Formation energy of TMs in In2S3

There are three nonequivalent crystallographic environments for In and S atoms in the β-phase of In2S3. The In substitutional sites in the crystalline matrix of In2S3 are tetrahedral 8e (In8e), octahedral 8c (In8c), and octahedral 16h (In16h) sites12, corresponding to Wyckoff notations of space-group 141. Both, In8c and In16h sites are six-fold coordinated by S and are different in their associated In–S bond lengths. In8e occupies two-thirds of the available tetrahedral cation sites, while the remaining one-third tetrahedral sites are unoccupied. Sulfur atoms are arranged on distorted cubic closed packed lattices, where the S positions are distinct due to the different occupation of the surrounding In sites. The atomic structure of the β−In2S3 unit cell is illustrated in Supplementary Fig. 1.

Figure 1 shows the calculated formation energy of TMs on 8e tetrahedral and 16h octahedral sites of In for In-rich and S-rich growth atmospheres. In this set of calculations, one TM was added to the crystalline matrix of In2S3. The formation energies are depicted from EF = 0 eV (corresponding to p-type material) to EF = 2.1 eV (corresponding to n-type material), covering the whole range of possible Fermi energies. The points, where the line slop changes represent transitions of the stable charge state of the impurity. Since the formation energies of TMs on In8c and In16h octahedral sites are close, here, we only show formation energies on In16h sites. Here, our goal is to identify the most favorable defect configuration and the charged data are shown for completeness. In order to discuss Ti and Nb as charged defects, one needs to consider the possible charge compensation mechanisms. Since the concentration of TMs are quite high, the charge compensation by means of intrinsic defects is extremely unlikely. Hence, oxidation of Ti and Nb would require presence of abundant acceptor dopants in the system. Since In2S3 is an n-type material, Ti and Nb can be essentially considered as solutes, i.e., in neutral charge state. Therefore, the only relevant finite size corrections would be due to the elastic interactions, which we safely ignored because the considered concentrations are comparable to what is experimentally achievable. Clearly, the calculated energies can not be used to predict equilibrium concentrations.

Fig. 1: Formation energy of Ti and Nb in In2S3.

The calculated formation energy of Ti and Nb on 8e tetrahedral and 16h octahedral sites of In as a function of the Fermi level under In-rich (a) and S-rich (b) growth conditions. Here, one TM was incorporated in an 80-atom cell.

For both, In- and S-rich samples, TMs prefer to occupy In octahedral sites rather than tetrahedral sites. Due to its very low formation energy (0.06 eV at CBM), ({{rm{Ti}}}_{{{rm{In}}}^{16{rm{h}}}}) can exist in In-rich samples in very high concentrations, whereas the formation energy of ({{rm{Ti}}}_{{{rm{In}}}^{16{rm{h}}}}) in S-rich samples is one order of magnitude larger (0.59 eV at CBM). The formation energy of ({{rm{Nb}}}_{{{rm{In}}}^{16{rm{h}}}}) is 0.68 and 1.22 eV for In-rich and S-rich conditions, respectively. Intuitively, one would expect a lower formation energy of Nb/Ti under In-poor conditions. However, one has to consider that here the thermodynamic constraints are on the possible boundary phases of the solutes and the substituted elements, which leads to the coupling of their chemical potentials Eqs. (5), (6), and (7). Therefore, with the increase of the chemical potential of In under In-rich conditions, the chemical potential of the solute increases as well. The origins of the different formation energies of Ti and Nb will be discussed in subsection “Heat of solution of Ti and Nb in In2S3”.

Electronic properties of In2S3 alloyed with TMs

The calculated partial and total densities of states (DOS) of pure In2S3 is shown in Fig. 2. The calculated electronic band structure of β−In2S3 with hybrid functional is given in ref. 25. The DOS at the uppermost part of the valence band is mainly derived from the S 3p states mixing with In 4d states. Due to the small contribution of S 3s states, they are not shown here. Figure 3 shows HSE spin-polarized densities of states of Ti- and Nb-substituted In2S3, where the energy level of 0 eV corresponds to the highest occupied state. Here, four Ti or Nb ions were either accommodated in four In16h or In8c octahedral sites, corresponding to 5% concentration of these solutes. The subsequent incorporation sites are separated from each other by about 8.6 Å, which is the largest distance achievable when using the chosen supercell. It is found that the energetic position and shape of the DOS of the impurity bands are different when the solute TMs substitute In16h and In8c sites. As mentioned earlier, there are three nonequivalent crystallographic environment for S atoms in the structure of β−In2S3. Here, we refer to them as S1, S2, and S3. S1 atoms are four-fold coordinated with two In8c, one In16h, and one In8e next neighbors; S2 atoms are three-fold coordinated with two In16h and one In8c next neighbors; and S3 atoms are four-fold coordinated with three In16h and one In8e next neighbors. TMs on In16h sites are bonding with one S1, two S2, and three S3. While the ones on In8c sites are bonding with four S1 and two S2 atoms. Therefore, the lattice distortions caused by TMs on In8c or In16h sites are different.

Fig. 2: Density of states of pure β−In2S3.

Spin-polarized partial and total densities of states of pure β–In2S3. Energy zero is equivalent to the VBM.

Fig. 3: DOS of TM-incorporated In2S3.

Spin-polarized local densities of states computed for Ti-substituted (a, b) and Nb-substituted (c, d) In2S3. The TMs are added on 16h and 8c octahedral sites. Energy zero is equivalent to the highest occupied state.

In 2008, Palacios et al.26 studied the electronic structure of V- and Ti-doped In2S3 using DFT. For Vanadium, they found a narrow half-filled intermediate band isolated from the VB and CB of In2S3. Recently, we revisited V-doped In2S320, using hybrid functional and explained that the erroneous DFT-predicted partially filled IB was caused by neglecting the strongly correlated nature of d-electrons and the antecedent Jahn–Teller effect. In fact, trivalent V ions do not form any IB at all. And only if V ions get reduced, then, the divalent V introduce a totally-filled triplet IB within the band gap of In2S320. For Ti, they showed that when Ti replaces In octahedral site (the different In octahedral sites were not distinguished in their study), the GGA approximation predicts an overlap between the d-bands of Ti and the CB of In2S3, which yields to an IB positioned at the bottom of the conduction band. They assumed that correcting the GGA band gap would shift the position of the CBM upward and the incorporation of Ti would eventually lead to a partially filled IB inside the gap. As illustrated in our previous work20, going from semi-local PBE to HSE functional, the band gap opens from 0.84 eV to 2.03 eV, and the downward shift of the VBM is about twice larger than the upward shift the CBM, which is in contrast to the assumption of Palacios et al. We see from Fig. 3 that when Ti atoms occupy In16h sites, an occupied singlet state occurs at 0.14 eV above the VBM and an unoccupied doublet state appears in the CB. On the ({{rm{Ti}}}_{{{rm{In}}}^{8{rm{c}}}}) configuration; however, no in-gap state appears and only sulfur p states near the band edges move into the gap and reduce the band gap to 1.57 eV. In case of Nb, Fig. 3 shows that inclusion of Nb on In16h and In8c sites leads to occupied doublet in-gap states at 0.62 eV and 0.45 eV above the VBM, respectively. The electronic band structure of ({{rm{In}}}_{28}{{rm{S}}}_{48}!:!{{rm{Ti}}}_{4}^{16{rm{h}}}) and ({{rm{In}}}_{28}{{rm{S}}}_{48}!:!{{rm{Nb}}}_{4}^{16{rm{h}}}) are illustrated in Supplementary Fig. 2.

In the octahedral geometry of In2S3, the six neighboring ligands are oriented along the z-axis and the diagonal directions of the xy plane. Therefore, the ({d}_{{z}^{2}}) and dxy orbitals are subject to a stronger electrostatic repulsion compared to the dxz, dyz, and ({d}_{{x}^{2}-{y}^{2}}) orbitals. Consequently, transition metal substituents occupying these sites experience a corresponding crystal field that pushes the dxz, dyz, and ({d}_{{x}^{2}-{y}^{2}}) orbitals (also called t orbitals) into a lower energy compared to the ({d}_{{z}^{2}}) and dxy orbitals (also called e orbitals). In addition, due to the spin splitting, these levels further split into spin-up (t+ and e+) and spin down (t and e) levels. As a result of crystal field and exchange-splitting, d levels of free Ti3+ and Nb3+ have respectively [({t}_{+}^{1}{e}_{+}^{0}{t}_{-}^{0}{e}_{-}^{0})] and [({t}_{+}^{2}{e}_{+}^{0}{t}_{-}^{0}{e}_{-}^{0})] electronic configurations.

When entering the octahedral In site, e+ and e levels of TM hybridize with S dandling-bond states (e(p)) and create two types of levels with e symmetry: one with a high contribution of six neighboring S atoms, referred to as dangling-bond hybrid (DBH) states and the other mostly localized on the TM atom, referred to as crystal field resonance (CFR) states27. Note that the eDBH and eCFR are bonding and antibonding levels, respectively. For the t+ and t levels, however, there is no host level with t symmetry in the similar energy range. Therefore, the t(d) states remain largely uncoupled and the amplitude of wavefunction is mainly localized on the TM, hence a CFR state27. Consequently, the order of electronic states of Ti and Nb on octahedral In sites are as:

$${e}_{+}^{{mathrm{DBH}}}to {e}_{-}^{{mathrm{DBH}}}to {t}_{+}^{{mathrm{CFR}}}to {t}_{-}^{{mathrm{CFR}}}to {e}_{+}^{{mathrm{CFR}}}to {e}_{-}^{{mathrm{CFR}}}.$$

We note that realizing an ideal IB alloy depends on the position and occupancy of the ({t}_{{mathrm{CFR}}}^{+}) state. The simplified electron-filling schemes of Ti and Nb on vacant In site (({{rm{Vac}}}_{{rm{In}}}^{3-})) are shown in Fig. 4. We see that for both Ti and Nb, the partially filled ({t}_{+}^{{mathrm{CFR}}}) state splits into a filled lower-energy and an empty upper-energy branch, i.e., ({t}_{+}^{{mathrm{CFR}}}to {a}_{+}^{1}+{e}_{+}^{0}) for Ti and ({t}_{+}^{{mathrm{CFR}}}to {e}_{+}^{2}+{a}_{+}^{0}) for Nb. The a and e notations refer to the singlet and doublet natures of these states, respectively. Note that the partially filled degenerate states are Jahn–Teller active centers, which necessiates removing the degeneracy of the intermediate state through splitting it into filled and empty states. Therefore, the onset of a partially filled IB which can trigger simultaneous VB → IB and IB → CB transitions is extremely unlikely. However, if well delocalized, the ({a}_{+}^{1}) and ({e}_{+}^{2}) in-gap states can increase the sub-band gap photon absorption, mostly through assisting in the IB → CB transition.

Fig. 4: The projected e and t DOS of TM-incorporated In2S3 and their corresponding electron-filling diagram.

The projected e and t DOS of Ti (a), Nb (c), and V (e) on In16h site, and their schematic electron-filling diagrams (b, d, and f). Energy zero in sub-figures a, c, and e is equivalent to the highest occupied state. Sub-figures e and f are reproduced from ref. 20 with permission from the Royal Society of Chemistry.

In 2011, Ho28 used chemical vapor transport method to grow Nb-doped In2S3 films and detected below gap transitions of 1.52 and 1.42 eV. Since these transitions were not detected in undoped In2S3, he assigned them to the creation of an IB. However, it was not clear whether the observed transitions stemmed from a deep defect level or a delocalized band. Thus, the question is whether the observed intermediate levels have an unwanted localized or a beneficial delocalized character. According to our calculations, In2S3:Nb allows IB to CB transitions of 1.55 and 1.66 eV, as 5% Nb ions occupy In16h and In8c sites, respectively. We note that although the spontaneous VB to IB and IB to CB transitions (which was the primary goal of constructing IB solar cells) can not be achieved, but if the wavefunctions of doped TM become delocalized throughout the lattice, we can have IB to CB sub-band gap transitions in addition to the VB to CB transitions.

Note that as we move from left to the right along the 3d row of the periodic table, we approach the heavier 3d metals, whose atomic t and e orbitals are deeper in energy. Due to this simple rule, by increasing the atomic number along the same row, the interactions between the inserted TM and anion dangling bonds become stronger. Therefore, the t orbital of the 3d transition metals after Ti should reside below the VBM. As discussed in our previous work, the t+ state of the trivalent V splits into e+ and a+ sublevels, which reside at the top of VB and bottom of the CB, respectively. Hence, no in-gap state forms20. Going down the group 5B of periodic table from V to dubnium (Db) (or in general all d-block groups), the energy of d orbitals increases. Consequently, the t and e energies of the heavy Db should reside far above the energy levels of the light V.

Effect of concentration on the intermediate states formed by the incorporation of Ti and Nb

As mentioned before, at low concentrations, transition metal impurities are likely to introduce undesired deep defect levels inside the band gap of the host semiconductor. However, at sufficiently high concentrations, their wavefunctions become delocalized and a beneficial IB forms. In the case of In2S3, we know from the previous subsection that Ti and Nb impurities induce respectively filled singlet and doublet states inside the band gap of In2S3. However, up to this point it is not possible to predict whether such states are localized defect levels or beneficial delocalized IBs. In order to gain insights on this issue, we present in Figs. 5 and 6 the concentration-dependent spin-polarized DOS of the In2S3 doped with Ti and Nb ions that are located on the In16h sites. Furthermore, we show the partial charge density isosurfaces associated to the defect-induced states as the concentration of the doped TM increases. Since the charge density is associated to the defect wavefunction, its degree of (non)localization is associated to creation of localized or delocalized defect-induced states.

As we can see in Fig. 5a, c, e, the Ti-induced features on the electronic structures of In2S3, highlighted by the dashed box in each figure, do not show variations on their width, shape, or position as the Ti concentration increases, and only their height changes. Furthermore, their narrow shape points out that even at higher concentrations, the Ti-induced defect states would remain localized. In order to confirm this conclusion, we rely on the charge density isosurfaces of the Ti-induced defect states shown in Fig. 5b, d, f. As shown, even for a concentration of 5%, there is a rather negligible overlap between the states induced by Ti ions and host ions. This means that Ti-doped In2S3, at least for doping concentrations up to 5%, would not favor the formation of a beneficial IB.

Fig. 5: Concentration-dependent DOSs of Ti in In2S3 and their corresponding charge density isosurfaces.

Spin-polarized DOS computed for Ti-substituted In2S3 along with the charge density isosurfaces associated with the features induced in the band gap for a, b one, c, d two, and e, f four impurities inside the supercell. The Ti atoms are located on the available 16h In sites in the supercell and the pink box in density of states plots points out the defect states for which the correspondent charge density isosurfaces are produced using VESTA41 (displayed at 3% of their maximum value). The Ti impurities are shown in red and marked with an arrow of the same color. In and S are shown as green and blue spheres, respectively.

On the other hand, as shown in Fig. 6a, c, e, Nb-induced states in In2S3 are concentration dependent. When changing the doping concentration from 1.25 to 2.5%, the situation is analogous to the case of Ti and only the height of the defect states varies. However, when the Nb concentration reaches a value of 5%, the shape of the defect-induced states changes, too. Carrying out the same charge density analysis done for the case of Ti, we obtained the results presented in Fig. 6b, d, f. As illustrated, at the concentration of 2.5 and 5%, we see that the overlap of the defect wavefunction is achieved in all three dimensions and the introduction of a delocalized IB becomes probable. This result agrees with the wider intermediate band observed in the DOS (Fig. 6) and band structure (Supplementary Fig. 2) of the Nb-doped In2S3, which amounts to 0.39 eV, compared to 0.15 eV obtained for the Ti case. Therefore, the potentially relevant Nb-doped In2S3 with its well positioned IB can exhibit beneficial features for below gap IB → CB transitions.

Fig. 6: Concentration-dependent DOSs of Nb in In2S3 and their corresponding charge density isosurfaces.

Spin-polarized densities of states computed for Nb-substituted In2S3 along with the charge density isosurfaces associated with the features induced in the band gap for a, b one, c, d two, and e, f four impurities inside the supercell. The Nb atoms are located on the available 16h In sites in the cell and the pink box in density of states plots points out the defect states for which the correspondent charge density isosurfaces are produced using VESTA41 (displayed at 3% of their maximum value). The Nb impurities are shown in red and marked with an arrow of the same color. In and S are shown as green and blue spheres, respectively.

In 2017, Wägele et al.29 used X-ray diffraction, Raman spectroscopy, and scanning electron microscopy to analyze structural properties of the V-doped In2S3 for various concentrations of V. They found that even with excessive amount of V (as high as 5.8 at%) in In2S3, no additional peak appeared in the X-ray spectra and only the lattice constant decreased, which is due to difference in atomic radii of V (1.35 Å) and In (1.55 Å). V and Nb are chemically similar elements and the atomic radii of Nb (1.45 Å) is closer to that of In. In this regard, we expect that In2S3 is also stable against Nb incorporation.

The reader must be warned that the onset of a filled IB shifts the quasi-Fermi level to the top of the IB, which consequently limits the open circuit voltage (VOC) of the device. Therefore, a key area of future research would be to identify possible mechanisms of depopulating the filled IB. Photo-filling or -emptying of an IB proposed by Strandberg and Reenass30 could be helpful here as well. We note that the filled IB can be used on a device with spatially varying occupation of intermediate-band states. In such a device, the absorber has a filled IB in half of the device and an empty IB on the other half31. The theoretical efficiency of a device with spatially decoupled VB-to-IB and IB-to-CB is 52.83%, which is lower than the efficiency of an ideal partially filled IB but still significantly higher than the efficiency of the best single band gap solar cell. Based upon current state-of-the-art in the field, below gap absorption of Nb-doped In2S3 can be also further improved through an effective non-compensated pn-doping approach, which has been successfully applied in TiO232.

Heat of solution of Ti and Nb in In2S3

The heat of solution (Hs) of TMs in In2S3 were calculated according to Eq. (7) and are shown in Fig. 7. Our results indicate that there is a difference between Hs of TMs in In-rich and S-rich samples. For both Ti and Nb, substituting an In site is energetically more favorable under In-rich condition and their inclusion on S-rich samples becomes energetically more expensive. In particular, we see that the heat of solution of Ti atoms substituting the In sites are lower than Nb.

Fig. 7: Heat of solution of TMs in In2S3.

Calculated heat of solution of Ti (a) and Nb (b) as a function of concentration under In-rich and S-rich growth conditions.

To understand the difference in solubilities of Ti and Nb in In2S3, we analyze the mechanical and chemical contributions to the calculated heat of solution33. The mechanical (or structural) effect refers to the energy stored due to the size misfit of the impurity in the host crystalline matrix. The chemical effect represents the energy stored or released due to the constitution of impurity-host bonds. Depending on the nature of the bonds, this term can be positive or negative. According to Lozovoi et al.33, we define configurations A, Av, B, and Bv as follows:

  • A: The relaxed structure of the In2S3 host;

  • Av: The same atomic structure as A, with an In atom removed from the 16h site;

  • B: The relaxed structure of ({{rm{TMIn}}}_{31}{{rm{S}}}_{48});

  • Bv: The same atomic structure as B, with the TM impurity removed from the 16h site.

Note that Av and Bv are hypothetical structures that we construct through using the relaxed A and B structures, respectively.

The change in energy due to structural effect consists of two steps; Host Removal (HR) and Structural Substitution (SS), which we define as following:

$${H}_{{rm{HR}}}=E({A}_{v})-E(A)+{mu }_{{rm{In}}},$$




The contribution of Chemical Effect (CE) to heat of solution is defined as:

$${H}_{{rm{CE}}}=E(B)-E({B}_{v})-Delta {H}_{f}(Ti{S}_{2})+2{mu }_{S}$$


for Ti, and

$${H}_{{rm{CE}}}=E(B)-E({B}_{v})-Delta {H}_{f}(Nb{S}_{2})+2{mu }_{S}$$


for Nb.

Table 1 lists the mechanical and chemical contributions to the heat of solution of Ti and Nb in In2S3. The energy values correspond to the incorporation of 1.25% TM in In-rich samples. We see that the removal of an In atom from the host configuration (the HHR mechanism) accounts for a large increase in the energy of the system. The effect of structural substitution (HSS) is modest for Nb (0.01 eV). In case of Ti, however, there is a considerably larger lattice distortion contributed to the SS mechanism, which is due to the smaller size of Ti compared to Nb. Another distinguishing feature can be observed when the Ti (or Nb) are back into the vacant site (HCE). Since both Ti and Nb are more electropositive than In, they form stronger bonds with the six surrounding S atoms and the term HCE causes a large decrease in the energy of the system containing the alloying TM. For Ti, contribution of HCE in decreasing the heat of solution is −8.33 eV. As a consequence, the chemical decrease almost cancels the mechanical increase (sum of HHR and HSS mechanisms) in the total Hs. This results in a very low, but still positive heat of solution for Ti (0.08 eV). Note that the mechanical increase still dominates the chemical decrease, resulting in a positive heat of solution for Ti. The bonding between Nb and S is weaker compared to Ti. Therefore, the HCE is less negative for Nb and the energy consuming host removal and structural substitution can not be cancelled out by forming the weaker Nb-S bonds. Hence, the heat of solution of Nb is larger compared to Ti.

Table 1 Heat of solution of TMs in In2S3.

Our results indicate that the heat of solution, Hs, of Ti and Nb are positive in both In-rich and S-rich atmospheres, suggesting that their dissolution into In2S3 is an endothermic reaction. Therefore, both In-rich and S-rich samples can accommodate Ti and Nb atoms on octahedral In sites without leading to the formation of secondary phases.

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