Calculation for High Pressure Behaviour of Potential Solar Cell Materials Cu₂FeSnS₄ and Cu₂MnSnS₄

Abstract

Exploring alternatives to the Cu₂ZnSnS₄ kesterite solar cell absorber, we have calculated first principle enthalpies of different plausible structural models (kesterite, stannite, $P\overline{4}$ and GeSb type) for Cu₂FeSnS₄ and Cu₂MnSnS₄ to identify low and high pressure phases. Due to the magnetic nature of Fe and Mn atoms we included a ferromagnetic (FM) and anti-ferromagnetic (AM) phase for each structural model. For Cu₂FeSnS₄ we predict the following transitions: $P\overline{4}$ (AM) $\stackrel{16.3\mathrm{GPa}}{\to }$ GeSb type (AM) $\stackrel{23.0\mathrm{GPa}}{\to }$ GeSb type (FM). At the first transition the electronic structure changes from semi-conducting to metallic and remains metallic throughout the second transition. For Cu₂MnSnS₄, we predict a direct AM (kesterite) to FM (GeSb-type) transitions at somewhat lower pressure (12.1 GPa). The GeSb-type structure also shows metallic behaviour.

1. Introduction

The impending exhaustion of fossil fuel has prompted the exploration and exploitation of alternative energy resources, with the solar energy harvesting through photovoltaic devices spearheading these efforts. In an attempt to overcome the restraints of Si-based materials, the direct optical band gap of chalcogenide-based solar cells offers the benefit of higher absorption in comparison to silicon. Given that the employed chalcogenides are composed of multiple elements, one can additionally optimise the photovoltaic properties of the respective material by appropriate metal or chalcogenide substitution. Among the various chalcogenides investigated for this purpose, the quaternary semiconductor Cu₂ZnSnS₄ has attracted considerable attention [1,2]. The suitability of this material for solar cell applications stems from its almost optimal band gap (${\mathit{E}}_g$≈ 1.5 eV), its high absorption coefficient in the visible range, and its earth-abundant, low-cost, and non-toxic constituents [3,4,5]. In a previous experimental and theoretical study on Cu₂ZnSnS₄ we have investigated the high pressure behaviour to probe its reaction to tensile stress [6]. One of the biggest issues with Cu₂ZnSnS₄ is that it suffers from Cu-Zn cationic disorder [7]. The main reason why Cu and Zn can be interchanged easily is their similar ionic radius. The analogues Cu₂FeSnS₄ and Cu₂MnSnS₄ have similarly favourable properties for use as a solar cell absorber. We expect cationic disorder to be less present due to the bigger difference in size of Fe and Mn in comparison to Cu. In this work, we want use density functional-based (DFT) first principle methods to investigate how Cu₂FeSnS₄ and Cu₂MnSnS₄ behave under tensile stress to understand the physical limitation of those materials.

2. Materials and Methods

2.1. Calculation Set-Up

The periodic density functional theory (DFT) calculations were performed with VASP 5.4.4 [8,9,10,11]. A plane wave basis set with an energy cutoff of 550 eV with the projector augmented (PAW) potentials [12,13] was used, whereby the 5s, 5p and 4d electrons of Sn, 3s, 3p electrons of S, and 4s, 3d electrons of Cu, Fe and Mn were explicitly considered. The electronic convergence criteria was set at least to ${10}^{-5}$ eV, whereby the Blocked-Davidson algorithm was applied as implemented in VASP. The structural relaxation of internal and external lattice parameters was set to a force convergence of ${10}^{-2}$ eV/Å${}^2$ while the conjugate-gradient algorithm implemented in VASP was used [14]. The freedom of spin polarisation was enabled and a Gaussian smearing approach with a smearing factor $\sigma$ of 0.01 eV was utilised. For all structures we simulated 16 atoms which corresponds to the number of atoms in the kesterite unit cell. The cells were fully optimised with a 8 × 8 × 4 k-grid constructed via the Monkhorst-Pack scheme [15] and centered at the $\Gamma$-point with the PBE functional [16]. The cells include two magnetic ions (Fe or Mn) which can be arranged in two different magnetic phases, ferromagnetic (FM) with parallel magnetic moments and anti-ferromagnetic (AM) with antiparallel ones. On top of the PBE-optimised structures, single point calculations for the band gap and DOS with the HSE06-functional [17,18,19,20] were performed with a 4 × 4 × 2 k-grid to account for an accurate electronic structure. The tetrahedron method with Blöchl corrections [21] was applied for band structure evaluation.

The pressure dependence was determined by selecting volume points in a range of about 50 Å${}^3$ below the minima. This corresponds to a pressure range of 0–30 GPa. We used a step size of 8 Å${}^3$ which lead to at least 10 volume points for each structural model. At each point we optimised the ionic positions and cell shape, while keeping the volume constant. We fitted the total energy versus volume to a Birch-Murnaghan Equation of State (B-M EoS) [22]. Then the pressure at each volume was obtained from the P(V) formulation of the same EoS (for details see Appendix A.3). Using the pressure we calculated the enthalpies ($\mathit{H}\left(P\right)=E+PV$) for each structural model and compared them over the investigated pressure range to identify the most stable structures.

All energy and enthalpy differences between different structural models in the following refer to the KS unit cell size, hence to two formula units.

2.2. Structural Models

In quaternary chalcogenide semiconductors the equilibrium structure at ambient pressures in most cases are kesterite (KS, Figure 1a) or stannite (ST, Figure 1b) structures [23,24]. We include both structures as potential low pressure phases for Cu₂FeSnS₄ and Cu₂MnSnS₄. Please note that in the cited work by Schorr et al. [23] besides KS and ST also three disordered structures are suggested which are very unlikely to occur in our systems due to the different ionic radii of the involved elements. In our high pressure study on Cu₂ZnSnS₄ we found the distorted rocksalt structure (GeSb, Figure 1c) to be the most stable phase beyond 16 GPa [6]. Therefore we include GeSb as a high pressure phase in this study. We also include a tetragonal $P\overline{4}$ structure (Figure 1d), which is discussed in literature as the thermodynamically most stable structure for Cu₂FeSnS₄ [25,26,27].

KS, ST and $P\overline{4}$ have a coordination number of 4, due to the same structural motif, they are close in formation energy and which structure forms depends on the crystallisation conditions. Unless the crystallisation is done carefully to enable the formation of the thermodynamic equilibrium the crystallisation process is kinetically driven, which we can not simulate in our DFT calculations. The high pressure phase GeSb has a coordination number of 6. Transitioning from a four to a six-fold coordination is a massive structural change associated with a large difference in the energy of formation. Due to the large energy difference, we can describe the pressure induced transition well in DFT.

3. Results and Discussion

3.1. Equilibrium Structures

Before discussing the enthalpies we will review the equilibrium structures at zero pressure of Cu₂FeSnS₄ and Cu₂MnSnS₄, obtained at the PBE level (

Table 1. Optimized lattice parameter a and c (in Å) for Cu₂FeSnS₄ ST $P\overline{4}$ and KS at the PBE level of theory in comparison to experimental (exp.) lattice parameter. The first column refers to the magnetic phase, $\Delta E$ denotes the energy difference to the most stable phase (in meV) and ${\mu }_{\mathrm{Fe}}$ refers to the magnitude of the magnetic moment at Fe (in ${\mu }_B$).

Mag.	ST				$\mathit{P}\overline{4}$				KS	Method
	$\mathit{a}$	$\mathit{c}$	${\mathbf{\mu }}_{\mathrm{Fe}}$	$\Delta \mathit{E}$	$\mathit{a}$	$\mathit{c}$	${\mathbf{\mu }}_{\mathrm{Fe}}$	$\Delta \mathit{E}$	$\Delta \mathit{E}$
AM	5.471	10.695	3.1	22	5.469	5.346	3.1	0	122	PBE
FM	5.467	10.724	3.1	82	5.469	5.365	3.1	152	139	PBE
AM	5.460	10.742			5.433	5.410				Exp. [27,29]

) and compare to published crystal structures. XRD (X-ray diffraction) analysis by Brockway dating back to 1934 revealed that natural Cu₂FeSnS₄ samples crystallise in ST structure [28]. Those results where subsequently confirmed in the 1970s by Ganiel et al., they furthermore studied the magnetic ordering via Mössbauer spectroscopy and found that it was anti-ferromagnetic [29]. In 1972 Springer studied the solution series Cu${}_2$Fe${}_{1-x}$Zn${}_x$SnS${}_4$. Beyond x = 0.4 and above 680 $°$C he observed an ST crystal structure he labeled $\beta$-Cu${}_2$(Fe/Zn)SnS${}_4$. Below x = 0.4 and 680 $°$C he found another tetragonal phase he labeled $\alpha$-Cu${}_2$(Fe/Zn)SnS${}_4$ [25,30]. In 2000 pure $\alpha$-Cu${}_2$FeSnS${}_4$ was synthesised [26]. After the solid state reaction of CuFeS₂ on SnS at 1323 K in sealed graphite crucible for 24 h the product was cooled slowly over 50 h. Through the slow cooling they obtained the thermodynamically most stable $\alpha$-phase. Through XRD measurements the space group of the sample was determined to be $P\overline{4}$ [26]. Those results where confirmed by Rincon et al. who furthermore studied the magnetic susceptibility and revealed that also $\alpha$-Cu${}_2$FeSnS${}_4$ exhibits an anti-ferromagnetic ordering [27].

Our PBE results for Cu₂FeSnS₄ agree very well with the experiments. We found the anti-ferromagnetic $P\overline{4}$ (corresponding to $\alpha$-Cu${}_2$FeSnS${}_4$) structure to be most stable (Table 1). But it is only 22 meV more stable than the naturally occurring anti-ferromagnetic ST structure. The anti-ferromagnetic KS structure is another 100 meV above the anti-ferromagnetic ST structure. The small energy difference between ST and $P\overline{4}$ may give an explanation why the $P\overline{4}$ phase is hard to obtain in pure form. From the experimental results we conclude that the formation of the ST phase has to be kinetically favoured. This aspect dominates the crystallisation process if the samples are rapidly cooled. We expect the thermodynamic equilibrium to build up slowly based on the small energy difference between the $P\overline{4}$ and ST structures, hence slow cooling is crucial to obtain $\alpha$-Cu${}_2$FeSnS${}_4$. In agreement with the experimental data we find the anti-ferromagnetic ordering favoured over the ferromagnetic ordering by 82 meV and 152 meV, for ST and $P\overline{4}$, respectively. The PBE lattice parameter for both Cu₂FeSnS₄ phases agree reasonably well with the XRD experiments. The lattice parameters change only slightly between the magnetic phases and are within the error bars of the functional applied.

Our PBE prediction for the most stable structure for Cu₂MnSnS₄ does not agree with experimental results. Magnetisation and neutron-diffraction measurements have shown that Cu₂MnSnS₄ has an anti-ferromagnetic ST structure [31]. We predict the anti-ferromagnetic KS structure to be 35 meV more stable (

Table 2. Optimized lattice parameter a and c (in Å) for Cu₂MnSnS₄ KS, ST and $P\overline{4}$ at the PBE level of theory in comparison to other simulated and experimental (exp.) lattice parameter. The first column refers to the magnetic phase, $\Delta E$ denotes the energy difference to the most stable phase (in meV) and ${\mu }_{\mathrm{Mn}}$ refers to the magnitude of the magnetic moment at Mn (in ${\mu }_B$).

Mag.	ST				KS				$\mathit{P}\overline{4}$	Method
	$\mathit{a}$	$\mathit{c}$	${\mathbf{\mu }}_{\mathrm{Mn}}$	$\Delta \mathit{E}$	$\mathit{a}$	$\mathit{c}$	${\mathbf{\mu }}_{\mathrm{Mn}}$	$\Delta \mathit{E}$	$\Delta \mathit{E}$
AM	5.498	10.895	4.1	35	5.468	11.020	4.1	0	52	PBE
FM	5.504	10.880	4.1	49	5.477	11.000	4.2	39	87	PBE
AM	5.407	10.678	4.0	37	5.369	10.832	4.0	0		PBEsol [32]
AM	5.519	10.795	4.5	0	5.468	11.038	4.4	15		HSE06 [32]
AM	5.517	10.806								Exp. [31]

). We also tested the $P\overline{4}$ structure and found it to be the least stable anti-ferromagnetic and ferromagnetic structure. The anti-ferromagnetic $P\overline{4}$ structure is 17 meV less stable than the anti-ferromagnetic ST phase at the PBE level. Our results on the relative KS and ST stability agree closely with PBEsol calculations by Scragg et al [32]. The same group has also carried out HSE06 calculations and found the ST structure to be more stable than the KS structure. The difference is only 15 meV per unit cell. The error in the lattice constants of the PBE calculations is in the range of the differences between the structures and magnetic phases. The PBE lattice constants for Cu₂MnSnS₄ show subtle interplay between magnetism and structure but all of them are close to the experimental lattice parameter of ST (AM).

In contrast to the KS equilibrium structure of Cu₂ZnSnS₄, the compounds Cu₂FeSnS₄ and Cu₂MnSnS₄ experimentally favour a ST and/or $P\overline{4}$ structure. The ST and $P\overline{4}$ structures are group theoretically related, both structures exhibit similar cationic layers. We can rationalise the formation of ST and or $P\overline{4}$ structures over KS by comparing the ionic crystal radii (defined according to Fumi and Tosi [33]) of the substituted bivalent elements (Zn, Fe and Mn) in chalcogenides determinded by Shannon [34].

Cu₂ZnSnS₄ in the KS structure consists of Cu⁺-Zn²⁺ layers and Cu⁺-Sn⁴⁺ layers. The crystal radius of Zn²⁺ (rc = 0.60 Å) is identical to the crystal radius of Cu⁺ (rc = 0.60 Å), which allows them to fit in the same layer. If we replace Zn²⁺ with the larger Fe²⁺ (rc = 0.63 Å) or Mn²⁺ (rc = 0.66 Å), a ST or $P\overline{4}$ structure is formed. In those structures pure Cu⁺ layers alternate with Sn⁴⁺-Fe²⁺/Mn²⁺ layers. Like this the largest and the smallest ion (Sn⁴⁺, r${}_{\mathrm{c}}$ = 0.55 Å) are paired in one layer. In a KS structure the large bivalent cation would have to be in the same layer as the second largest Cu⁺ ion. We suspect that this is the main reason which drives the formation of the ST and/or $P\overline{4}$ structures for the magnetic derivates.

In contrast to Cu₂FeSnS₄, we do not find the ST structure to be more stable than KS in our calculations for Cu₂MnSnS₄, although Mn²⁺ has an even larger ionic crystal radius than Fe²⁺. We found that PBE fails do describe the electron density around Mn²⁺, the charge is more smeared out than for Fe²⁺ which leads to larger positive charge at Mn (for details see

Table A3. Bader charges for B = Fe or Mn (in $e^{-}$) the listed anti-ferromagnetic Cu₂BSnS₄ structures at the PBE and HSE06 level.

Struc	Q${}_{\mathbf{PBE}}$(Fe)	Q${}_{\mathbf{HSE}06}$(Fe)	Q${}_{\mathbf{PBE}}$(Mn)	Q${}_{\mathbf{HSE}06}$(Mn)
KS	0.86	1.05	1.02	1.18
ST	0.86	1.05	1.02	1.19

in Appendix A.1).

The PBE lattice parameter a and c of the ST and $P\overline{4}$ structure (2c for the $P\overline{4}$ structure) for Cu₂FeSnS₄ differ by less than 0.1%. The differences in Cu₂MnSnS₄ of a and c between ST and KS are larger, particularly in c where the difference is 2%. We think that the reason must be the different composition of the cationic layers. If we compare the ST structures of both materials, we find that the size of the lattice parameters corresponds to the crystal radius of the bivalent cation. All presented structures for Cu₂FeSnS₄ and Cu₂MnSnS₄ have lattice parameter within 2% of the values for the Cu₂MnSnS₄ KS (5.485 Å and 10.94 Å [23]).

3.2. Pressure-Dependent Enthalpies

If we plot and compare the enthalpies for Cu₂FeSnS₄ (Figure 2a,b) we find the anti-ferromagentic $P\overline{4}$ structure to be most stable up to 16.3 GPa. Throughout this pressure range the anti-ferromagentic ST structures is only 22 to 40 meV less stable. The anti-ferromagentic KS structure is even less stable than the ferromagnetic ST structure which is about 100 meV above the $P\overline{4}$ structure. The energy splitting between AM and FM increases from the KS over the ST to the $P\overline{4}$ structure. As a consequence the ferromagnetic $P\overline{4}$ structure is as unstable as the ferromagnetic KS structure in the investigated pressure range. At 16.3 GPa, we find a transition from the $P\overline{4}$ structure to the GeSb structure. Its purely a structural transition, the magnetic phase remains anti-ferromagnetic. The cell volume decreases by 13% (

Table 3. Predicted transitions for Cu₂FeSnS₄ and Cu₂MnSnS₄ in comparison to experimental transition for Cu₂ZnSnS₄. The table contains the transition pressure ($p_T$ in GPa) with the corresponding cell volumina before ($V_1$ in Å${}^3$) and after ($V_2$ in Å${}^3$) the transition, $\Delta V$ denotes the relative volume change. All volumina refer to two formula units.

Composition	${\mathit{p}}_{\mathit{T}}$	Transition	${\mathit{V}}_1$	${\mathit{V}}_2$	$\Delta \mathit{V}$
Cu2FeSnS4	16.3	$P\overline{4}$(AM) → GeSb(AM)	272	236	−13.1%
Cu2FeSnS4	23.0	GeSb(AM) → GeSb(FM)	226	225	−0.6%
Cu2MnSnS4	12.1	KS(AM) → GeSb(FM)	289	248	−14.0%
Cu2ZnSnS4 [6]	16.0	KS → GeSb	280	240	−15.2%

) through the transition. At 23.0 GPa we predict a magnetic phase transition of GeSb to ferromagnetic. The energy difference between the two different magnetic phases is very small, at the structural transition pressure it is 15 meV and decreases up to the magnetic transition pressure. Afterwards it increases again, but the difference remains small, at 25 GPa it amounts to 75 meV. The anti-ferromagnetic modification for KS and ST is more stable than the ferromagnetic modification over the whole pressure range. As pointed out for the equilibrium structures, the anti-ferromagnetic $P\overline{4}$ structure and the ST structure can be observed experimentally, depending on the preparation method. Due to the small energy difference between both structures also at the transition pressure, we predict that an ST-to-GeSb transition would also appear at a very similar pressure as the $P\overline{4}$-to-GeSb transition.

The enthalpy for Cu₂MnSnS₄ (Figure 3a,b) indicates that anti-ferromagnetic KS is most stable at ambient conditions and up to 12.1 GPa, where we predict a KS-to-GeSb phase transition. The structural phase transition is accompanied by a magnetic phase transition from anti-ferromagnetic to ferromagnetic. The cell volume decreases by 14% (Table 3) through the transition. At the transition pressure the ferromagnetic GeSb modification is 66 meV more stable than the anti-ferromagnetic modification. With increasing pressure the difference is nearly constant, at 20 GPa it amounts to 73 meV. For the KS structure the anti-ferromagnetic modification remains more stable than the ferromagnetic modification by over 40 meV throughout the whole pressure range. For the ST structure the ferromagnetic modification becomes more stable around 12 GPa. The difference between both magnetic phases is much lower than for KS and remains below 20 meV throughout the whole pressure range. The anti-ferromagnetic $P\overline{4}$ has a very similar stability as the ferromagnetic ST structure. The splitting between AM and FM is the largest, rendering the ferromagnetic $P\overline{4}$ structure the least stable through the whole pressure range. As pointed out above, the relative stability of KS and ST is wrong at the PBE level. Over the whole pressure range the difference in energy between KS and ST stays below 50 meV. In comparison to the energy change induced by the structural phase transition (already 1 eV at 5 GPa above phase transition), this energy difference is small. That is why we think we can still predict a phase transition around 12 GPa. But experimentally we expect a ST(AM)-to-GeSb(FM) transition instead of the KS-to-GeSb phase transition our calculations suggest.

3.3. Electronic Structure

We investigated the electronic band structure for equilibrium and high-pressure structures for both compounds at the equilibrium and at the transition pressure with the HSE06 hybrid functional [17].

For Cu₂FeSnS₄, we predicted the band gap of the anti-ferromagnetic ST structure to be 1.3 eV (

Table 4. Calculated HSE06 band gaps (in eV) for Cu₂FeSnS₄ and Cu₂MnSnS₄ for KS and ST strucutral models (struc.) in comparison to experimental (exp.) results. For DOS plots please refer to Appendix A.4.1.

Composition	Struc.	AM	FM	Exp. (AM)
Cu2FeSnS4	$P\overline{4}$	1.0	0.8	-
Cu2FeSnS4	ST	1.3	0.9	1.6 eV [35]
Cu2MnSnS4	KS	1.3	1.0	-
Cu2MnSnS4	ST	1.1	1.0	1.42–1.79 eV [32]

). Based on absorption spectra, the band gap was determined to be 1.6 eV [35]. We think that the difference from our prediction is largely due to the fact that we only optimised our structures at the PBE level but also partly due to the error of the HSE06 functional in reproducing band gaps. In an earlier study within our group we obtained similar results for Cu₂ZnSnS₄ KS. Experimentally the Cu₂ZnSnS₄ KS band gap is determined to be 1.5 eV [6]. The HSE06 band gap for the PBE optimised structure is 1.2 eV. Only if the structure is also optimised at the HSE06 level, we obtain the experimental band gap [36]. The HSE06 band gap deviation for Cu₂ZnSnS₄ of the PBE structure is −0.3 eV. We expect it to have similar magnitude for Cu₂FeSnS₄ and Cu₂MnSnS₄.

The HSE06 band gap for the most stable anti-ferromagnetic $P\overline{4}$ structure of Cu₂FeSnS₄ is 1 eV. For $P\overline{4}$ and ST Cu₂FeSnS₄ the anti-ferromagnetic modification has an 0.2 and 0.3 eV larger band gap than the ferromagnetic modification.

For Cu₂MnSnS₄ we predicted an equilibrium band gap for ST of 1.1 eV (Table 4). Experimental measurements by Raudsich et al. of Cu₂MnSnS₄ indicate a band gap of 1.42 to 1.79 eV. All measured samples contained Cu₂MnSn₃S₈ as a secondary phase. They also calculated the band gap at the HSE06 level which they reported to be 1.5 eV. Again the deviation to our result must be due to the fact that they also carried out the optimisation at the HSE06 level while we restricted ourselves to PBE optimisations.

To understand the change of the electronic structures under pressure, we also calculated the DOS at the transition pressures for both systems. For the $P\overline{4}$ structure of Cu₂FeSnS₄ at the transition pressure the band gap is widened to 1.4 eV (Figure 4). For the naturally occurring ST structure (AM) the band gap is widened to 1.5 eV (for DOS see Appendix A.4.2). After the transition to the anti-ferromagnetic GeSb structure the band gap closes completely. In the DOS plot for anti-ferromagnetic GeSb at the transition pressure we can see that all bands from the valence band now extend in the region from 0 eV to 1.5 eV which is the band gap region for the $P\overline{4}$ structure. Thus we predict a change from semi-conducting to metallic behaviour. The band gap stays zero with the second magnetic transition from anti-ferromagnetic to ferromagnetic.

For Cu₂MnSnS₄ we find the same behaviour concerning the electronic structure at the transition pressure, a closing of the gap and a metallic character above the transition pressure (Figure 4). During the structural transition, the magnetic structure changes from AM to FM. We are confident that this also holds for the ST (AM) to GeSb (FM) transition we expect based on the observation that the experimental equilibrium structure for Cu₂MnSnS₄ is the anti-ferromagnetic ST structure. To verify this assumption we also calculated the DOS for ST at the transition pressure, its band gap is 1.2 eV (for DOS see Appendix A.4.2). This confirms that also for the ST (AM) to GeSb (FM) transition the electronic structure would change from semi-conducting to metallic.

Both materials show similar electronic structure changes at the transition pressure as Cu₂ZnSnS₄, which changes from semi-conducting to metallic at 16 GPa (for details see Appendix A.4.2).

3.4. Mechanical Properties

Finally we want to analyse how the bulk modulus changes due to the phase transitions in the magnetic materials Cu₂FeSnS₄ and Cu₂MnSnS₄ and compare to Cu₂ZnSnS₄. In the used equation of states the equilibrium volume, the bulk modulus and its pressure dependence are fit parameters.

First of all it strikes that regardless of the composition all tetragonal anti-ferromagnetic structures (KS, ST or $P\overline{4}$) have very similar bulk moduli ranging within 2 GPa around the value for Cu₂ZnSnS₄ KS (

Table 5. PBE bulk modulus ($B_0$ in GPa) and first derivative ($B_0^{{}^{\prime }}$ in GPa/m${}^3$) for the listed structural models (Struc.) with the given magentic phase (Mag.) for the listed compositions. Derived from the Birch–Murnaghan EoS fit. For all fit paramters please refer to Appendix A.3.

Composition	Struc. (Mag.)	${\mathit{B}}_0$	${\mathit{B}}_0^{{}^{\prime }}$
Cu2FeSnS4	$P\overline{4}$ (AM)	68.32	4.38
Cu2FeSnS4	ST (AM)	69.24	4.27
Cu2FeSnS4	GeSb (AM)	85.77	3.84
Cu2FeSnS4	GeSb (FM)	80.38	4.25
Cu2MnSnS4	ST (AM)	67.25	4.16
Cu2MnSnS4	GeSb (AM)	80.49	3.83
Cu2ZnSnS4 [6]	KS	68.63	4.64
Cu2ZnSnS4 [6]	ST	68.77	4.64
Cu2ZnSnS4 [6]	GeSb	82.16	4.57

). The first derivative $B_0^{{}^{\prime }}$ shows larger differences, the values for Cu₂FeSnS₄ and Cu₂MnSnS₄ are 7% and 10% lower than for the KS Cu₂ZnSnS₄ material. Without an error analysis we can not determine whether the differences in $B_0^{{}^{\prime }}$ are significant. At zero pressure the $P\overline{4}$ (AM) structure of Cu₂FeSnS₄ has a higher bulk modulus than the ST (AM) structure, but at higher pressures eventually it flips due to the larger first derivative for the ST (AM) structure. If we compare the same structure ST(AM) for all three materials, the bulk modulus of Mn over Zn to Fe are slightly increasing, but only in a range where it would not be measurable experimentally.

All bulk moduli for the anti-ferromagnetic GeSb structures are in the range of 77.8–85.8 GPa, thus each about 15% larger than their tetragonal counterparts. In all cases the phase transition leads to stiffer materials. The bulk modulus of GeSb is smallest for Cu₂MnSnS₄ (AM), followed by Cu₂ZnSnS₄ and largest for Cu₂FeSnS₄ (AM). This is the same ordering we observe for the ST (AM) phases.

4. Conclusions

We calculated first principle enthalpies with PBE for different structural models for Cu₂FeSnS₄ and Cu₂MnSnS₄ to identify low and high-pressure modifications. Thereby, we probed ferromagnetic and anti-ferromagnetic phases.

In agreement with experimental findings, we found the anti-ferromagnetic $P\overline{4}$ structure to be the most stable for Cu₂FeSnS₄ at ambient pressure. We additionally confirmed that the naturally occurring ST (AM) is nearly as stable until the following transition. At 16.3 GPa, we predict a structural transition to the anti-ferromagnetic GeSb structure, thereby, the coordination number of the metal ions changes from 4 to 6. The structural transition is accompanied by a change of the electronic structure from semi-conducting to metallic. At 23.0 GPa, we found a magnetic phase transition from anti-ferromagnetic to ferromagnetic, the electronic structure remains metallic.

Due to the deficits of the used density functional, we failed to identify the correct equilibrium structure for Cu₂MnSnS₄. All possible low pressure phases are in a small energy window, and PBE predicts the KS (AM) structure as most stable. Experimental data and HSE06 optimisations indicate that anti-ferromagnetic ST structure is present under ambient conditions. At the HS06 level the difference is only 15 meV per unit cell [32], and also at the PBE level the difference is small (under 50 meV over the whole pressure range). All four-fold coordinated anti-ferromagnetic structures show a structural and magnetic phase transition to GeSb (FM) around 12 GPa. This transition also leads to a change of the electronic structure from semi-conducting to metallic.

The results for both materials are similar to our findings for Cu₂ZnSnS₄, where we observe a KS-to-GeSb transition around 16 GPa [6]. Also in Cu₂ZnSnS₄ this transition leads to a change of the electronic structure from semi-conducting to metallic. Only taking into account the band gaps and the predicted transition pressures, we conclude that the magnetic material Cu₂FeSnS₄ is similarly suited for the use in thin film solar cells as Cu₂ZnSnS₄. Cu₂MnSnS₄ also has a band gap in the desired range for a solar cell absorber but is less resistant against tensile stress than Cu₂ZnSnS₄ and Cu₂FeSnS₄.