagnetism with $T_{c}$ close to room
temperature.

##  I Introduction

In the nascent era of spintronics, the studies of localized impurity spins in
the low-dimensional systems have become increasingly important [1, 2, 3]. At
the large inter-spin distances (as compared to a lattice constant), the
coupling between magnetic impurities in metals and semiconductors is primarily
due to the indirect Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction
via free electrons and holes (see Ref. [4] and references therein). The
indirect character of this interaction manifests itself in the fact that the
actual coupling occurs via Friedel oscillations of the free-carrier charge
density in a host material (see, e.g., Refs. [5, 4]). Accordingly, it is very
sensitive to the details of the electronic energy spectrum and spatial
dimensionality of the problem. The manipulation of electronic spectrum
parameters such as the energy gap, spin-splitting at the different points of
the Brillouin zone, and the spin-orbit interaction constant can generate
nonstandard collective properties in the impurity spin ensemble (e.g., the
long-range ferromagnetic (FM) ordering), leading potentially to a range of
optoelectronic, spintronic, and energy harvesting applications [1, 7, 8, 6].
For instance, the indirect exchange interaction, mediated by near-surface
electrons, was shown to couple local spins and facilitate the spatial spin
correlations [9].

Naturally, the spin density generated by an impurity magnetic moment in a two-
dimensional (2D) electron or hole gas can act on another impurity moment or
their cluster such that the resulting collective state becomes very complex.
This complexity can be well captured phenomenologically in terms of the
Landau-Lifshitz-Gilbert (LLG) equation [10, 11]. The LLG equation can describe
the systems with different long-range magnetic orders (i.e., FM,
antiferromagnetic, helical, etc.) and corresponds to the mean-field
approximation (MFA). From the microscopic point of view, the MFA amounts to
the average of the internal magnetic field over the different impurity spin
configurations (with respect to their indirect exchange interaction), which is
identical for each magnetic ion (i.e., no spatial fluctuations). This mean
field generates the spatially uniform charge carrier and magnetic ion
magnetizations. Subsequent splitting in their mutual spin spectra is sustained
at temperatures $T<T_{c}$, where $T_{c}$ is the FM phase-transition
temperature [12].

While the MFA is valid for sufficiently large magnetic ion concentrations
$n_{i}$ (e.g., $n_{i}k_{F}^{3}\sim 1$ in 3D systems, where $k_{F}$ is the
Fermi wavevector [14, 13, 12]), the composition and spin fluctuations in the
magnetic ion ensemble can become substantial at smaller densities (more
precisely, smaller $n_{i}k_{F}^{3}$ for 3D), leading eventually to its
failure. This physical picture indicates qualitatively that at a given
$n_{i}$, there should exist a critical free charge-carrier concentration
$n_{e}$ (an areal density in the 2D case) such that at $n_{e}<n_{e,cr}$, the
phase-transition temperature $T_{c}$ becomes zero and the long-range FM order
ceases to exist. As the 2D Fermi wavevector $k_{F}=(2\pi n_{e})^{1/2}$ is
related to the free carrier density, $n_{e,cr}$ can be well expressed through
$k_{F}$ and then through the Fermi energy $E_{F}$ in a parabolic energy band
with an effective mass $m^{\ast}$. Moreover, the constant density of states in
the 2D case leads to the essential equivalence of $E_{F}$ and the chemical
potential $\mu$ when the underlying electron gas is sufficiently degenerate.
This conveniently permits us to use $\mu$ as a control parameter for the
manipulation of FM order characteristics (like local magnetization, spin
polarization of charge carriers, etc.) in the 2D semiconductor structures.
Unlike the metallic counterparts, $n_{e}$ (and thus $\mu$) in a dilute
magnetic semiconductor (DMS) [14, 15, 16] can be controlled independently of
$n_{i}$ via a number of methods (such as an external bias or additional
doping), highlighting its versatility in applications.

The purpose of the present paper is to analyze theoretically the effect of the
random distribution of magnetic impurities on the formation of the long-range
FM order in the 2D DMS structures. The geometric confinement of the structures
under consideration enables the application of the Ising model for the spin-
spin interaction of the magnetic impurities when it is mediated by the free
carriers experiencing a spin-orbital field directed normal to the 2D plane.
Our analysis based on the RKKY formalism clearly illustrates that randomizing
the spin-spin interaction results in the gradual suppression (down to complete
elimination) of magnetic order at the relatively short periods of Friedel
oscillations compared to the mean inter-ion distance (thus, in the regime of
high carrier concentrations). Similarly, it is also revealed that the thermal
distribution of the free carriers makes the FM order impossible at/below the
low values of the chemical potential. These findings clearly indicate the
existence of a limited range of free carrier densities favorable for the FM
order unlike in the MFA. The investigation further highlights the optimum
conditions to achieve the maximum critical temperature $T_{c}$. A numerical
calculation is provided by using a DMS quantum well (QW) as an example along
with a brief discussion on another magnetically doped 2D system, namely, the
few-layered van der Waals materials.

##  II Theoretical model

As discussed above, it is convenient to express everything in terms of the
chemical potential $\mu$. Since $\mu$ is directly proportional to $n_{e}$
($\mu\approx E_{F}\sim n_{e}$), the problem can be classified into two
regimes. The first corresponds to a small charge carrier concentration, where
the spatial dependence of the 2D RKKY potential (see below) is unimportant.
Thus, the mean-field treatment can be used. Moreover, the MFA in this case is
well described by the Kondo-like Hamiltonian averaged over the spin states of
localized spin moments [17]. As $n_{e}$ grows, the Friedel oscillations of the
free carrier density become important, causing the fluctuations in the
magnetic ion subsystem and subsequently precluding the application of the
simple (essentially single-impurity) Kondo-like approach. The collective
behavior of the magnetic ions can instead be described by the random Ising
Hamiltonian with the exchange energy $J(r)$ [18, 19] in the form of the 2D
RKKY interaction.

We begin with the case of a relatively small $n_{e}$, corresponding to the
transition from a nondegenerate 2D carrier gas to a degenerate one. Here, the
Kondo-like Hamiltonian of the carrier-ion exchange interaction takes the usual
form

$\mathcal{H}_{K}=\frac{I}{N_{0}}\sum_{j}\mathbf{S}_{j}\cdot\mathbf{s}~{}\delta(\mathbf{r}-\mathbf{R}_{j}),$
(1)

where $I$ is the carrier-ion exchange constant (in units of energy,
characterizing the confinement effect in our 2D structure), $N_{0}$ is the
areal density of cation sites, and $\mathbf{S}_{j}$ denotes the impurity spin
at site $j$ (positioned at $\mathbf{R}_{j}$ in a host lattice) interacting
with an itinerant spin $\mathbf{s}$ at location $\mathbf{r}$. To be specific,
let us apply $\mathcal{H}_{K}$ to the lowest heavy-hole subband, which is
separated from the light-hole band due to the spin-orbit interaction. As the
latter interaction quantizes the spin along the direction normal to the 2D
plane (say, the $z$ axis), a fictitious spin operator $S^{*}=\pm 3/2$
represents the carrier spin in the basis of heavy-hole eigenfunctions [20].
This transformation leads to the effective Hamiltonian

$\mathcal{H}_{em}=\frac{I}{3N_{0}}\sum_{j}S_{j,z}S_{z}^{\ast}\delta(\mathbf{r}-\mathbf{R}_{j}),$
(2)

where the interaction is reduced to the coupling of spin $z$-components, i.e.,
the Ising form of the carrier-ion exchange interaction. Note that the
interaction with light holes may modify Eq. (2) involving the terms
proportional to the transversal spin components. However, their contributions
can be neglected when the separation between two hole subbands is sufficiently
larger than the thermal energy. Further, $\mathcal{H}_{em}$ can be made to
resemble the Kondo Hamiltonian in Eq. (1) by defining the valence band spin
operator $S_{e}$ as $\frac{1}{3}S_{z}^{*}$.

The mean-field treatment of the Hamiltonian $\mathcal{H}_{em}$ starts from the
introduction of mean free carrier $\langle S_{e}\rangle$ and magnetic ion
$\langle S_{i}\rangle$ spin polarizations. Supposing a simple heavy-hole band
structure with an isotropic in-plane effective mass $m^{\ast}$, the mean
carrier-spin polarization $\langle S_{e}\rangle$ can be written in terms of
the spin subband hole populations $n_{\pm}$ with a chemical potential $\mu$ as

$\langle S_{e}\rangle=\frac{1}{2}\frac{n_{+}-n_{-}}{n_{+}+n_{-}},$ (3)

where

$n_{\pm}=\sum_{\mathbf{k}}\frac{1}{e^{(\varepsilon_{\mathbf{k}_{\pm}}-\mu)/T}+1}.$
(4)

By convention, temperature $T$ is expressed in units of energy. A finite spin
polarization $\langle S_{e}\rangle$ arises due to a finite polarization
$\langle S_{i}\rangle$ of localized spins, which induces a Zeeman-like energy
of the homogeneous Weiss field modifying the energy of free carriers with 2D
wavevector $\mathbf{k}$

$\varepsilon_{\mathbf{k}_{\pm}}=\frac{\hbar^{2}k^{2}}{2m^{\ast}}\pm\frac{1}{2}Ix_{i}\langle
S_{i}\rangle.$ (5)

Here, $x_{i}=n_{i}/N_{0}$ (i.e., the fraction of impurity magnetic ions in the
host lattice) and $\langle S_{i}\rangle$ is the thermally averaged impurity
spin $S_{j,z}$.

Equations (3)-(5) describe the dependence of $\langle S_{e}\rangle$ on
$\langle S_{i}\rangle$. To determine the phase-transition temperature $T_{c}$,
at which the infinitesimal magnetization appears, the expression for $\langle
S_{e}\rangle$ needs to be linearized in $\langle S_{i}\rangle$. This yields

$\langle
S_{e}\rangle=\frac{Ix_{i}}{2T\left(1+e^{-\xi}\right)\ln\left(1+e^{\xi}\right)}\langle
S_{i}\rangle,$ (6)

where $\xi=\mu/T$. Similarly, the linear approximation for $\langle
S_{i}\rangle$ results in

$\langle
S_{i}\rangle=\frac{S(S+1)}{3\pi}\frac{m^{\ast}I}{\hbar^{2}N_{0}}\ln(e^{\xi}+1)\langle
S_{e}\rangle,$ (7)

where $S$ denotes the spin state of the magnetic impurities. The above set of
relations [i.e., Eqs. (6) and (7)] describe the mutual influence of carrier
and magnetic-ion spin polarizations which become nonzero below a certain
critical temperature $T_{c}$. The condition for $T_{c}$ can also be obtained
from Eqs. (6) and (7) as

$T_{c}\left(1+e^{-\mu/T_{c}}\right)=T_{0},\
T_{0}=\frac{S(S+1)}{6\pi}\frac{m^{\star}}{\hbar^{2}N_{0}}I^{2}x_{i}.$ (8)

As shown, $T_{0}$ is a characteristic temperature (in energy units) which
depends on the type of a host material and magnetic impurities. It actually
corresponds to the FM phase-transition temperature in the limit of high
carrier densities $\mu\gg T_{0}$ in the present mean-field treatment (thus,
with no consideration of the fluctuations in the magnetic impurity ensemble).
The transcendental equation for $T_{c}$ given in Eq. (8) can be solved as a
function of $\mu$ and $T_{0}$ numerically.

It is instructive to compare Eq. (8) with $T_{c}^{3d}$ obtained for a 3D DMS
with the corresponding volume density of cation sites $N_{0}^{3d}$ [17]:

$T_{c}^{3d}=\frac{S(S+1)}{3}\left(\frac{I}{N_{0}^{3d}}\right)^{2}\frac{\chi_{e}^{(1)}}{g^{2}\mu_{B}^{2}}n_{i}n_{e},$
(9)

where $\chi_{e}^{(1)}$ denotes the carrier magnetic susceptibility per
particle and $g$ and $\mu_{B}$ stand for the Landé $g$-factor and Bohr
magneton, respectively. Applying $\chi_{e}^{(1)}=(3/8)g^{2}\mu_{B}^{2}/E_{F}$
for the degenerate carriers allows us to estimate the ratio

$\frac{T_{0}}{T_{c}^{3d}}=2\left(\frac{\pi}{3}\right)^{1/3}(x_{e}^{3d})^{-1/3}$
(10)

in terms of the 3D carrier density $x_{e}^{3d}=n_{e}^{3d}/N_{0}^{3d}$ provided
that the other parameters in 3D and 2D cases coincide. Since the free carrier
density $n_{e}^{3d}$ is normally much smaller than $N_{0}^{3d}$ in a realistic
DMS sample (e.g., $x_{e}^{3d}<10^{-3}$) [14], $T_{c}^{3d}$ is likely to be
significantly lower than $T_{0}$. With $T_{c}$ approaching $T_{0}$ as
discussed above [Eq. (8)], the confinement in a 2D DMS system appears to
provide a clear advantage over the 3D counterpart.

Now let us turn to the second case of degenerate charge carrier gases. To
account for explicitly the disorder in the magnetic impurity subsystem, it is
convenient to eliminate the charge-carrier spin variables from Eq. (2) in
favor of an effective spin-spin interaction between the localized spin
moments. By following the well-known procedure [4, 5, 21, 17], this can be
achieved with Eq. (2) rewritten in terms of an effective Ising-like
Hamiltonian

$\mathcal{H}=\sum_{j<j^{\prime}}J(\mathbf{r}_{ij})S_{zj}S_{zj^{\prime}},$ (11)

where $J(\mathbf{r}_{ij})\equiv J_{ij}$ is the interaction potential in energy
units. The Hamiltonian in Eq. (11) contains two ”sources of randomness”. The
first is the thermal disorder, which means that the spin has a random
projection on the specific $i$-th site of a 2D host lattice. Likewise, the
spatial disorder is the second source as the spin can be randomly present or
absent at a host lattice site. It’s worth noting that our formalism works for
any form of $J_{ij}$ so that the effects like spin splitting at the corners of
the Brillouin zone in some 2D crystal monolayers (see, e.g., Ref. [8] and
references therein) can easily be incorporated.

The indirect interaction of localized spins in a metallic host is usually
thought of in the RKKY form [4, 21]. In the bulk semiconductors with
degenerate electron/hole gases, such an interaction results in the FM ordering
[14, 13, 18, 19]. While the particularities of the electronic band structure
in a specific 2D substance can certainly influence the form of $J_{ij}$ (see,
e.g., Ref. [22]), these details are neglected in demonstrating the universal
features as it does not change the results qualitatively. In the simplest case
of a one-band carrier structure, the RKKY interaction in 2D can be expressed
as [23, 24]

$J(r)=-U_{0}\bigg{[}J_{0}(x)Y_{0}(x)+J_{1}(x)Y_{1}(x)\bigg{]},\ x=k_{F}r\,,$
(12)

where $J_{0,1}$ and $Y_{0,1}$ are Bessel and Neumann functions of the zeroth
and first order, respectively [25], and

$U_{0}=J_{0}x_{e},\ J_{0}=\frac{m^{\ast}I^{2}}{4\pi\hbar^{2}N_{0}},\
x_{e}=\frac{n_{e}}{N_{0}}.$ (13)

$I$ and $N_{0}$ are as defined earlier Eq. (1).

As the impurity ferromagnetism has already been studied for bulk
semiconductors [18, 19], it is illustrative to compare the properties of 2D
and 3D range functions that essentially determine the macroscopic
characteristics in the present treatment (such as the FM phase-transition
temperature). The 3D RKKY potential reads

$J_{3D}(r)=J_{03D}(x_{e}^{3d})^{4/3}F(2k_{F}r),\ F(x)=\frac{x\cos x-\sin
x}{x^{4}},$ (14)

where $J_{0,3D}=I^{2}m^{\ast}(N_{0}^{3d})^{-2/3}(3/\pi)^{1/3}(3/2\hbar^{2})$.
This expression clearly has a form much simpler than that in Eq. (12). At a
small $x$, the range function $F(x)\approx-1/3x$, i.e., it is divergent. At a
large $x$, on the other hand, the range function decays like $x^{-3}\cos x$,
which is rather rapid. In comparison, the asymptotics of the 2D range function
at the small and large values of $x$ become [25]

$\displaystyle\frac{J(r)}{U_{0}}$ $\displaystyle\approx$
$\displaystyle-\frac{2}{\pi}\ln\frac{x}{2},\ x<<1$ (15)
$\displaystyle\frac{J(r)}{U_{0}}$ $\displaystyle\approx$
$\displaystyle\frac{\sin 2x}{\pi x^{2}},\ x>>1.$ (16)

It can be shown that at the lower end of $x$, the 2D range function has a
weaker, logarithmic divergence than that in the 3D case ($\sim 1/x$).
Similarly, the decay to zero at the other end is also slower in the case of
the 2D range function. Figure 1 provides a numerical evaluation of these
functions for the full range of $x=k_{F}r$. As expected, the 3D range function
decreases much faster than its 2D counterpart at a large $x$. More
specifically, its amplitude at $x>4$ is approximately ten times smaller than
that for the 2D range function. The observed weaker divergence ($x\rightarrow
0$) and slower decay ($x\rightarrow\infty$) (thus, the enhanced indirect
exchange interaction) is the condition desirable for a higher FM phase-
transition temperature, indicating further the potential advantage of the 2D
structures over the 3D systems. This fact also follows from quantitative
estimation of Eq. (10).

Figure 1: Comparison of the range functions for the RKKY interaction
potentials in the 2D (black, dashed line) and 3D (red, solid line) spatial
dimensions. The inset provides a magnified view of the 3D RKKY range function.

With an explicit form of the interaction in place [Eq. (12)], we are now in a
position to take advantage of the random-field method that has initially been
developed for bulk 3D samples [18, 19]. In this approach, any spin $S_{zj}$ is
treated as a source of random field $H_{zi}\equiv\sum_{j\neq
i}J(\mathbf{r}_{ij})S_{zj}$ which acts on other similar spins. Then, all
observable properties of the system are determined by the distribution
function $f(H)$ of the random field $H$. More precisely, any spin average
$\overline{\langle A\rangle}$ has the form $\int A(H)f(H)dH$, where the bar
stands for the averaging over the spatial disorder. In addition, $A(H)$ is a
single-particle thermal average with an effective form of the Hamiltonian
${\cal H}$ [18, 19],

${\cal H}_{\mathrm{eff}}=\sum_{i}H_{zi}S_{zi}.$ (17)

The explicit expression for the distribution function $f(H)$ reads

$f(H)=\left\langle\overline{\delta\left(H-\sum_{j(\neq
i)}J(\mathbf{r}_{ij})S_{zj}\right)}\right\rangle.$ (18)

As the configurational averaging (i.e., over the spatial disorder) and the
thermal averaging cannot be achieved exactly in Eq. (18), we apply an
alternative approach, i.e., the self-consistent averaging in the spirit of the
statistical theory of magnetic resonance line shape [26]. By using the
spectral representation of the $\delta$ function, a set of self-consistent
equations can be obtained for the spin averages
$m_{l}=(-1)^{l}~{}\overline{\langle S_{z}^{l}\rangle}$, $l=1,2,$… (analogous
to the $l$-th order moment in a sense). The macroscopic magnetization
$\mathcal{M}$ simply becomes $\mathcal{M}=g\mu_{B}m$ (with $m\equiv m_{1}$).

For an arbitrary spin $S$, the explicit form of this set reads [18, 19]

$\displaystyle
f(H)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{iH\rho+\mathcal{G}(\rho)}d\rho,$
(19a)
$\displaystyle\mathcal{G}(\rho)=\left\langle\overline{n_{i}\int\left(e^{-iJ(\mathbf{r})\sigma\rho}-1\right)d^{2}r}\right\rangle$
$\displaystyle=\sum_{\sigma=1/2}^{S}\bigg{\\{}a_{\sigma}\mathcal{F}_{0}(\sigma\rho)+ib_{\sigma}\mathcal{F}_{1}(\sigma\rho)\bigg{\\}},$
(19b) $\displaystyle
a_{\sigma}+ib_{\sigma}=\int_{-\infty}^{\infty}\bigg{[}A_{\sigma}(h)+iB_{\sigma}(h)\bigg{]}f(H)dH,$
(19c)
$\displaystyle\mathcal{F}_{0}(z)+i\mathcal{F}_{1}(z)=n_{i}\int[\exp\left(iJ(\mathbf{r})z\right)-1]d^{2}r,$
(19d) $\displaystyle A_{\sigma}(h)+iB_{\sigma}(h)=\frac{2}{Z_{S}}[\cosh(\sigma
h)+i\sinh(\sigma h)],$ (19e) $\displaystyle
Z_{S}=\sum_{\sigma=-S}^{S}e^{-\sigma
h}=\frac{\sinh[(S+1/2)h]}{\sinh(h/2)};h=\frac{H}{T}.$ (19f) The self-
consistency is achieved by inserting Eq. (19a) into Eq. (19c) and integrating
over $H$. This yields  $\displaystyle
a_{\sigma}+ib_{\sigma}=\int_{-\infty}^{\infty}\left[\mathcal{A}_{\sigma}(z)+i\mathcal{B}_{\sigma}(z)\right]$
$\displaystyle\times\exp\bigg{[}\sum_{\sigma=1/2}^{S}\bigg{\\{}a_{\sigma}\mathcal{F}_{0}(z_{\sigma})+ib_{\sigma}\mathcal{F}_{1}(z_{\sigma})\bigg{\\}}\bigg{]}dz,$
(20a)
$\displaystyle\mathcal{A}_{\sigma}(z)+i\mathcal{B}_{\sigma}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}[A_{\sigma}(h)+iB_{\sigma}(h)]$
$\displaystyle\times\exp(izh)dh,\ z_{\sigma}=\frac{\sigma z}{T}.$ (20b) The
above equations are valid for Ising spin of arbitrary magnitude $S$. Below we
apply these equations to the representative case of spin 1/2 as well as S=5/2.
A typical example is Mn ions which are ubiquitous magnetic impurities in 2D
and 3D DMSs (see Refs. [8, 6, 14] and references therein).

For $S=1/2$, we have $\sigma=\pm 1/2$ and the governing equations result in
the dimensionless magnetization

$m=\int_{-\infty}^{\infty}\tanh\left(\frac{H}{2T}\right)f(H)dH,$ (21)

where $f(H)$ is defined by Eq. (19a) with

$\displaystyle\mathcal{G}(\rho)$ $\displaystyle=$
$\displaystyle\mathcal{F}_{0}\left(\frac{\rho}{2}\right)+i\mathcal{F}_{1}\left(\frac{\rho}{2}\right),$
(22a) $\displaystyle\mathcal{F}_{0}\left(\frac{\rho}{2}\right)$
$\displaystyle=$ $\displaystyle 2\pi
n_{i}\int_{0}^{\infty}\bigg{[}\cos\left(J(r)\frac{\rho}{2}\right)-1\bigg{]}rdr,$
(22b) $\displaystyle\mathcal{F}_{1}\left(\frac{\rho}{2}\right)$
$\displaystyle=$ $\displaystyle 2\pi
n_{i}\int_{0}^{\infty}\sin\left(J(r)\frac{\rho}{2}\right)\ rdr.$ (22c) Then,
Eq. (21) assumes the form
$m=T\int_{0}^{\infty}\frac{e^{\mathcal{F}_{0}(\rho)}}{\sinh\frac{\pi\rho
T}{2}}\sin\left[m\mathcal{F}_{1}(\rho)\right]d\rho$ (23)

following the integration over $H$ [19]. This expression defines the
dimensionless magnetization $m$ in a self-consistent manner.

The MFA asymptotics of Eq. (23) corresponds to $n_{i}\rightarrow\infty$ [19],
which reduces to $\rho\rightarrow 0$ as well as
$\mathcal{F}_{0}(\rho)\rightarrow 0$. In this case, we can obtain from Eq.
(23)

$\displaystyle m$ $\displaystyle=$ $\displaystyle
T\int_{0}^{\infty}\frac{\sin\left[m\rho W_{0}\right]}{\sinh\frac{\pi\rho
T}{2}}d\rho,$ (24) $\displaystyle W_{0}$ $\displaystyle=$ $\displaystyle 2\pi
n_{i}\int_{0}^{\infty}rJ(r)dr\equiv T_{0}.$ (25)

Here, $W_{0}$ actually corresponds to $T_{0}$ defined earlier in Eq. (8),
which is the Curie temperature $T_{c}$ in the degenerate regime based on the
so-called homogeneous Weiss field approximation [4]. This coincidence between
the results of two different approaches is not accidental. It actually stems
from the fact that the RKKY model implicitly takes into account the first-
order contribution in the carrier-ion exchange coupling (i.e., the homogeneous
Weiss field) along with the fluctuating second-order component to ensure the
convergence of integral over the carrier wavevectors [17]. As such, the
average over the RKKY oscillations [see the integration in Eq. (25)] cancels
out the second-order term, leaving the contribution from the first-order
intact.

Evaluation of the integral in Eq. (24) (i.e., the MFA asymptotics) yields, as
expected, the well-known expression of the mean-field magnetization for the
spin 1/2 Ising model

$m=\tanh\frac{mT_{0}}{T}.$ (26)

To obtain the phase-transition condition from Eq. (26), we apply the usual
procedure $m\rightarrow 0$, which generates once more $T=T_{0}$. This
procedure can be regarded as a consistency check for our approximation.

The same procedure $m\to 0$, when applied to the more accurate relation of Eq.
(23), leads to the following random-field expression for $T_{c}$

$1=T_{c}\int_{0}^{\infty}\frac{\mathcal{F}_{1}(\rho)e^{\mathcal{F}_{0}(\rho)}\
d\rho}{\sinh\frac{\pi\rho T_{c}}{2}}.$ (27)

Contrary to the MFA shown in Eq. (26), this relation indicates the existence
of a critical condition associated with the case $T_{c}=0$. More specifically,
at $T_{c}\to 0$, Eq. (27) can be reduced to

$\frac{2}{\pi}\int_{0}^{\infty}\frac{\mathcal{F}_{1}(\rho)}{\rho}e^{\mathcal{F}_{0}(\rho)}\
d\rho=1.$ (28)

The resulting condition is a complex function of $n_{i}$ and $n_{e}$ (thus,
$\mu$). For a given host material and the magnetic ion density $n_{i}$, it
specifies the free carrier concentration beyond which the long-range order is
impossible in the system even at zero temperature.

The situation for $S=5/2$ is qualitatively the same, while the derivations are
much more cumbersome. In fact, it is difficult even to write a closed form
expression for the dimensionless magnetization. After some algebra, we arrive
at the following equation for $T_{c}$

$\displaystyle 1$ $\displaystyle=$
$\displaystyle\frac{2T_{c}}{7}\int_{0}^{\infty}\mathcal{F}_{11}(\rho)e^{\mathcal{F}_{01}(\rho)}L_{5/2}\left(\pi\rho
T_{c}\right)d\rho,$ (29a) $\displaystyle\mathcal{F}_{01}(\rho)$
$\displaystyle=$ $\displaystyle\frac{2\pi
n_{i}}{3}\int_{0}^{\infty}\bigg{(}\cos\frac{5}{2}\zeta+\cos\frac{3}{2}\zeta+\cos\frac{1}{2}\zeta-3\bigg{)}rdr,$
$\displaystyle\mathcal{F}_{11}(\rho)$ $\displaystyle=$ $\displaystyle 2\pi
n_{i}\int_{0}^{\infty}\bigg{(}\sin\frac{5}{2}\zeta+\frac{3}{5}\sin\frac{3}{2}\zeta+\frac{1}{5}\sin\frac{1}{2}\zeta\bigg{)}rdr,$
$\displaystyle\zeta$ $\displaystyle=$ $\displaystyle\rho J(r).$ (29b) Here
$L_{5/2}(z)=\coth\frac{z}{6}-\coth z$. The expression for the critical
concentration can be derived from Eq. (29a) via the asymptotic relation
$L_{5/2}(z\rightarrow 0)=5/z$ to yield
$\frac{10}{7\pi}\int_{0}^{\infty}\frac{\mathcal{F}_{11}(\rho)}{\rho}e^{\mathcal{F}_{01}(\rho)}d\rho=1.$
(30)

Equations (29a) and (30) are solved numerically in Sec. III.

##  III Results and Discussion

For the numerical calculation of the phase-transition temperature, it is
convenient to express both $T_{c}$ and $\mu$ in units of $T_{0}$. In these
units, Eq. (8) assumes the form

$y\left(1+e^{-\frac{\xi_{0}}{y}}\right)=1,$ (31)

where $y=T_{c}/T_{0}$ and $\xi_{0}=\mu/T_{0}$. Equation (31) indicates $y\to
1$ as $\xi_{0}\gg 1$; i.e., $T_{c}$ cannot exceed $T_{0}$. Furthermore,
$T_{c}$ appears to attain its asymptotic value by $\mu\approx 5T_{0}$. Being
based on the mean-field treatment, the dependence $y(\xi_{0})$ in Eq. (31) is
expected to remain valid up to a moderately degenerate carrier gas in a 2D
semiconducting host (obviously including the nondegenerate case). By contrast,
Eq. (27) or (29a) can be used to describe $y(\xi_{0})$ for high values of
$\xi_{0}$ (thus, $\mu$). Assuming magnetic ions Mn2+ with $S=5/2$ as an
example, the numerical solution of Eq. (29a) similarly shows that
$T_{c}\approx T_{0}$ at $\mu\approx 5T_{0}$ when approaching from the
opposite, heavily degenerate regime. Thus, the chemical potential around
$5T_{0}$ comprises the crossover between the two $n_{e}$ regimes, enabling the
interpolation between them.

Figure 2 illustrates the combined outcome of $T_{c}$ vs. $\mu$ in the full
range of $\mu$ (with $S=5/2$). Evidently, the maximal $T_{c}$ (or its close
vicinity) can be achieved only in a relatively narrow range of $\mu$ near the
crossover point. This is different from the mean-field model which predicts
$T_{c}=T_{0}$ once $\mu$ becomes sufficiently large. As $\mu$ is lowered,
$T_{c}$ shows a rather rapid but continuous decrease to a value $\approx
0.22T_{0}$ (denoted as $T_{c,tr}$) corresponding to
$\mu_{min}\approx-0.28T_{0}$ and then the solution ceases to exist abruptly.
This threshold behavior originates from the minimal density of free carriers
needed to mediate the indirect exchange interaction. The latter restriction
qualitatively distinguishes a 2D case from the 3D one, where $T_{c}^{3d}$ can
be arbitrarily small but does not vanish even at an infinitesimal carrier
density [27].

In the heavily degenerate regime, $T_{c}$ also reveals a gradual decrease but
this time to zero. The $T_{c}=0$ condition from Eq. (30) gives the critical
value $\mu_{cr}\approx 16.7T_{0}$. This decay to zero is because the averaging
over the spatial disorder cancels out for $\mu\geq\mu_{cr}$ due to the rapid
oscillations of the RKKY range function. In fact, the above observation
reflects the fact that the ratio $\mu/T_{0}$ represents the geometric factor
proportional to $(\bar{R}/\bar{r})^{2}$, where $\bar{R}$ ($\sim n_{i}^{-1/2}$)
is a mean distance between magnetic ions and $\bar{r}$ ($\sim
k_{F}^{-1}\sim\mu^{-1/2}$) approximates the RKKY oscillation period shown in
Fig. 1. Interestingly, its inverse $(\bar{r}/\bar{R})^{2}$ can be interpreted
as the mean number of magnetic ions interacting coherently with each other by
the dominant FM spin-spin coupling. As increasing $\mu$ reduces $\bar{r}$ and
the number of ions interacting coherently, the FM order becomes unsustainable
beyond a certain critical value (i.e., $\mu_{cr}$). Combined with the analysis
in the non-/weakly degenerate regime discussed earlier, our model predicts
that the FM ordering can be achieved only for $\mu_{min}<\mu<\mu_{cr}$ with
the optimum condition around $5T_{0}$.

Figure 2: FM phase-transition temperature $T_{c}$ vs. the chemical potential
$\mu$ (both in units of $T_{0}$) in a 2D DMS with $S=5/2$. The abscissa axis
also corresponds to a ratio between the free-carrier and magnetic ion
densities $n_{e}$ and $n_{i}$ (except a constant). In the mean-field
treatment, $T_{c}$ would stay at $T_{0}$ with no dependence on $\mu$ in the
highly degenerate regime. The dashed line projects the condition at which the
FM ordering ceases to exist in the nondegenerate regime.

Note that the overall behavior of $T_{c}(\mu)$ for $\mu\gtrsim 5T_{0}$ appears
similar to that in 3D bulk samples [18]. This is to be expected judging from
the comparable characteristics of the RKKY range functions (except the
magnitude) described in Fig. 1. The only difference is the exact form of the
normalization factor $T_{0}$ which shows disparate functional dependences in
the 2D and 3D cases. On the other hand, this very difference in $T_{0}$
illustrates a distinct feature of the 2D system in the non-/weakly degenerate
regime. As the curve $T_{c}(\mu)$ in the bulk samples has shown qualitative
accord with the experiments and Monte Carlo simulations in the degenerate
regime [28], it is reasonable to anticipate a similar level of agreement in
the 2D structures under consideration. Of course, it should be noted that the
current 2D model is limited by consideration of Ising impurity spins as
described above.

For numerical estimation of $T_{c}$ in a realistic case, a QW of Cd0.9Mn0.1Te
is chosen as a specific example. The values of the relevant parameters found
in the literature [29] are the carrier-ion exchange constant $I=0.88$ eV and
the hole effective mass $m^{\ast}=0.8m_{0}$, where $m_{0}$ is the free
electron mass. In addition, the 2D hole density $n_{e}$ is estimated to be
$\approx 10^{11}$ cm-2 that leads to $\mu/T_{0}\cong 8$ at $x_{i}=10\%$.
Substituting all these values to the expression predicts $T_{c}\approx 180$ K,
which is a high value for a DMS. This analysis further suggests that $T_{c}$
can be increased by another 30 % or so (to $\sim 240$ K) if the free hole
density is lowered (not raised contrary to the conventional perception) by
about 40 % to the desired $\mu/T_{0}\approx 5$. Controlling $n_{e}$ (thus,
$\mu$) independent of $n_{i}$ is clearly possible, which is particularly so in
the 2D structures. Note that our estimation of $T_{c}$ is rather rough as the
values of the material parameters are temperature, pressure and other external
stimuli dependent. Nevertheless, the results strongly indicate that the FM
ordering can be achieved even at/above room temperature when the 2D DMS
systems are properly optimized. For instance, recent ab initio calculations
predicted the carrier-ion exchange constant significantly larger than 1 eV in
magnetically doped 2D transition-metal dichalcogenides along with comparable
hole effective masses [30, 31]. Hence, it is not unreasonable to expect a
substantial enhancement of $T_{c}$ in these structures, where the modulation
of free carrier concentrations over a wide range can be readily achieved [32].

##  IV Summary and Outlook

Possible magnetic long-range order in the doped planar semiconductor
structures with Ising impurity spins exhibits a large body of interesting
physical effects, making them promising candidates for spintronic, electronic
and even photovoltaic applications [1, 3, 33]. In this work, we demonstrate
that the magnetic impurities, realizing the Friedel oscillations of the
constituent 2D free carrier gas, may generate room temperature FM order in a
host structure. The conditions suitable to reach the maximum possible $T_{c}$
is elucidated, which can provide a useful guideline for experimental
realization. It is noted that the onset of ferromagnetism considered here is
due only to the RKKY interaction, while there are evidently other mechanisms
(like direct ferro- or antiferromagnetic exchange between the close pairs of
impurities) that can also promote the appearance of magnetic order in the 2D
semiconductor structures [15, 16, 14]. Further, there is another important
effect which is present in all 2D structures except graphene. This effect is
related to the synergy between the RKKY indirect exchange coupling and the
Rashba spin-orbit interaction, leading to the interesting phenomena such as
the strong anisotropy in the resulting $J(r)$ [34]. These and other higher-
order effects are outside the scope of the current study.

###### Acknowledgements.

This work was supported, in part, by the National Science Center in Poland as
a research project No. DEC-2017/27/B/ST3/02881 and by the US Army Research
Office (W911NF-16-1-0472).

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# Achieving 45% efficiency of CIGS/CdS Solar Cell by adding GaAs using
optimization techniques

Satyam Bhatti, Habib Ullah Manzoor, Ahmed Zoha, and Rami Ghannam  James Watt
School of Engineering, University of Glasgow, United Kingdom
Email<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

This paper proposes an efficient three-layered p-GaAs/p-CIGS/n-CdS (PPN), a
unique solar cell architecture. Copper indium gallium selenide (CIGS)-based
solar cells exhibit substantial performance than the ones utilizing cadmium
sulfide (CdS). On the contrary, CIGS-based devices are more efficient,
considering their device performance, environmentally benign nature, and
reduced cost. Therefore, our paper proposes a numerical analysis of the
homojunction PPN-junction GaAs solar cell structure along with n-ZnO front
contact that was simulated using the Solar Cells Capacitance Simulator
(SCAPS-1D) software. Moreover, we investigated optimization techniques for
evaluating the effect of the thickness and the carrier density on the
performance of the PPN layer on solar cell architecture. Subsequently, the
paper discusses the electronic characteristics of adding GaAs material on the
top of the conventional (PN) junction, further leading to improved values of
the parameters, such as the power conversion efficiency (PCE), open-circuit
voltage (VOC), fill factor (FF) and short-circuit current density (JSC) of the
solar cell. The most promising results of our study show that adding the GaAs
layer using the optimised values of thickness as 5 ($\mu$m) and carrier
density as $1\times 10^{20}$ (1/cm) will result in the maximum PCE, VOC, FF,
and JSC of 45.7%, 1.16 V, 89.52% and 43.88 $(mA/m^{2})$, respectively, for the
proposed solar cell architecture.

###### Index Terms:

GaAs, CIGS, CdS, ZnO, SCAPS-1D, Optimization, Simulation, Efficiency,
Homojunction, Thin film solar cells Semiconducting GaAs compounds.

##  I Introduction

With the rapid increase in electricity demand, solar cells are playing a vital
role in producing green, reliable, and efficient energy sources to meet the
United Nation’s sustainable development goals-7 [1]. One of the promising
solar cell architectures involves the thin film CIGS (Copper Indium Gallium
Selenide) and CdS (Copper Sulfide) solar cell, which is well known for
yielding higher Power Conversion Efficiencies (PCE) and generates efficient
Levelized Cost of Electricity (LCOE) with manufacturing costs minimal as
compared to other solar cells architectures [2, 3]. This paper proposes
simulation-based modelling of thin film layer CIGS/CdS solar cells followed by
the optimization of the nanowire GaAs (Gallium Arsenide) layer, which was
added to the top layer of the baseline multi-junction solar cell architecture.
Moreover, the simulations presented in this paper were performed using the
Solar Cells Capacitance Simulator (SCAPS-1D), a one-dimensional solar
simulation tool that provides an in-depth investigation of electronics and
information systems [4, 5, 6]. Moreover, SCAPS-1D evaluates one of the vital
electrical characteristics such as $PCE$, Fill factor $(FF)$, Current Density
curves $(J_{SC})$ and the open circuit voltage $(V_{OC})$ of the solar cells
[7].

In the recent past, an exponential interest in CIGS and CdS solar cells has
recently increased owing to their attractive properties and applications [8].
For the CIGS semiconductor material, the wide direct bandgap range lies
between 1.0 to 1.7 eV, whereas that of the CdS lies within the 2.2 and 2.4 eV
range and is an important parameter in terms of determining the range
wavelengths of light that could be possibly absorbed by the solar cell and
help in predicting the PCE of converting the sunlight into electricity [9].
These factors, specifically the CIGS and CdS semiconductor materials highly
favourable materials for photovoltaic energy conversion applications [10, 11].
Furthermore, CIGS and CdS materials contribute towards high efficiency, low
cost of manufacture, thin-film fabrication technique, environmentally
friendly, varied spectral response, high radiation tolerance and absorption
coefficient [12]. Therefore, research in these materials has led to the
advanced development and fabrication of semiconductor devices,
optoelectronics, and nanotechnology from photovoltaic to biosensors and are
widely used as catalysts in chemical reactions involving hydrogen production
from water [13].

On the contrary, one limiting factor in developing more efficient solar cells
using semiconductor materials lies in their incapability to absorb light from
the solar spectrum [14]. Following this, CdS also led to a concern about
toxicity and stability because of the availability of Cadium in the solar
cells [15]. However, mechanical stacking of multijunction of semiconductor
materials can help to mitigate these concerns whilst enhancing the overall
solar cell efficiency [16]. In our paper, we run a number of simulations
involving the traditional single-junction CIGS/CdS solar cell layers and then
later add the GaAs layer to study the properties of the multijunction solar
cell. From the literature, several attempts have been made to optimize the
overall efficiency of the solar cell and were conducted by [17, 18, 19]. The
main idea behind this analysis is the improvement of the device efficiency
using materials cheaper than conventional CIGS [20]. A 5 $\mu$m of a new layer
p-GaAs has been added for that purpose. Various thicknesses of the CIGS
absorber layer ranging from 0.5 to 5 $\mu$m have been used.

Accordingly, the findings of our study showcase that an increase in the
absorber layer thickness improves the performance and overall power conversion
efficiency of the new CIGS solar cells. Initially, the study incorporated a
window layer of ZnO, a buffer layer (CdS), an absorbent layer (CIGS) and a
GaAs layer with varied values of thicknesses between, 0.5, 1, 1.5, 2, 2.5, 3,
3.5, 4, 4.5 and 5 $\mu$m. Also, the optimisation of the GaAs material is
performed with the help of the heatmap confusion matrix in order to analyse
the most optimized thickness and carrier density of different layers of solar
cells. In addition, a comparison of results including the performance and
efficiency of the PN-junction (optimized CIGS and CdS) solar cell was
estimated. The most promising results of the paper revealed that a thin top
layer of p-GaAs on the conventional solar cell (p-CIGS and n-CdS) with an
optimized thickness layer and high carrier density had a considerable
influence on the performance of the solar cell architecture. Adding a p-GaAs
layer as thin as 5 $\mu$m on the top of the PN-junction solar cell
substantially improved the conversion efficiency of the solar cell from 20.07%
(unoptimized PN) to 45.47 % (optimized PPN). The results showed that the new
ultra-thin CIGS solar cells structure has performance parameters that are
comparable to those of the conventional ones with reduced cost.

Therefore, our paper proposes a simple three-layered p-GaAs/p-CIGS/n-CdS (PPN)
solar cell with a practical thickness and a high conversion efficiency. The
PPN solar cell comprises a high p-GaAs composition on top of p-CIGS and n-CdS
multi-junction solar cells. In addition, computing a thin p-GaAs layer created
a graded energy bandgap and the solar PN junction could only absorb photons
having an energy equal to or greater than the energy bandgap. Herein, it is
worth mentioning that a graded energy bandgap has a wider energy bandgap to
absorb more photons. Moreover, adding an extra thin layer increased the number
of holes in the solar cells, ultimately increasing power conversion
efficiency. The behaviour and performance of the solar cells were evaluated by
performing a comprehensive set of simulations under different configurations
(i.e., thickness and carrier density of the layer) using the SCAPS-1D
software.

###  I-A Organisation of the Paper

Our paper is divided into 8 Sections. Section 2 of the paper discusses the
significance, need and limitations of the CIGS/CdS multijunction solar cell
architecture. Followed by, Section 3 describes the process of setting up our
simulation environment in the SCAPS-1D. Section 4 includes the efficiency
optimization of the CIGS/CdS solar cell with the help of thickness and carrier
density optimization using the heatmap confusion matrix. Section 5 presents
the results of electrical parameters after adding a GaAs layer on top of the
CIGS/CdS solar cell. Next, Section 6 discusses a critical comparison of IV,
PV, and QE characteristics and the high-temperature effect. Lastly, section 7
discusses the obtained results in detail, and concluding remarks are presented
in Section 8.

Figure 1: Figure represents the baseline structure of the proposed multi-
junction solar cell architecture, with simple three p-CIGS/n-CdS/n-ZnO, where
the ZnO layer acts as a window/buffer layer. The solar light in the structure
is incident from the right contact (front) using the simulation tool SCAPS-1D.
The baseline values for the given solar cell architecture are set as mentioned
in the simulation setup whilst the thickness and current density (doping)
values for the baseline are set as 0.5 $\mu$m and $1\times 10^{10}$
(1/$cm^{3}$), respectively.

###  I-B Contributions to the Research Paper

The key contributions to the research article are presented as follows:

TABLE I: The table below depicts the input parameters that were set for the simulation setup. Input Electrical Parameters | Measurement | Layer 1: | Layer 2: | Layer 3: | Layer 4:
---|---|---|---|---|---
| Unit | p-GaAs | p-CIGS | n-CdS | n-ZnO
Bandgap | (eV) | $1.420$ | $1.100$ | $2.450$ | $3.300$
Electron Affinity | (eV) | $4.070$ | $4.500$ | $4.400$ | $4.600$
Dielectric Permittivity | (relative) | $12.900$ | $13.600$ | $10.000$ | $9.000$
Conduction Band Effective Density of states | (1/$cm^{3}$) | $2\times 10^{18}$ | $2.2\times 10^{18}$ | $2.2\times 10^{18}$ | $2.2\times 10^{18}$
Valence Band Effective Density of states | (1/$cm^{3}$) | $1\times 10^{19}$ | $1.8\times 10^{19}$ | $1.8\times 10^{19}$ | $1.8\times 10^{19}$
Electron Thermal Velocity | (cm/s) | $1\times 10^{7}$ | $1\times 10^{7}$ | $1\times 10^{7}$ | $1\times 10^{7}$
Hole Thermal Velocity | (cm/s) | $1\times 10^{7}$ | $1\times 10^{7}$ | $1\times 10^{7}$ | $1\times 10^{7}$
Electron Mobility | ($cm^{2}$/Vs) | $1\times 10^{3}$ | $1\times 10^{2}$ | $1\times 10^{2}$ | $1\times 10^{2}$
Hole Mobility | ($cm^{2}$/Vs) | $1\times 10^{2}$ | $1\times 10^{1}$ | $1\times 10^{1}$ | $2.5\times 10^{1}$

  * •

Discussed the significance, need and limitations along with a thorough
literature review of the CIGS and CdS multijunction solar cell.

  * •

Implemented simulations on the SCAPS-1D tool for designing the most optimized,
high-efficiency and robust solar cell architecture.

  * •

Performed efficiency optimization for the CIGS/CdS multi-junction solar cell
architecture by adjusting the thickness and carrier density using the heatmap
confusion matrix.

  * •

Whilst performing simulations, critical optimization techniques are applied to
the CIGS/CdS solar cell as a first step and then, for CIGS/CdS/GaAs as a
second step with the help of the heatmap confusion matrix ranging from 0.5
$\mu$m to 5 $\mu$m for thickness and 10 (1.00En) (1/$cm^{3}$) and 20 current
density, respectively.

  * •

In the second step, introduced a novel multi-junction solar cell architecture
by adding an n-GaAs layer on top of the pre-existing CIGS/CdS multi-junction
solar cell and carried out the thickness and carrier density optimization to
achieve the highest PCE value.

  * •

Further, investigated the electronic and electrical characteristics such as
PCE, fill factor, current density and open circuit voltage of the
p-CIGS/p-CdS/n-GaAs multijunction solar architecture.

  * •

Lastly, a thorough comparative analysis is presented, showing the IV, PV and
QE characteristics graphs for the most optimized solar cell architecture with
their respective optimized thickness and current density values.

##  II CIGS and CdS Multijunction Solar Cell

Initially, a multijunction solar cell architecture using the CIGS and CdS
semiconductor materials is designed. CIGS and CdS materials are widely used
for manufacturing purposes, mainly because of key parameters such as low
production cost and high solar energy yield. The literature suggests that the
materials have achieved higher PCE values than their counterpart Silicon solar
cells installed on household rooftops. Moreover, the CIGS solar cells use a
thin layer of CIGS as an absorbing layer and have high absorption coefficient
values, which eventually results in high-efficiency solar cells. The flexible
nature of the CIGS semiconductor materials broadens their application in
various industries, research and industry perspectives such as photovoltaic,
biosensors, portable electronic devices, off-grid power systems, utility-scale
solar power plants, for powering remote sensing and for communication devices
[21].

Subsequently, the semiconductor CdS is used as the window or also known as the
buffer layer in the solar cells. The window layer is defined as a thin,
transparent allowing the sunlight to pass through the absorber layer and also
restricts the unwanted recombination of the holes and electrons generated by
the p-CIGS (absorbed photons) layer. Moreover, the window layer is typically
made of a transparent conducting oxide (TCO), such as indium tin oxide (ITO),
fluorine-doped tin oxide (FTO), or zinc oxide (ZnO). These materials have high
transparency to visible light and low electrical resistance, which allows them
to efficiently collect and transport the electrons generated by the absorbed
photons. Furthermore, the choice of window layer material depends on several
factors, including the specific absorber material used in the solar cell, the
desired efficiency, and the cost of the materials. Different TCOs have
different properties that can affect the performance of the solar cell, such
as their work function, surface roughness, and doping level.

Therefore, the window layer plays a critical role in the performance of a
solar cell by controlling the flow of charge carriers and minimizing energy
losses due to recombination, while also allowing sunlight to pass through to
the absorber layer. Accordingly, in the multijunction solar cell of
p-CIGS/p-CdS, CdS is often used as the window layer in the respective solar
cell architecture. Additionally, the transparent window layer sits on top of
the absorbing layer and allows light to pass through to reach the CIGS layer.
CdS has a high optical transmission, which allows it to efficiently transmit
light to the CIGS layer, while also helping to protect the CIGS layer from
degradation due to exposure to air and moisture. However, it’s important to
note that cadmium is a toxic substance, raising concerns about CdS solar
cells’ environmental impact. Subsequently, efforts are being made to develop
alternative materials for window layers, such as zinc oxide (ZnO), a non-toxic
alternative to CdS.

###  II-A CIGS/CdS: Significance and Need

Previous research studies indicate that the combination of CIGS and CdS
multijunction solar cells is known to provide promising electrical and
electronic characteristics even in harsh environments along with achieving
considerably higher efficiencies. CIGS and CdS solar cell layers are used due
to the fact that they offer numerous advantages over the widely existing
materials manufactured in the solar industry. Not only, CIGS has high
efficiency in converting sunlight into electricity but also it has the
capacity to generate more power (Ws) per unit of surface area (A) in
comparison with other materials. Accordingly, this makes CIGS and CdS
materials ideal for applications in solar cells where either the space is
limited or reduced size is targeted, for example, rooftops or portable
devices. In addition, CIGS materials are known to be flexible, which allows
them to be used in various applications, including curved or irregular
surfaces. Moreover, this property also makes the design of solar cells
suitable for building-integrated photovoltaics (BIPV) where they are
integrated into the design of buildings and other complex structures. Lastly,
CdS materials proved to be an excellent window or buffer layer material due to
their high optical transmission, which allows them to efficiently transmit
light to the CIGS layer while also providing protection from environmental
factors that can degrade the solar cell [22]. Overall, the combination of CIGS
and CdS offers a high-efficiency, flexible, and durable solution for solar
cells that can be used in a wide range of applications.

Figure 2: Figure represents the optimization technique used to evaluate the
most efficient value of the thickness for the p-CIGS and n-CdS semiconductor
materials using the heatmap confusion matrix. In addition, (a) Outputs the
Efficiency, $\eta$ (%), (b) Outputs the Fill Factor, FF (%), (c) Current
Density (A/$cm^{3}$), (d) Open Circuit Voltage ($V_{OC}$), (e) Comparison of
the IV and PV characteristics after the thickness optimization, (f) Quantum
Efficiency (%) Curve.

###  II-B CIGS/CdS: Limitations

While CIGS and CdS solar cells offer many advantages, there are also some
challenges associated with their use. One challenge is the cost of producing
CIGS solar cells. The materials used to make CIGS solar cells are relatively
expensive, making the manufacturing process more costly than other types of
solar cells. This has limited the widespread adoption of CIGS solar cells,
especially in utility-scale applications. Another challenge is the potential
environmental impact of using CdS as a window layer material. Cadmium is a
toxic substance, and concerns have been raised about the possibility of
environmental contamination from CdS solar cells during their production, use,
and disposal. Efforts are being made to develop non-toxic alternatives to CdS
for use in window layers, such as zinc oxide (ZnO), but these materials are
not yet widely used in commercial applications. Finally, CIGS solar cells are
susceptible to degradation over time due to exposure to moisture and air. This
can reduce the efficiency of the solar cell and limit its lifespan. However,
research is ongoing to develop new encapsulation methods and materials that
can protect the CIGS layer from degradation and extend the lifespan of CIGS
solar cells. Overall, while there are some challenges associated with using
CIGS and CdS solar cells, ongoing research and development efforts are aimed
at addressing these challenges and improving the efficiency and durability of
these solar cell technologies [23].

Figure 3: Figure represents the optimization technique used to evaluate the
most efficient value of the thickness for the p-CIGS and n-CdS semiconductor
materials using the heatmap confusion matrix. In addition, (a) Outputs the
Efficiency, $\eta$ (%), (b) Outputs the Fill Factor, FF (%), (c) Current
Density (A/$cm^{3}$), (d) Open Circuit Voltage ($V_{OC}$), (e) Comparison of
the IV and PV characteristics after the thickness optimization, (f) Quantum
Efficiency (%) Curve.

The overall efficiency of CIGS and CdS solar cells has improved significantly
over the past few decades, making them competitive with other types of solar
cells in terms of efficiency. The current record for CIGS solar cell
efficiency is around 23.35 which is close to the efficiency of traditional
silicon-based solar cells. The high efficiency of CIGS solar cells is due to
their ability to absorb a broad range of wavelengths of light, including the
blue and green parts of the spectrum, which traditional silicon solar cells
cannot absorb efficiently. The CdS window layer in CIGS solar cells also plays
a critical role in their overall efficiency. CdS has high optical
transmission, which allows it to efficiently transmit light to the CIGS layer,
resulting in higher conversion efficiencies. However, as mentioned earlier,
the use of CdS raises concerns about its potential environmental impact.
Overall, the efficiency of CIGS and CdS solar cells is highly competitive with
other types of solar cells, and ongoing research and development efforts are
focused on improving their efficiency and reducing their cost.

The efficiency of CIGS and CdS solar cells can be evaluated in both
simulation-based environments and real-world environments. In simulation-based
environments, the efficiency of solar cells can be modeled using computer
simulations that take into account various factors, such as material
properties, cell design, and environmental conditions. These simulations can
provide insights into the theoretical efficiency of solar cells and can help
guide the design and optimization of solar cell materials and structures. In
real-world environments, the efficiency of solar cells can be evaluated by
measuring their performance under actual operating conditions. Factors such as
temperature, humidity, and shading can affect the performance of solar cells
in real-world environments, leading to variations in efficiency compared to
simulation-based results. Generally, the efficiency of CIGS and CdS solar
cells in real-world environments is lower than in simulation-based
environments due to various factors such as partial shading, thermal losses,
and other environmental factors. However, ongoing research and development
efforts are focused on improving the real-world performance of CIGS and CdS
solar cells, including developing new encapsulation techniques, optimizing the
cell design, and developing new materials for use in the window layer.
Overall, the efficiency of CIGS and CdS solar cells is evaluated in both
simulation-based and real-world environments, and ongoing research is focused
on improving their efficiency and performance in both environments.

Figure 4: Figure represents an optimization of the carrier charge density,
also known as doping concentration (1/$cm$) for the p-CIGS/n-CdS/n-ZnO multi-
junction solar cell architecture incorporating the Heatmap confusion matrix.
Likewise to the thickness optimization, Doping optimisation values are as
follows: (a) Outputs the Efficiency, $\eta$ (%), (b) Outputs the Fill Factor,
FF (%), (c) Current Density (A/$cm^{3}$), (d) Open Circuit Voltage ($V_{OC}$),
(e) Comparison of the IV and PV characteristics after the thickness
optimization, (f) Quantum Efficiency (%) Curve.

##  III Simulation Setup

For the analysis of the different solar cell architectures, a simulation
environment was developed on the SCAPS-1D tool for examining the electronic
and electrical parameters along with measuring the PCE, FF, VOC and JSC
values. Initially, a solar cell was designed using the three-layered P-N-N
semiconductor layers. With light incident from the right contact (front),
followed by a layer of n-ZnO, n-CdS and p-GaAs to the left contact (back)
formed a multi-junction solar cell architecture. The incident light plays a
pivotal role in determining the efficiency of the solar cell and thus, in our
simulation setup, we introduced the incident light from the right contact
(front) throughout the simulations settings. Furthermore, the initial working
point of the simulations was set as follows: Temperature - 300K, Voltage - 0V,
Frequency - $10^{6}$ Hz and the number of points as 5.

In addition to this, for plotting the electrical characteristics curves, the
settings for Pause at each stop were applied as V1 (0V to -0.8V), V2 (0.8V to
0.8V), frequency (f1: $10^{2}$ Hz to f2: $10^{6}$ Hz) and the wavelength (WL1:
300nm to WL2: 900nm). Moreover, at each step, the number of points was set as
41, 81, 21, and 61 with an increment of 0.02V, 0.02V, 5 points per decade, and
10nm, respectively. It is worth mentioning that the author later in the study
also introduced a layer of GaAs with the aim of achieving the highest
efficiency and accordingly, all the settings of the working point remain the
same for different multi-junction solar cell architecture. Subsequently,
figure 1 represents the direction of the incident light along with the
p-CIGS/n-CdS/n-Zno multi-junction solar cell architecture and Table 1 depicts
the input electrical parameters such as the bandgap, electron affinity (eV),
dielectric permittivity (relative), conduction band effective density of
states, valence band effective density of states, electron thermal velocity,
hole thermal velocity, electron and hole mobility of the respective, 4 layered
solar cell architecture.

Figure 5: Figure represents an optimization of the carrier charge density,
also known as doping concentration (1/$cm$) for the p-CIGS/n-CdS/n-ZnO multi-
junction solar cell architecture incorporating the Heatmap confusion matrix.
Likewise to the thickness optimization, Doping optimisation values are as
follows: (a) Outputs the Efficiency, $\eta$ (%), (b) Outputs the Fill Factor,
FF (%), (c) Current Density (A/$cm^{3}$), (d) Open Circuit Voltage ($V_{OC}$),
(e) Comparison of the IV and PV characteristics after the thickness
optimization, (f) Quantum Efficiency (%) Curve.

##  IV Efficiency Optimization

After incorporating the necessary input electrical values and setting up the
simulation environment, in the next step, the authors conducted the efficiency
optimization consisting of three important steps of the study. Accordingly, to
meet the objective of this paper, a critical optimization of different solar
cell architectures is performed. The first step of optimization involves
varying the thickness of the CIGS and CdS layers in the multi-junction solar
cell of p-CIGS/n-CdS/n-ZnO. Followed by, the second step included the same
solar cell architecture, however, the efficiency optimization is evaluated
with the help of changing the doping values. Lastly, the third step introduces
a GaAs layer to the existing solar cell architecture and accordingly, both the
thickness and doping optimization is performed simultaneously.

###  IV-A STEP 1: Optimization of Thickness for CIGS and CdS layer

Initially, a baseline solar cell architecture is proposed consisting of
p-CIGS/n-CdS/n-ZnO and as a first step, the thickness optimization is
performed for estimating the overall efficiency of the solar cell at different
values of thickness that range from 0.5 to 5.0 $\mu$m, at an increment of 0.5
$\mu$m. Accordingly, the increment in the values of thickness is performed in
such a manner that an increment is made for each 0.5 $\mu$m value for both the
CIGS and CdS semiconductor material of the solar cell independently.
Therefore, to present a relation between the two parameters’ increment of
thickness, the authors used the heatmap correlation matrix to provide a clear
indication of the values of the overall performance of the solar cell at
varied values of thickness of both materials.

The estimation of thickness is highly essential for designing the most
optimized solar cell due to cost saving of materials and subsequently, the
solar cell manufacturers could use the materials optimally. The results from
the heatmap confusion matrix showcase four parameters as Efficiency, fill
factor, current density and open circuit voltage. For thickness optimization
of the materials p-CIGS/n-CdS, the heatmap confusion matrix indicates a
thickness value of 0.5 $\mu$m for CdS and 5.0 $\mu$m for CIGS, giving a
maximum efficiency of solar cell design as 20.07%. Followed by a thickness of
1 $\mu$m for CdS and 5.0 $\mu$m for CIGS yields a maximum Fill factor of
80.88%. Whereas the maximum values of parameters current density and open
circuit voltage output a 43.59 $A/m^{2}$ and 0.57 V, respectively for
thickness values of 4.5 $\mu$m for CdS and 5.0 $\mu$m for CIGS.

In addition, for the electrical performance of the proposed solar cell, PV, IV
and QE characteristics are plotted to measure the maximum power point tracking
of the solar cell. Accordingly, the maximum value as observed from the IV
curve is 45 $A/m^{2}$ and the maximum power as analysed from the PV curve is
32 W. Subsequently, the maximum quantum efficiency of the baseline proposed
solar cell architecture is 95%.

###  IV-B STEP 2: Optimization of Carrier Density for CIGS and CdS layer

After optimizing the thickness values of the p-CIGS/n-CdS/n-ZnO multi-junction
solar cell architecture, the authors conducted the optimization of the carrier
density parameter of the solar cell using the same input values as set in the
working simulation environment. However, the authors used the most optimized
value of the thickness of the solar cell as calculated in the above
subsection, i.e., 0.5 $\mu$m for CdS semiconductor material and 5.0 $\mu$m for
CIGS material. It is worth mentioning that the authors used this value of
thickness for the remaining simulations of the study.

Figure 6: Figure represents the introduction of the p-GaAs layer to the
proposed baseline multi-junction solar cell architecture. The GaAs layer is
added on top of the p-CIGS layer next to the left contact (back) of the
proposed solar cell architecture for analysing the electrical and electronic
performance in the real-world environment with the help of the SCAPS-1D
simulation tool. Moreover, the light in this scenario is incident from the
right contact (front) to achieve the maximum possible efficiency of the
proposed solar cell architecture.

Furthermore, the optimization of the acceptor density of the CIGS $1.0\times
10^{n}$ ($A/m^{2}$) and the donor density of the CdS $1.0\times 10^{n}$
($A/m^{2}$) is evaluated using the heatmap confusion matrix which gives the
values of the doping concentration for both the CdS and CIGS. Likewise, to the
previous subsection, herein, also the authors measured the electrical
characteristics such as Efficiency (%), Fill Factor (%), Current Density
($A/m^{2}$) and the open circuit voltage (V) for the critical optimization of
the doping concentration. Accordingly, figure 4 (a), (b) and (d) indicate the
optimization of the doping concentration at $1\times 10^{20}$ (1/cm) for both
CIGS and CdS materials yields a maximum value of efficiency, fill factor and
open circuit voltage as 32.12 %, 86.93 % and 0.87 V, respectively. On the
contrary, the optimized value of doping concentration for donor density of CdS
lies in the range of $1\times 10^{10}$ (1/cm) to $1\times 10^{20}$ (1/cm) at
an acceptor density of CIGS as $1\times 10^{10}$ (1/cm) resulted in a maximum
value of current density equivalent to 43.89 ($A/m^{2}$).

Figure 7: Figure represents the output results for the simulations performed
by introducing the GaAs layer to the proposed baseline solar cell
architecture. Herein, the optimization for the thickness and the charge
current density (doping concentration) is evaluated simultaneously using the
same heatmap confusion matrix. Accordingly, electrical characteristics are
presented as: (a) Outputs the Efficiency, $\eta$ (%), (b) Outputs the Fill
Factor, FF (%), (c) Current Density (A/$cm^{3}$), (d) Open Circuit Voltage
($V_{OC}$), (e) Comparison of the IV and PV characteristics after the
thickness optimization, (f) Quantum Efficiency (%) Curve, that includes the
GaAs layer to the proposed multi-junction solar cell.

In addition to this, the electrical characteristics are calculated from the
SCAPS-1D simulation tool. Figure 5 showcases the IV, PV and quantum efficiency
for the most optimized values of the doping concentration values at $1\times
10^{10}$ (1/cm) for the donor density of CdS and $1\times 10^{20}$ (1/cm) for
the CIGS semiconductor materials. Accordingly, the curves indicate that the
maximum value of current density is 43.2 $A/m^{2}$ and the maximum power is
32.91 W. Subsequently, the maximum value of the quantum efficiency at the
optimized inputs of doping concentration results in 96%.

Figure 8: Figure represents the output results for the simulations performed
by introducing the GaAs layer to the proposed baseline solar cell
architecture. Herein, the optimization for the thickness and the charge
current density (doping concentration) is evaluated simultaneously using the
same heatmap confusion matrix. Accordingly, electrical characteristics are
presented as: (a) Outputs the Efficiency, $\eta$ (%), (b) Outputs the Fill
Factor, FF (%), (c) Current Density (A/$cm^{3}$), (d) Open Circuit Voltage
($V_{OC}$), (e) Comparison of the IV and PV characteristics after the
thickness optimization, (f) Quantum Efficiency (%) Curve, that includes the
GaAs layer to the proposed multi-junction solar cell.

##  V Introducing GaAs Layer

One of the most common semiconductor materials used in the multi-junction
solar cell is Gallium Arsenide (GaAs) which is primarily stacked on the top of
either each layer or at the top of all layers proposed in a solar cell
architecture.

Each layer is designed to absorb a specific portion of the solar spectrum,
allowing the cell to capture a broader range of sunlight and convert it into
electricity more efficiently. One common material used in multi-junction solar
cells is Gallium Arsenide (GaAs). GaAs is a semiconductor material with unique
properties that make it well-suited for solar cell applications. Here’s why
GaAs are used as a semiconductor layer in multi-junction solar cells:

  * •

High energy conversion efficiency: GaAs have a relatively high energy
conversion efficiency compared to other semiconductor materials used in solar
cells. It has a direct bandgap, which means it can efficiently convert
sunlight into electricity without losing much energy as heat.

  * •

Wide bandgap: GaAs has a wide bandgap, which allows them to absorb higher-
energy photons from the solar spectrum. By incorporating GaAs in the solar
cell stack, it can absorb photons from the blue and green regions of the
spectrum, which are not efficiently absorbed by other materials such as
silicon (commonly used in single-junction solar cells).

  * •

Tandem cell configuration: In a multi-junction solar cell, the semiconductor
layers are arranged in a tandem configuration, with each layer tuned to absorb
a specific part of the solar spectrum. GaAs are often used as the top layer in
the stack because it has a higher bandgap than other materials, making it
suitable for capturing the higher-energy photons. The layers beneath the GaAs
layer can be designed to absorb lower-energy photons, ensuring efficient use
of the entire solar spectrum.

  * •

Temperature stability: GaAs have excellent temperature stability, allowing
them to maintain their high performance even at elevated temperatures. This
characteristic is crucial for solar cells, as they can heat up under intense
sunlight.

  * •

Mature technology: GaAs has been extensively researched and developed for
various applications, including solar cells. It benefits from a well-
established manufacturing process and has a proven track record in high-
performance photovoltaic devices [24].

  * •

High electron mobility: GaAs have a higher electron mobility compared to other
common semiconductor materials like silicon. This property makes GaAs suitable
for high-speed electronic devices, such as field-effect transistors (FETs) and
integrated circuits, where fast switching and high-frequency operation are
required.

  * •

Low noise characteristics: GaAs exhibit low noise characteristics, making them
ideal for applications in low-noise amplifiers and microwave devices. This
property is particularly advantageous in high-frequency communication systems
and radar technology.

  * •

Wide frequency range: GaAs exhibit excellent performance across a wide
frequency range, including microwave and millimetre-wave frequencies. It
enables the development of devices and circuits for wireless communications,
satellite communications, radar systems, and high-frequency electronics.

  * •

High power handling capability: GaAs materials can handle high power levels
without significant degradation in performance. This property makes GaAs
suitable for power amplifiers and other high-power electronic devices,
including those used in telecommunications and defence applications.

  * •

Optoelectronic applications: GaAs is widely used in optoelectronic devices
such as light-emitting diodes (LEDs), laser diodes, and photodetectors. GaAs-
based LEDs and laser diodes have superior performance in terms of efficiency,
brightness, and wavelength range, making them valuable for applications in
lighting, optical communications, and optical sensing.

  * •

Compatibility with complementary metal-oxide-semiconductor (CMOS) technology:
GaAs can be integrated with CMOS technology, allowing for the development of
hybrid circuits and systems that leverage the advantages of both GaAs and
CMOS. This integration enables the fabrication of high-performance, mixed-
signal devices and integrated circuits with diverse functionality.

  * •

Radiation hardness: GaAs exhibit inherent radiation hardness, meaning they can
withstand the effects of ionizing radiation without significant degradation in
performance. This characteristic makes GaAs suitable for applications in space
technology, nuclear power plants, and high-energy physics experiments.

By incorporating a GaAs semiconductor layer in a multi-junction solar cell,
the overall efficiency of the cell can be significantly increased. GaAs help
capture a broader range of sunlight, including higher-energy photons, and
convert them into electricity more effectively, leading to improved solar cell
performance.

While GaAs materials offer numerous advantages, there are also some
limitations associated with their use. One limitation is the higher cost of
GaAs compared to other semiconductor materials, primarily due to the complex
manufacturing processes involved. This cost factor restricts the widespread
adoption of GaAs in certain applications where cost-effectiveness is a
critical consideration. Additionally, GaAs is a brittle material, making it
more prone to cracking and breakage during handling and fabrication. Moreover,
GaAs-based devices may face challenges in scaling down to smaller dimensions
due to material properties and technological constraints. Despite these
limitations, ongoing research and advancements aim to address these issues and
further enhance the capabilities and cost-effectiveness of GaAs materials for
broader application domains.

###  V-A STEP 3: Optimization of Thickness and Carrier Density for GaAs layer

Furthermore, figure 6 represents the proposed solar cell architecture which
includes another layer of the p-GaAs layer to the baseline multi-junction
solar cell. Subsequently, the GaAs layer is added on top of the p-CIGS layer
next to the left contact (back) of the proposed solar cell architecture for
analysing the electrical and electronic performance in the real-world
environment with the help of the SCAPS-1D simulation tool. Moreover, the light
in this scenario is incident from the right contact (front) to achieve the
maximum possible efficiency of the proposed solar cell architecture. In
addition to this, all working environments of the simulation setup were kept
to same so as to avoid any discrepancy in the analysis of the output results
and thus, the overall performance of the multi-junction solar cell.

Consecutively, the optimization of the GaAs semiconductor material was
performed in terms of the thickness and doping concentration with the value
ranging between 0.5 $\mu$m to 5.0 $\mu$m and $1\times 10^{11}$ (1/cm) to
$1\times 10^{20}$ (1/cm), respectively. Accordingly, figure 7 showcases that
the maximum efficiency and open circuit voltage of the proposed solar cell
architecture is achieved at 45.47 and 1.16 V, respectively, at a thickness of
5.0 $\mu$m and acceptor density of GaAs material of $1\times 10^{20}$ (1/cm).
One of the most promising results in the heatmap confusion matrix indicates
that the current density of the solar cell remains unaffected i.e., 43.88
($A/m^{2}$) even by changing the thickness and doping concentration values of
the solar cell. On the contrary, the percentage of the maximum fill factor is
89.52% at the thickness and doping concentration values of 5.0 $\mu$m and
$1\times 10^{16}$ (1/cm), respectively.

Additionally, the current density (blue curve) as obtained from the IV curve
outputs a peak value of 44.8 ($mA/Cm^{2}$) as shown in Figure 8 and thus, the
maximum power observed is equivalent to the 46 W (the orange curve) at a
thickness and doping concentration value of 5.0 $\mu$m and $1\times 10^{20}$
(1/cm), respectively. Followed by, the maximum quantum efficiency achieved for
the proposed solar cell with GaAs with the most optimized values of thickness
and doping concentration is equal to 99.2 % as shown in Figure 8. It is worth
mentioning that the quantum efficiency curve consists of the quantum
efficiency (%) vs the wavelength curve (nm) with a range between 300 (nm) to
1130 (nm).

##  VI Discussions

In this section of the manuscript, we discuss the results of the comparison of
various IV, PV and QE characteristics of the proposed baseline solar cell,
thickness optimization curve, and doping concentration curve and a thorough
comparison is made for the electrical properties of the solar cell after
adding the p-GaAs layer on top of the proposed baseline solar cell
architecture. The comparison gives a clear indication of the most optimized
thickness and doping concentration values that need to be set whilst
manufacturing the different solar cell architectures at a mass level.

Figure 9: Figure represents a critical comparison of IV Characteristics of the
proposed baseline multi-junction solar cell (p-CIGS/n-CdS/n-ZnO) architecture
after the thickness optimization (blue). Followed by, the orange curve shows
the results after the optimization of the doping concentration of the proposed
solar cell. Lastly, the grey curve shows the results of the optimization of
both thickness and doping concentration after introducing the GaAs layer to
the proposed solar cell architecture.

###  VI-A Comparison of IV Characterisitcs

Figure 9 represents the IV characteristics in terms of the current density,
JSC ($mA/cm^{2}$) of the baseline - after thickness optimization (blue), the
optimized curve for doping concentration (orange) and the optimization results
after introducing the GaAs layer (green) on the proposed solar cell with
varied values of the open circuit voltage, i.e., 0.5V, 0.83V and 1.17V,
respectively. The curve indicates that the optimization of the solar cell
architecture along with the thickness and current density led to an increase
in the open circuit voltage of the solar cell architecture. In addition, an
enhancement in a solar cell’s performance or a change in its operating
circumstances is often indicated by an increase in the VOC of the solar cell
on the IV curve. This improvement of VOC is attributable to circumstances like
better material quality, less charge carrier recombination, lower series
resistance, or different operating environment like temperature. However, to
appropriately evaluate the effectiveness and overall performance of the solar
cell, it’s crucial to take into account the entire IV curve, including
elements like the short circuit current (Isc), fill factor (FF), and power
output which is discussed in the subsequent subsection of the manuscript.

Figure 10: Figure represents the comparison of PV Characteristics of the
proposed baseline multi-junction solar cell (p-CIGS/n-CdS/n-ZnO) architecture
after the thickness optimization (blue). The orange curve shows the results
after the optimization of the doping concentration of the proposed solar cell
and the grey curve shows the results of the optimization of both thickness and
doping concentration after introducing the GaAs layer to the solar cell.

###  VI-B Comparison of PV Characterisitcs

Another comparison of power output is presented in Figure 10 showing a
relation of the power density ($mW/cm^{2}$) vs voltage (V) characteristics,
where like the previous step, the thickness optimization (blue), doping
concentration (orange) and the p-GaAs layer optimization (green) are
evaluated. The maximum of the power curve, represented as $P_{MP}$, is the
point at which the solar cell should be operated to produce the most
electricity. It occurs at a voltage of $V_{MP}$ and a current of $I_{MP}$ and
is also referred to as $P_{MAX}$ or maximum power point (MPP). Figure 10
showcases that the power output for the thickness optimization of the proposed
baseline solar cell architecture has a peak of 25 W, whereas that of the
doping concentration is measured as 32 W. On the contrary, introducing the
p-GaAs layer on the top of the proposed solar cell yields a peak value of
power at 45.47 W.

Therefore, adding a p-GaAs layer highly affects the output power of the solar
cell and accordingly, it also results in an increased highest efficiency of
the solar cell to 45.47% which is thus so far the maximum efficiency recorded
for a multi-junction solar cell consisting of p-GaAs/p-CIGS/n-CdS
semiconductor materials. Moreover, the solar cell shows an overall improvement
in functionality and effectiveness. The increased power output is due to the
solar cell’s material quality has improved, allowing for better light
absorption and electrical conversion with optimized thickness and doping
concentration values performed on the SCAPS-1D simulation tool. Additionally,
improved operating conditions, such as temperature and illumination levels,
decreased recombination losses and lower series resistance are the other
miscellaneous reasons that contribute to an increase in the overall power
output.

Figure 11: Figure showcases the results of the comparison of Quantum
Efficiency (QE) for the proposed solar cell (p-CIGS/n-CdS/n-ZnO) architecture
after the thickness optimization (blue). Accordingly, the orange curve shows
the results after the optimization of the doping concentration of the proposed
solar cell and the grey curve depicts the results of the optimization of both
thickness and doping concentration after introducing the GaAs layer to the
solar cell.

###  VI-C Comparison of QE Characterisitcs

Subsequently, an improvement in the proposed solar cell’s overall capacity to
convert photons of various wavelengths into the electrical current is
evaluated using the quantum efficiency vs. lambda (wavelength) curve. The
solar cell is growing more effective at absorbing a larger variety of photons
across the electromagnetic spectrum as QE rises. The increased light-trapping
techniques, improved material characteristics, decreased recombination losses,
and improved device topologies are one of the major contributing factors that
led to a sudden rise in the overall quantum efficiency. Therefore, an increase
in the quantum efficiency (%) in the quantum efficiency vs Lambda (wavelength
in nm) curve depicts that the solar cell is performing efficiently in regards
to capturing a wide range of wavelength spectrum which further results in an
exponential rise in the total power conversion efficiency of the proposed
solar cell. Accordingly, the authors presented a comparative analysis of the
quantum efficiency (%) for baseline thickness optimization (blue), doping
concentration optimization (orange) and the optimization after introducing the
GaAs layer. As analysed from Figure 11, the quantum efficiency for baseline
thickness, doping concentration and the GaAs layer is 82%, 94% and 99.48% at
the 1020 nm of wavelength.

##  VII Conclusions

This paper proposes an efficient three-layered p-GaAs/p-CIGS/n-CdS (PPN), a
unique solar cell architecture. Copper indium gallium selenide (CIGS)-based
solar cells exhibit substantial performance than the ones utilizing cadmium
sulfide (CdS). On the contrary, CIGS-based devices are more efficient,
considering their device performance, environmentally benign nature, and
reduced cost. Therefore, our paper proposes a numerical analysis of the
homojunction PPN-junction GaAs solar cell structure along with n-ZnO front
contact that was simulated using the Solar Cells Capacitance Simulator
(SCAPS-1D) software. Moreover, we investigated optimization techniques for
evaluating the effect of the thickness and the carrier density on the
performance of the PPN layer on solar cell architecture. Subsequently, the
paper discusses the electronic characteristics of adding GaAs material on the
top of the conventional (PN) junction, further leading to improved values of
the parameters, such as the power conversion efficiency (PCE), open-circuit
voltage (VOC), fill factor (FF) and short-circuit current density (JSC) of the
solar cell. The most promising results of our study show that adding the GaAs
layer using the optimised values of thickness as 5 ($\mu$m) and carrier
density as $1\times 10^{20}$ (1/cm) will result in the maximum PCE, VOC, FF,
and JSC of 45.7%, 1.16 V, 89.52% and 43.88 $(mA/m^{2})$, respectively, for the
proposed solar cell architecture.

This paper proposes an efficient three-layered p-GaAs/p-CIGS/n-CdS (PPN), a
unique solar cell architecture. Copper indium gallium selenide (CIGS)-based
solar cells exhibit substantial performance than the ones utilizing cadmium
sulfide (CdS). On the contrary, CIGS-based devices are more efficient,
considering their device performance, environmentally benign nature, and
reduced cost. Therefore, our paper proposes a numerical analysis of the
homojunction PPN-junction GaAs solar cell structure along with n-ZnO front
contact that was simulated using the Solar Cells Capacitance Simulator
(SCAPS-1D) software. Moreover, we investigated optimization techniques for
evaluating the effect of the thickness and the carrier density on the
performance of the PPN layer on solar cell architecture. Subsequently, the
paper discusses the electronic characteristics of adding GaAs material on the
top of the conventional (PN) junction, further leading to improved values of
the parameters, such as the power conversion efficiency (PCE), open-circuit
voltage (VOC), fill factor (FF) and short-circuit current density (JSC) of the
solar cell. The most promising results of our study show that adding the GaAs
layer using the optimised values of thickness as 5 ($\mu$m) and carrier
density as $1\times 10^{20}$ (1/cm) will result in the maximum PCE, VOC, FF,
and JSC of 45.7%, 1.16 V, 89.52% and 43.88 $(mA/m^{2})$, respectively, for the
proposed solar cell architecture.

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