Structural and Optoelectronic Behaviour of Copper Doped $Cs_{2}AgInCl_{6}$ Double Perovskite: A DFT Investigation

Perovskite-based solar cells (PSCs) have recently been promoted as a renewable technology option for conventional solar cell technology capable of tackling global energy demands and climate change challenges owing to their economic and environmental viability [1]. Their emergence as one of the most promising emerging technologies has aroused the interest of the photovoltaic community, owing to their increasing power conversion efficiency (PCE) from $3.8$% in 2009 [2] to $25.2$% [3], materials availability, low production cost and ease of fabrication process [4, 5, 6, 7, 8, 9, 10].
One of the core elements governing PSC performance is the perovskite materials, typically $CH_{3}NH_{3}PbI_{3}$ or MAPbI, serving both as light harvesters and charge carrier mediators [2, 5, 11]. Over the years, numerous studies have shown the material (MAPbI) to possess appealing qualities needed for photovoltaic and optoelectronic applications, such as suitable bandgap ($\sim 1.5$ eV), good photoconductivity, considerable lifetime diffusion length ($>100$ $\mu$m), high optical absorption coefficient, great bipolar transporting capability, defect tolerance ability, and low carrier effective masses with high mobility [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Despite having these exceptional properties, MAPbI still faces some fundamental issues; such as structural instability, toxicity associated with lead ($Pb$), photocurrent hysteresis and scalability [26, 27, 28, 29, 30, 31], which have hampered their large scale commercialization as viable PSCs.
Several attempts have been made to address the most fundamental issues of instability and toxicity inherent in halide perovskites; such as multi-cation substitution, hydrophobic moieties incorporation (e.g. hydrophobic polymer), surface passivation of perovskite absorber, carbon encapsulation, low dimensionality scheme/treatment and lead replacement with non-toxic elements [26, 32, 33, 34, 35, 36, 37, 38]. Reports have shown that substituting $Pb$ with: (i) non-toxic group IVA elements ($Sn$, $Ge$) results to chemical instability and poor device performance owing to oxidation to their 4+ states [39, 40, 41, 42]; (ii) isovalent elements ($Bi$, $Sb$) leads to reduced device efficiency [43]. This has thus created the need to develop novel classes of materials that are capable of tackling the instability and toxicity issues while still retaining the appealing properties of the $Pb$-based halide perovskites (LHPs).
As of late, double perovskites (DPs) are beginning to gain popularity in the photovoltaic community owing to their ability to tackle issues of $Pb$ toxicity and structural instability [44, 45]. DP is represented with the general $A_{2}M^{{}^{\prime}}M^{{}^{\prime\prime}}X_{6}$ stoichiometry, where $A$ denotes large cation like $Cs,Rb$; $M^{{}^{\prime}}$ and $M^{{}^{\prime\prime}}$ represent monovalent and trivalent cations respectively ($M^{{}^{\prime}}=Ag^{+},Cu^{+},Na^{+};M^{{}^{\prime\prime}}=In^{3+},Bi^{3+}$) and $X$ halide ($X=Cl^{-},I^{-},Br^{-}$) [44, 45]. The materials properties of cation-ordered and vacancy-ordered DPs have been investigated experimentally and theoretically to determine their suitability for photovoltaic and optoelectronic applications. Research findings have shown that most DPs exhibit considerable thermal and mechanical stability compared to $Pb$-based halide perovskites, but possess large bandgap with indirect nature, which have limited their usage in solar cell applications [44, 46, 47, 48, 49].
Given this, attentions are now drawn towards direct bandgap DPs. Direct bandgap DPs are in the forefront following the pioneering work by Volonakis and co-workers in 2017 where $Cs_{2}AgInCl_{6}$ (CAIC) DP was proposed, synthesized and identified as a potential, environmentally-benign replacement for $Pb$-based halide perovskites for photovoltaic and other optoelectronic applications [50]. CAIC is a direct-bandgap DP with high thermal and mechanical stability, which crystallizes in the face-centred cubic structure with space group $Fm\bar{3}m$, and has an experimental lattice parameter of $10.469-10.481$ Å and bandgap of $2.5-3.3$ eV [50, 51, 52]. These have made CAIC of huge research interest and a potential candidate for LHPs. However, pure bulk CAIC crystal or powder are characterized with low photoluminescence quantum yield (PLQY) and photo-absorption coefficient compared to CAIC nanocrystals (NCs) and these are as a result of parity-induced forbidden transition [51, 53, 54].
In a quest to tuning and optimizing the optoelectronic properties of DPs, experimental findings have identified doping engineering as a viable way of achieving this and thus, has the potential of enabling their widespread usability beyond photovoltaic applications [55, 56]. Very recently, reports had shown the synthesis of doped-CAIC NC, treated with transition ($Mn$) and post-transition ($Bi$) metals, exhibiting high PLQYs, enhanced photo-absorption and other related optical properties when compared with pure CAIC either in powder or NC forms [53, 57, 58, 59].
Numerical simulation has widespread application to a variety of problems [60, 61, 62, 63]. In particular, Monte Carlo simulation and mathematical modelling has been widely applied in the quest for more cost-effective fabrication of devices [64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75], whereas properties of new materials have been studied with quantum mechanical calculations using density functional theory for decades [45, 38]. However, theoretical studies based on Density Functional Theory (DFT) on $M$-cation doping in double perovskites are scarce. In a recent study, Jiao and associates used DFT scheme to investigate material properties of metal-alloying DP $Cs_{2}AgM_{x}Br_{6}$ ($M_{x}=Sb,In,Bi$), where indirect to direct bandgap transition was observed [76]. Transition metal doping can lead to enhancement of material properties, especially their optoelectronic properties.
Based on the scope of our literature search, $Cu$-doping in CAIC is yet to be explored both theoretically and experimentally. Consequently, this paper seeks to investigate the effect of $Cu$-doping on the structural and optoelectronic properties of CAIC ($Cs_{2}Ag_{(1-x)}Cu_{x}InCl_{6}$) using the virtual crystal approximation (VCA) approach within the framework of DFT. VCA is a first-principles technique in modelling disordered solid solutions via pseudopotential averaging and effective in treating disordered systems [77, 78, 79]. This work only focus on the bulk optoelectronic properties of perovskite materials.
The remaining part of paper is organized in the following order. In Section 2, the computational methods employed for the calculations are described. Section 3 is devoted to the presentation and discussion of our results. Finally, a brief summary of the work is given.

In this work, the ab-initio calculations for $Cs_{2}AgInCl_{6}$ (CAIC) and $Cs_{2}Ag_{(1-x)}Cu_{x}InCl_{6}$ (CAIC:Cu) solid solutions were performed using the pseudopotential plane-wave technique based on density functional theory (DFT) as implemented in Quantum ESPRESSO (QE) software package [80, 81]. Within the framework of DFT, the structural and optoelectronic properties with the electronic exchange-correlation (XC) potential were calculated using Perdew-Berke-Ernzerhof (PBE) [82] based on the generalized gradient approximation (GGA). The van der Waals functional (vdW-DF-OB86) [83] and the hybrid PBE0 functional [84] were employed to treat the electronic exchange-correlation potential for the calculations of the lattice parameters and band structures respectively.
For the electron-ion interaction, the Optimized Norm-Conserving Vanderbilt (ONCV) pseudopotentials [85] were used for all calculations and construction of the virtual atoms ($Ag_{(1-x)}-Cu_{x}$). The virtual crystal approximation (VCA) method [77, 78] was used to generate the pseudopotentials of the virtual atoms ($Ag_{(1-x)}-Cu_{x}$), where the mixing ratio $x$ was varied from 0 to 1 in the step of 0.1. The plane wave energy cut-off of $100$ Ry and Monkhorst-Pack special [86] k-points of $6\times 6\times 6$ were used for optimization calculations and the calculations of the electronic band structure and optical properties while a denser k-points of $12\times 12\times 12$ were used for density of state (DOS) calculations. While the convergence threshold for self-consistent-field (SCF) iteration was set at $10^{-10}$ eV, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization method [87] was employed for the geometry optimization of the perovskites. The entire atomic positions were relaxed until the Helmann-Feynman forces on each atom become less than $20$ meV/Å.
To examine the optical properties of the perovskites, density functional perturbation theory (DFPT) [88, 89, 90] as implemented in QE was used to determine the complex frequency-dependent dielectric functions, $\varepsilon(\omega)$:

From Eq. 1, $\omega$ denotes the photon frequency, $\varepsilon_{1}(\omega)$ the real and $\varepsilon_{2}(\omega)$ the imaginary parts of the dielectric function $\varepsilon(\omega)$ respectively. To determine the light-harvesting capability of the perovskites, the absorption coefficients $\alpha(\omega)$ was calculated using Eq. 2

where

Other optical parameters of interest are optical conductivity, refractive index, extinction coefficient and energy-loss function. The optical conductivity $\sigma(\omega)$ and refractive index $n(\omega)$ of the materials were computed using the relations in Eq. 3 and 4 respectively.

In terms of the complex dielectric function in Eq. 1, the extinction coefficient $k(\omega)$ and the energy-loss function $L(\omega)$ were determined by using Eq. 5 and 6 respectively.

In this work, the effect of $Cu$-doping on the structural and optoelectronic properties of CAIC has been studied using first-principles DFT calculations and VCA approach. The ab-initio VCA method was used to model the solid solutions. The PBE0 functional was used for the band structure calculations after assessing the exchange-correlation functional of GGA-PBE. With increasing $Cu$-content, the crystal lattice shrinks following a linear function $a(x)=10.5128-0.1395x$, bulk modulus increases with a quadratic function of $B_{0}(x)=34.0636+1.4576x-0.7576x^{2}$, while the bandgap decreases quadratically with a function of $E(x)=3.2698-0.6463x-0.6936x^{2}$. The photo-absorption coefficient, optical conductivity and other optical parameters of interest are calculated using the DFPT method. The spectra obtained show enhanced absorption and conductivity at higher $Cu$-content. The variation tendencies, as a result of $Cu$-doping, in the structural and optoelectronic properties of the materials under study have shown $Cu$ to be an efficient dopant in treating double perovskites. This work presents $Cs_{2}Ag_{(1-x)}Cu_{x}InCl_{6}$ (CAIC:Cu) solid solutions as potential candidates for photovoltaics and optoelectronics.