Introduction

In the study of crystalline solids, an emphasis is often placed on the highest possible degree of crystalline order. Fewer elements and simple stoichiometry make things simpler. Defects are unavoidable evils to be minimized. Cation and anion substitutions are necessary evils, used for doping and for tuning properties. Recently though, a new kind of simplicity has emerged when sufficient disorder is introduced that high-symmetry structures are stabilized by entropic considerations. This emergence of high-symmetry structures when multiple metals share the same site has been extensively discussed in the context of high-entropy alloys (HEA), with five or more elements sharing a site1,2. HEA have attracted attention due to two factors: the stability of high-symmetry phases not stable with fewer number of elements on the same site, and the unique properties that emerge in these materials2.

This has recently been expanded to high-entropy carbides3,4,5, high-entropy oxides6, and high-entropy fluorides7, where a simple anionic sub-lattice coexists with a cation site shared by five or more cations. In all of these cases, entropy comes from different elements sharing the same site, where each distinct element contributes to the configurational entropy equally, resulting in the stability of high-symmetry phases at high temperature, when the entropy term dominates the enthalpy term in the Gibbs free energy, ΔG = ΔH − TΔS, resulting in a single phase that can be quenched to ambient conditions. Here we extend this idea into new territory by considering entropy coming from different charge states of the constituent elements. This opens a large range of possibilities by reducing the number of elements needed for entropy-stabilization to be relevant.

One way to introduce elements with different charge states is to use valence skip elements in materials with intermediate nominal oxidation. This is an important class of materials hosting phenomena such as superconductivity8, negative thermal expansion9, and the charge-Kondo effect10,11. Such materials come in two categories: one where the average valence is a fraction of an electron and one with an integer number average valence, where the difference between the most stable states is an even number of electrons. In the former, when the charge is modulated in space we find the development of charge ordering, and when modulated in time, we find valence fluctuations. In some materials, these can overlap and differ only in the temperature at which the state is stable12. In the case of a whole number average valence, we find the emergence of disproportionation of ions into different sites, typically thought of as charge disproportionation8. Depending on the relative effective charge-transfer energy of the cation and valence band energy of the anion the electron may be shared along a covalent bond between the cation and anion13,14. Two distinct crystallographic sites may arise resulting in a bond-disproportionated material with uniform cation charges and holes on the coordinating anions15,16. One can find fluctuations in these compounds by inducing a metallic state either through forced orbital overlap between neighboring atoms, by applying pressure, or the substitution of some elements to offset the whole number average charge such that a well-ordered lattice becomes unstable, leading to delocalization of the electrons.

Valence skip elements have stable oxidation states that vary by two charges where the middle oxidation state is not stable, such that M(n−1)+ and M(n+1)+ are stable, but the M(n)+ is not stable or metastable. Materials with valence skip elements, such as Au, Tl, Sn, Pb, Sb, Bi, can have a whole number average nominal oxidation8. An insulating disproportionated state can emerge when we have an ordering of the charges, such is the case of Ba2Bi3+Bi5+O617 and Cs2Au1+Au3+Br618. If the ordering of charges is destabilized, then electrons are delocalized, and electron conduction involves two-electron valence fluctuations on the valence skip element in these materials. These valence fluctuations have long been discussed in relation to superconductivity, where they are thought to contribute to pairing interactions through correlation-enhanced electron-phonon interaction19 or negative-U superconductivity8,20. Here we consider the fluctuations on Sn in (Ag,Sn)Se, and the effect they have on the superconductivity and the stability of the cubic rocksalt structure. The role of charge and bond entropy in the stability of solids has been previously considered for LixFePO421 and LiNiO222, but the definitive role of charge/bond entropy in stabilizing a single phase remains an open question. We consider charge entropy on the Ag/Sn site in (Ag,Sn)Se, and the role of charge entropy in stabilizing the cubic structure. Unlike the high-entropy alloys and oxides, with five cations sharing the same site, a discussion of distinct ionic states when we consider two constituent elements is a simpler problem to solve.

We relate the charge-entropy in relation to the Sn-fluctuations to the stability of the rocksalt structure in AgxSn1−xSe. X-ray photoemission spectroscopy (XPS) reveals a single peak with constant binding energy regardless of the composition x and a small shoulder at higher energy that increases in intensity with increasing x. Strong hybridization of Se-p and Sn-d, consistent with our DFT calculations, allows for a shift in spectral weight to the lower energy peak in our XPS spectra. We find evidence of a single distinct peak in the Sn-Mössbauer spectra for each composition of (Ag,Sn)Se down to 4 K. However, the position of this peak shifts for different samples with different composition of Ag in AgxSn1−xSe, consistent with the average oxidation moving towards Sn2+ as x approaches zero. We provide a statistical argument for the distribution of Sn2+ and Sn4+ for a particular composition x to explain the shifting of the Sn-Mössbauer peak in each of the samples.

Results

Rocksalt AgxSn1−xSe and the state of Sn

AgxSn1−xSe is stable in the rocksalt structure for 0.38 ≤ x ≤ 0.50, shown in Fig. 1(a), where the lattice constant a increases with increasing Sn content23. To obtain single phase samples it is necessary to quench from high-temperature23. Superconductivity is found in all AgxSn1−xSe samples in the rocksalt structure, where the Tc increases with higher Sn content, smaller x24. We attempt the synthesis of samples with higher Sn content with the aid of high-pressure high-temperature synthesis, but are unable to extend the stability of AgxSn1−xSe for x ≤ 0 .38 even under high-pressure conditions. We find the same trend of increasing lattice constant and Tc with increasing Sn content in high-pressure grown samples as shown in the Supplemental Material, Fig. S5 (Tc = 6.0 K for x = 0.38, Tc = 4.0 K for x = 0.50). Fluctuations inside the superconducting state have been reported in AgxSn1−xSe for x = 0.50, with evidence of field-induced quantum fluctuations observed in resistivity measurements at low temperature25.

Fig. 1: Entropy of cation site in rocksalt structure (Ag,Sn)Se2−.
figure 1

a The rocksalt structure of (Ag,Sn)Se shown on the left side, and the two possible charge distributions on the Ag/Sn site are shown on the right. Top right shows the Sn2+/Sn4+ distribution, and bottom right shows the Sn2+/Sn3+ distribution, along the formulae dependence on x. b The entropy of mixing δSmix for different numbers of components, n = 2, 3, sharing the same site as a function x for a single component with all other components being equal. The mixing δSmix considering the charge entropy as a function of x for the case of Sn2+/Sn4+/Ag+1 and Sn2+/Sn3+/Ag+1 are shown for comparison. c A zoomed in result of (b) where the stability region of AgxSn1−xSe is highlighted with filled region around the peak of the curve for Sn2+/Sn4+/Ag+1 entropy.

Full size image

Earlier work on the Sn Mössbauer of AgxSn1−xSe with different compositions demonstrates a peak that lies between that expected for 2+ and 4+ state26, and this was explained by fluctuations of the Sn state where the Mössbauer probes the average of these two states due to the fast fluctuations relative to the energy difference of the probed states in Mössbauer26,27. A similar situation is expected when probing the states with NMR, where indeed a single peak is observed for x = 0.5 samples, which was attributed to a novel “\({{{{\rm{Sn}}}}}^{3+}\)” state28. Although it is reasonable considering the consistency of DFT calculations with the measured state in NMR28, the calculations presented in the study only rule out the presence of a static charge disproportionation into differentiated sites. Furthermore, the study lacks any discussion of dependence of the Sn state on composition x and ignores previous XPS work demonstrating two distinct states for samples with different Sn content.

We bring to attention the importance of the stability region 0.38 ≤ x ≤ 0.5 for the rocksalt structure of AgxSn1−xSe, and relate this to the state of Sn in the material in Section “Charge-Entropy in AgxSn1−xSe”. In Section “XPS on AgxSn1−xSe” we discuss the oxidation state of Sn in the same samples measured with XPS, and in Section “Low-temperature Mossbauer on AgxSn1−xSe” we discuss the state measured with Mössbauer spectroscopy. Mössbauer measurements were carried out at low temperature showing persistent fluctuations down to 4 K, which has implications for the superconductivity.

Charge-entropy in AgxSn1−xSe

We consider the entropy in the system as discussed for high-entropy alloys and compounds with the equation for configurational entropy S1,29. Here we utilize the idea of configurational entropy, and first consider the entropy on the cation site coming from two distinct atoms in the case of AgxSn1−xSe:

$$\Delta {S}_{{{{\rm{atom}}}}}=R\left[xln(x)+(1-x)ln(1-x)\right]$$
(1)

where x is the composition of Ag. This is plotted with a orange curve (n = 2) in Fig. 1(b). For comparison, we examine the entropy from three constituent cations (n = 3) on the same site, where x corresponds to the content of one component while the other two components have equal composition with the green curve (n = 3) in Fig. 1(b).

Going beyond the entropy coming from the individual atomic species on the cation sites, we examine the charge-entropy. By considering two charge states on Sn we are simultaneously tuning three parameters, albeit coupled, by adjusting the Ag composition x. The charge-entropy on the cation site depends on the composition of Ag in the sample and the resulting Sn states in the two fluctuation scenarios. Pauli paramagnetism in AgxSn1−xSe would be much larger for Sn3+ with its unpaired electron, but we find the susceptibility to be fairly small, providing strong evidence for 2+/4+ fluctuations27. Nevertheless, we have considered two scenarios for the charge-entropy in AgxSn1−xSe.

If we take the x = 0.5 sample as representing that of a true Sn3+ state, then as we increase the amount of Sn in the sample we must have an increase in the Sn2+ in the sample. The amount of Ag1+ and Sn3+ must be equal, such that the average charge of one Ag1+ and one Sn3+ is 2 + to balance with Se2− anions, while the remaining Sn will be in the 2+ state. For this case of Sn3+/Sn2+ fluctuations we have the following charge-entropy, \(\Delta {S}_{{{{{\rm{Sn}}}}}^{2+}/{{{{\rm{Sn}}}}}^{3+}}\), dependence on the Ag-content x:

$$\Delta {S}_{{{{{\rm{Sn}}}}}^{2+}/{{{{\rm{Sn}}}}}^{3+}}=R\left[xln(x)+xln(x) +(1-2x)ln(1-2x)\right]$$
(2)

where the first and second term correspond to the entropy from Ag1+ and Sn3+, while the third term corresponds to the entropy from Sn2+.

The more likely scenario considering the low paramagnetism, is the 4+/2+ fluctuations. Here, the amount of Sn4+ in the sample corresponds to half the amount of Ag1+, such that the average charge of one Sn4+ cation and two Ag+1 is 2+ to balance with Se2− anions, while the remaining Sn atoms are Sn2+. For the case of Sn4+/Sn2+ we have the following charge-entropy, \(\Delta {S}_{{{{{\rm{Sn}}}}}^{2+}/{{{{\rm{Sn}}}}}^{4+}}\), dependence on x:

$$\Delta {S}_{{{{{\rm{Sn}}}}}^{2+}/{{{{\rm{Sn}}}}}^{4+}}=R\left[xln(x)+\left(\frac{x}{2}\right)ln\left(\frac{x}{2}\right)+\left(1-\frac{3x}{2}\right)ln\left(1-\frac{3x}{2}\right)\right]$$
(3)

where the first term corresponds to the entropy from Ag1+, the second term is for entropy from Sn4+, and the third term corresponds to the entropy from Sn2+.

In Fig. 1(a) and (b), we compare the curves for \(\Delta {S}_{{{{{\rm{Sn}}}}}^{2+}/{{{{\rm{Sn}}}}}^{3+}}\) (purple curve) and \(\Delta {S}_{{{{{\rm{Sn}}}}}^{2+}/{{{{\rm{Sn}}}}}^{4+}}\) (blue curve). We find that each curve representing the charge-entropy touches at one point the curve for the configurational entropy of a 3-component system (n = 3) defined above. However, the maximum entropy for the \(\Delta {S}_{{{{{\rm{Sn}}}}}^{2+}/{{{{\rm{Sn}}}}}^{3+}}\) curve coincides with the maximum of the 3-component curve at x = 0.33, while the maximum of the \(\Delta {S}_{{{{{\rm{Sn}}}}}^{2+}/{{{{\rm{Sn}}}}}^{4+}}\) curve is pushed to higher value of x = 0.436. The maximum entropy reached in \(\Delta {S}_{{{{{\rm{Sn}}}}}^{2+}/{{{{\rm{Sn}}}}}^{4+}}\) is lower than that of \(\Delta {S}_{{{{{\rm{Sn}}}}}^{2+}/{{{{\rm{Sn}}}}}^{3+}}\). However, the stability region for AgxSn1−xSe coincides well with the peak of the charge-entropy based on the 2+/4+ fluctuation scenario, \(\Delta {S}_{{{{{\rm{Sn}}}}}^{2+}/{{{{\rm{Sn}}}}}^{4+}}\). We highlight the stability region of x around the maximum of the curve. Furthermore, quenching from high-temperature is important to stabilize a single phase sample as previously mentioned. This strongly hints at the stability of the phase being related to the domination of the entropy term at high temperature in the Gibbs free energy. Next, let us compare the observed oxidation state of Sn from XPS and Mössbauer spectroscopy.

XPS on AgxSn1−xSe

In Fig. 2 we show the core level spectra of the AgxSn1−xSe samples measured on cleaved surfaces with hard x-ray photoelectron spectroscopy (HAXPES), in order to obtain results representative of the bulk of the compound. For both Ag 3d and Sn 3d we observe single sharp assymetric peaks, split by the spin orbit coupling (SOC). As for the Se 3d, we observe a clear two peak structure overlapping due to the smaller SOC of the Se 3d of around 1 eV. In order to understand better the trends in the spectral lineshape as well as position, we show a close-up comparison with normalized intensity of the Sn and Ag 3d5/2 spectra in Fig. 3.

Fig. 2: Bulk-sensitive x-ray photoemission spectra.
figure 2

Hard x-ray photoemission spectra of Ag, Sn, and Se in cleaved AgxSn1−xSe samples are shown in (a–c). The splitting between the main peaks is consistent with that expected due to spin-orbit-coupling (SOC) for the Ag, Sn, and Se 3d states, respectively.

Full size image
Fig. 3: Comparison of cation photoemission spectra.
figure 3

Close-up of the hard x-ray photoemission spectra of the Ag-3d5/2 (a) (Sn-3d5/2) (b) in cleaved AgxSn1−xSe samples, normalized to their maximum intensities, together with the spectra from metallic Ag (Sn) from refs. 30, 31 shifted in binding energy to match the peak positions of AgxSn1−xSe in order to provide a comparison of the spectral lineshape.

Full size image

First, we observe in Fig. 3(a) that the asymmetry of the Ag 3d5/2 peaks fully matches that of Ag metal (obtained from30, shifted in energy to match the peak position of the AgxSn1−xSe spectra), as could be expected from the metallicity of the system with strong contributions from all Ag, Sn and Se at the Fermi level, as shown in Fig. S7 of Supplemental Material. The position of the peak remains constant in energy for all x. Thus, our Ag 3d core level spectra are consistent with a single valent Ag1+ metallic state that remains stable with increasing x.

As for the Sn 3d5/2 (Fig. 3(b)), we observe that while most of its spectral weight can be explained by the Doniach-Sunjich asymmetric shape as seen in Sn metal (obtained from31, shifted in energy to match the peak position of the AgxSn1−xSe spectra), there is an additional small component at around 486.5 eV. The single dominant component in the core level spectra rules out the possibility of a static (e.g., ordered) charge disproportionate, where 2 separate peaks from the Sn2+ and Sn4+ with spectral weight ratios directly reflecting their proportion in the ground state would be expected. Instead, it indicates that the Sn ions must be either all single valent or highly fluctuating state.

Under a single valent assumption, one would expect the nominal valence changes of Sn resulting from changing x, from 3+ to around 2.6+, to be reflected in a core level energy position shift. However, the Sn 3d peak remains in all cases at a constant position within our experimental resolution, ruling out also this possibility. This leaves us with the highly fluctuating valence scenario, with a fluctuation of two electrons due to an effective attractive U on the Sn atoms8. Considering the Robin-Day classification scheme32, AgxSn1−xSe falls into class IIIB where even a fast technique such as HAXPES does not show two distinct peaks corresponding to two distinct states.

Nevertheless, the presence of the additional component observed at around 486.5 eV matches with the expected “antibonding” peak that would arise from the two fluctuating states, as discussed in more detail in Supplemental Material, Sec. A. In particular, we observe that the experimental spectra can be well described by a fixed main Doniach-Sunjic peak at 485.5 eV and a small second contribution at 486.5 eV, as highlighted in the inset of Fig. 3(b), which systematically increases as a function of x (Fig. 4(b)), in agreement with the behavior predicted for the “antibonding” peak in the qualitative core level model presented in Supplemental Material, Sec. A.

Fig. 4: Mössbauer spectra and the average Sn state as a function of composition x.
figure 4

a The Mössbauer spectra of AgxSn1−xSe samples with different Ag-content, x. The spectra were measured at room temperature, and we found a shift in the peak position with Ag/Sn ratio in the samples. For x = 0.5 sample we find the peak in the isomer shift lies between that expected for 2+ and 4+, while an increase in Sn-content shifts the peak position towards that expected for a 2+ oxidation state. b The Ag-content, x in AgxSn1−xSe, dependence of the ratio of the Sn-3d5/2 fitted peaks in XPS and the isomer shift of Sn in Mössbauer spectra. A clear trend is observed in both as a function of x.

Full size image

The core level spectra discussed in this section differ significantly from those reported in the literature33, where additional components are observed both in the Sn 3d as well as Se 3d. In order to understand the reported differences, we present the XPS spectra on as-grown uncleaved surfaces in Supplemental Material, Fig. S1. While the Ag 3d results are similar to those obtained on the cleaved samples with HAXPES, the Sn 3d as well as Se 3d show significantly different spectral lineshapes. In the Sn 3d we observe a clear 2-component structure, with one peak at around 485.5 eV that would correspond to the bulk contribution observed in the cleaved HAXPES measurements, and a new, larger component at around 487.2 eV. The binding energy of this second larger component clearly differs from the 486.5 eV component discussed above, indicating the 487.2 eV peak in the uncleaved XPS spectra is not a mere change in the Sn2+-Sn4+ behavior but rather arises from a new contribution not present in the cleaved HAXPES measurements, namely the oxidized surface. Similarly, in the Se 3d, the experimental spectra no longer display the clear two SOC-split structure, but instead, have a more complex shape resulting from one additional set of surface components mixed in. These results indicate that the surface stability is affected by the Ag/Sn ratio in the samples, which results in different levels of surface oxidation and a different ratio of the surface to bulk peaks depending on x. As such, bulk-sensitive measurements on fresh, clean surfaces are crucial to obtain results representative of the intrinsic electronic structure of AgxSn1−xSe.

Low-temperature Mössbauer on AgxSn1−xSe

The Sn-Mössbauer for AgxSn1−xSe samples with different x are shown in Fig. 4(a), where we find a single peak for each composition x. The isomer shift of AgxSn1−xSe corresponds to an intermediate state between divalent and quadrivalent-tin compounds. The Sn2+ and Sn4+ impurity ions in PbSe should be compared with the state in AgxSn1−xSe to allow for a comparison where the environment is that of 6-coordinated Se in all cases27. The position of the peak in the isomer shift moves to higher values with decreasing x, which suggests fluctuations between two different states of Sn where the Mössbauer picks up the average of these fluctuations. The shift with composition suggests an increase in 2+ contribution to the fluctuations on average resulting in the position moving towards that of Sn2+, such is the state in SnTe27. This further supports the nature of the fluctuations deduced from the stability region, as discussed in Section “Charge-Entropy in AgxSn1−xSe”.

Mössbauer level lifetime determines the natural spectral-line width, which is Gnat = h/t0 = 0.324mm/s for Sn. The experimental width always exceeds 2Gnat, where the broadening is attributed to unresolved quadrupole splitting, finite absorber surface density, and electron exchange between two tin-charge states. Fluctuations on Sn in AgxSn1−xSe may be related to an activation energy, where the lifetime of each Sn increases such that t > > t0. In such a scenario, a single peak at high temperature should split into two peaks corresponding to each of these states. The slowing down of fluctuations and splitting of Mössbauer spectra has been observed in Eu3S4, where a single peak in the isomer shift at room temperature splits into peaks corresponding to 2+ and 3+ at low temperature34. We compare as a function of x the isomer shift and the relative peak ratio in XPS in Fig. 4(b), and we find a decrease in the isomer shift and an increase in the relative peak ratio as expected for an increase in Sn4+ contribution with increasing x.

We measure the Mössbauer spectra of three different samples, with x = 0.38, 0.44, and 0.50, down to 4.0 K. Comparing the Mössbauer spectra at room temperature for all three compositions in Fig. 4(a), we find the isomer shift decreases with increasing x. For all three samples, a single peak is observed down to 4.0 K, shown in Fig. 5, which means that if the peak position is a consequence of fluctuations, then these fluctuations persist down to 4.0 K. We do find some evidence for peak broadening at low temperature, and we fit the low temperature Mössbauer spectra to multiple peaks as shown in Supplemental Material, Fig. S3. The change in the isomer shift as a function of temperature, shown in Supplemental Material, Fig. S4, shows no evidence of a splitting of one peak into two peaks with the slowing down of fluctuations, as seen in Eu3S434. This is likely related to the disorder, lack of long range order, of Ag and Sn on the same site of AgxSn1−xSe, which hinders the distortion of the structure in a lower symmetry structure to localize the Sn2+ and Sn4+ states.

Fig. 5: Low temperature Mössbauer spectra of Sn for different x.
figure 5

The Mössbauer spectra of AgxSn1−xSe samples with different Ag-content, for x = 0.50, 0.44, 0.38 from left to right. An increase in Sn-content shifts the peak position towards that expected for a 2+ oxidation state. The spectra were measured at different temperatures down to 4 K, and we find a single dominant peak position for all x values down to 4 K.

Full size image

Using DFT simulations, we find that forcing different charges on Sn atoms in ordered lattices, followed by allowing the relaxation of the structure and the charge results in redistribution to almost equal charge on all Sn atoms. This hints at strong correlation in (Ag,Sn)Se evident by the shift of spectral weight in the Sn XPS from the higher binding energy peak to the lower binding energy. Also, we have performed calculations on supercells with disordered Ag and Sn atoms, and find the charge on all atoms varies within  ± 0.1e for different Ag:Sn ratios, as shown in Supplemental Material, Fig. S6.

We consider the statistical distribution of the nearest neighbor and next-nearest neighbor atoms to explain the shift in the position of the isomer shift of fluctuating Sn with x, as demonstrated in Fig. 6. With a Se2− lattice and constrained by Ag+1, we consider the effect of randomly distributed Ag/Sn on the likelihood of the central Sn being in the 4+ or 2+ state. We examine the nearest neighbor Se atoms bonded to Sn, n, (Fig. 6(a)) and next-nearest neighbor Ag/Sn atoms, \({n}^{{\prime} }\), (Fig. 6(b)) around a center Sn atom. For the case of high \(n/{n}^{{\prime} }\) on the right side of Fig. 6(a) and (b) we expect a Sn2+ state, and for the case of low \(n/{n}^{{\prime} }\) we expect a Sn4+ state. We compare the statistical distribution for the case of x = 0.50 and x = 0.38, and we find a higher Sn2+ state contribution to the fluctuations for the lower Ag content (x = 0.38) as expected, consistent with the change in the isomer shift in Mössbauer. We note that the x = 0.50 sample (Tc = 6.0 K) is well inside the superconducting state at 4.0 K, which suggests that these fluctuations on Sn persist inside the superconducting state. If these fluctuations on Sn are important to the emergence of superconductivity, then they may be related to the quantum fluctuations previously reported in resistivity measurements25.

Fig. 6: Statistical distribution of Sn environment for different x.
figure 6

a Statistical distribution of the environment of Sn environment, where we consider the nearest Se atoms and whether they are bonded to Sn or Ag. Based on the value x for Ag/Sn, we find a shift in the distribution of bonded Se atoms moving the distribution towards Sn2+ for smaller x. b Similar statistical analysis as (a), but considering the next-nearest Ag/Sn distribution around the center Sn atom. More Sn2+ like state is expected for more Sn in the environment.

Full size image

We consider the degree of short-range order (SRO) in AgxSn1−xSe associated with fluctuations on Sn. We find that SnSe distorts into an orthorhombic phase Pnma due to the lone-pair on Sn2+ ions35. However, the addition of Ag in AgxSn1−xSe results in cation site being occupied by two cation species without lone-pairs, Ag1+ and Sn4+, that limit distortion of the structure due to the remaining Sn2+ ions. Local distortions associated with the Sn2+ ions in AgxSn1−xSe may be present and should be investigated in future studies that consider the degree of SRO. The degree of SRO can be quantified by evaluating the Warren-Cowley parameters typically discussed for disordered alloys36,37. The emergence of SRO in rocksalt PbTe at high temperature38 and the Ni-Ni bonding driven SRO in KNiSe239 also at high temperature will be interesting to compare with AgxSn1−xSe. These investigations will deepen our understanding of a single-phase rocksalt structure the emerges from cation disorder on a single crystallographic site.

Conclusion

We report the HAXPES of AgxSn1−xSe, where for all x values a single symmetric peak with constant energy is found for the Ag-3d spectra and a single peak with constant energy maximum and varying asymmetry in the Sn-3d spectra. We find that the degree of asymmetry varies as a function of x consistent with a change in the ratio of Sn2+ to Sn4+, where higher Sn4+ is expected for higher x. In contrast, the isomer shift in the Mössbauer spectra shows a single peak that shifts towards the value of Sn4+ with increasing x. Taking the single peak in the HAXPES for Ag-3d as evidence for a Ag+1, we evaluate the charge entropy from cation species in AgxSn1−xSe considering the three cation species. We compare the charge entropy as a function of x considering these three cations Ag+1/Sn2+/Sn4+ and compare this with the model for having Ag+1/Sn2+/Sn3+, and find that the highest entropy region for the former model coincides with the stability region ranging from 0.38 to 0.5 for x of AgxSn1−xSe.

The stability region of AgxSn1−xSe demonstrates that crystalline order can emerge from high charge disorder. We believe that the stability region of AgxSn1−xSe validates the Sn-fluctuations long suspected and the nature of the competition that gives rise to a correlated metal in the normal state. Furthermore, the consideration of charge-entropy in this valence skip compound is successful in explaining the formation of a single phase that can be quenched from high temperature, and opens up the possibility of exploiting the charge entropy in search of other phases that include valence skip elements. However, one should still consider the nature of the individual charged species when looking for new charge-entropy stabilized compounds, which can result in distortion of the high symmetry structure or the formation of an impurity phases that limit the applicability of the charge-entropy model.

Finally, the two-electron mechanism related to the negative-U on Sn is likely related to the superconductivity in AgxSn1−xSe. One may even attribute the higher Tc in Sn-rich samples to a higher concentration of negative-U centers, but this will need further clarification with additional experiments. Once high crystalline order emerges the virtue of charge disorder results in a tension between charge states on the valence skip atoms that can promote superconductivity.

Methods

Synthesis and basic characterization

AgxSn1−xSe samples were synthesized with a high-pressure, high-temperature method utilizing a belt-press. Ag, Sn, and Se powders were well mixed in a mortar and pestle, then loaded into a gold capsule. The capsule was then pressed to 6 GPa and heated at 730 K for 1 hour before quenching and retrieving the polycrystalline product. A finely ground sample was loaded into a 0.3 mm capillary, and powder x-ray diffraction (XRD) patterns of AgxSn1−xSe were collected while rotating the capillary using Ag radiation (0.5594 Å). Samples synthesized under high-pressure were used for Mössbauer and XPS measurements. AgxSn1−xSe samples for HAXPES measurements were synthesized using Bridgman method by placing the elements Ag,Sn, and Se in different ratios for each x in vacuum-sealed tube that were sealed inside another tube and heated to 1000 °C over 15 h and remaining at 1000 °C for 40 h. Then the tube was cooled to 450 °C at the rate of 3 °C/h, followed by quenching the tubes in air.

Mössbauer spectroscopy

Mössbauer was performed using a CaSnO3 with 119mSn as the γ-ray source (Ritverc GmbH, 740 MBq). The absorption of 57Fe was used to calibrate the velocity of the source, and the isomer shift of BaSnO3 at room temperature was set as the origin. A 75 μm thick palladium film was placed between the source and the sample in order block the x-ray fluorescence of Sn. The synthesized samples were first characterized using X-ray diffraction and magnetization measurements. The sample was ground into a powder ( ~ 50 mg) and mixed under nitrogen atmosphere with boron nitride powder ( ~ 70 mg, Kishida Chemical, 99.5%) and polyethylene powder ( ~ 8 mg Beckman RIIC, polyethylene powder for IR spectroscopy) to improve the homogeneity of the sample. Then the sample was pressed into a pellet (10 mm diameter). All measurements, from room temperature down to 4 K, were carried out inside a 4He cryostat (Janis Research, ST-400). The time for each measurement was about 12 h.

X-ray photoemission

The XPS data on uncleaved samples were collected using a commercial Kratos AXIS Ultra spectrometer and a monochromatized Al Kα source (photon energy: 1486.6 eV). The base pressure during XPS was in the low 10−10 mbar range. The spectra were collected using an analyzer pass energy of 20 eV. XPS spectra were analyzed using the CasaXPS software. To fit the peaks, a combination of multiple Gaussian-Lorentzian mixture functions and a Shirley background was used. For the Ag-3d and the Sn-3d the binding energy separation and the area ratio of the doublets were not constrained, but the results are within experimental error of the tabulated values40, while for the Se-3d5/2 and Se-3d3/2 the binding energy separation and area ratio were constrained to 0.86 eV and 3:2, receptively. The sum of all the fitted functions to XPS peaks is shown with a black curve in Supplemental Material, Fig. S1.

The HAXPES data on cleaved samples were collected at the Max-Planck-NSRRC HAXPES end-station at the Taiwan undulator beamline BL12XU at SPring-8, Japan41 using an MB Scientific A-1 HE analyzer. The photon beam was linearly polarized with the electrical field vector in the plane of the storage ring (i.e., horizontal), and the photon energy was set at about 6.5 keV, with an overall energy resolution of around 280 meV, determined by measuring an Au reference sample. The samples were cleaved in a preparation chamber adjacent at a base pressure in the low 10−9 mbar, and transferred in-situ to the measuring chamber, where the base pressure was in the low 10−10 mbar.

Density functional theory

Electronic structure calculations were performed within the framework of density functional theory (DFT) as implemented in the Quantum Espresso package42. The generalized gradient approximation with the PBEsol for the exchange-correlation functional was used43,44 and PAW pseudopotentials. Calculations were converged with respect to the plane wave cutoff and reciprocal space grid. The face-centered cubic crystal structure (a = 5.675 Å) was used23. We performed calculations on large supercells (3 × 3 × 3) with disordered Ag and Sn atoms, placed randomly with our own built Python code to generate a disordered supercell, to evaluate the distribution of charge on atoms.

Statistical methods

The statistical distribution of atoms was calculated using combination statistics45. We consider total number of cations to choose from t and x is percentage of Ag atoms in t. Then, the total number of Sn atoms is s = t × (1 − x) and total number of Ag atoms is a = t × x. The probability Pn of n distinct surrounding atoms (nearest neighbor Se bonded to Sn or next-nearest neighbor Sn) around a central Sn atom is calculated with:

$${P}_{n}=\frac{C(s,n)\times C(a,m)}{C(t,r)}$$
(4)

where r is total number of nearest/next-nearest atoms, such that r = 6 when considering nearest neighbor Se atoms and r = 12 when considering next-nearest neighbor Ag/Sn atoms. We consider integer values of n between 0 and r, and the number of remaining atoms (nearest-neighbor Se not bonded to Sn or next-nearest neighbor Ag) is m = r − n. C is the combination functions defined as \(C(x,y)=\frac{x!}{y!(x-y)!}\). The calculation was repeated with increasing t until the calculated probability did not change to 3 significant digits.