Introduction

Thermoelectric (TE) technology has various applications, such as electric power generation from temperature differences or pollution-free cooling, and has recently attracted attention because it is environmentally friendly and sustainable.

The dimensionless figure of merit, ZT, is commonly used to evaluate the performance of TE materials. Higher ZT values indicate better TE performance. ZT is defined as:

$${ZT}=\frac{\left({S}^{2}\sigma T\right)}{\left({k }_{{{{\rm{C}}}}}+\,{k }_{{{{\rm{L}}}}}\right)}$$
(1)

where S, σ, T, kC, and kL, respectively, represent the Seebeck coefficient, electrical conductivity, absolute temperature, carrier thermal conductivity, and lattice thermal conductivity.

However, increasing the ZT value is not simple, because the physical parameters of ZT (S, σ, and kC) are functions of carrier concentration and depend on one another, thus exhibiting a trade-off relationship. The optimized carrier concentration has been reported as ~1025 m−3 1. In contrast, lattice thermal conductivity is the only parameter independent of carrier concentration. Therefore, controlling the carrier concentration and reducing the lattice thermal conductivity are fundamental strategies in designing efficient TE materials.

Among various TE materials, binary chalcogenide compounds Cu2-x(S, Se) exhibit extremely low thermal conductivities of ~0.5 W m−1K−1 2,3,4,5,6,7,8,9 and high ZT values (Cu2Se: ZT ≈1.8 at 1000 K; Cu1.97S: ZT ≈1.7 at 1000 K). Both Cu2-xS and Cu2-xSe possess similar structural and physical properties. At high-temperature regions, these compounds are classified as non-stoichiometric, maintaining the same crystal structure over a wide compositional range and enabling significant Cu-vacancy defects; 0 ≤ x < ~0.25 in Cu2-x(S, Se), as shown in the phase diagram provided by ref. 10. At high temperatures region, Cu2-x(S, Se) adopts an antifluorite structure in which anions form an fcc framework and Cu cations occupy the interstitial sites. Moreover, Cu2-x(S, Se) exhibits high Cu ionic conductivity due to the hopping of Cu ions from one site to neighboring sites11,12.

Based on the phonon-liquid electron-crystal concept, ref. 4 proposed that the extremely low thermal conductivity in Cu2Se is induced by the liquid-like ionic fluidity of Cu cations, and numerous subsequent studies adopting this approach for Cu2-x(S, Se) have been reported2,13,14,15,16,17,18,19,20,21,22,23,24 However, ref. 25 conducted inelastic and quasi-elastic neutron scattering experiments on Cu2Se to demonstrate that the Cu-hopping process is too slow to suppress phonon propagation. Although they suggested that low thermal conductivity can be ascribed to anharmonicity, no direct experimental evidence has been presented. However, strong anharmonicity in the various superionic compounds has been reported26,27,28,29,30 More recently, ref. 31 discussed the anharmonic phonon frequency of Cu2Se using self-consistent phonon calculations, which introduced anharmonicity up to the fourth order. They could calculate low lattice thermal conductivity values.

Meanwhile, the Cu defects in Cu2-x(S, Se) would affect lattice thermal conductivity according to Eq. 2:

$$k_{{\mbox{L}}} = \frac{1}{3} \sum C_i v_i l_i$$
(2)

where Ci, vi, and li are the heat capacity, phonon velocity, and phonon mean free path of mode i, respectively. The Cu vacancy defect induces point defect-phonon scattering and reduces l and kL. The presence of vacancy defects in Al-based Mackay-type 1/1-cubic approximants has been demonstrated to reduce the mean free path, thereby reducing kL32 Ref. 33,34 demonstrated that increasing point defects in materials reduces kL. Klemens formulated a model for the phonon-point-defect scattering rate along these lines35 As presented in Supplementary Note 1, the results are largely equivalent.

Furthermore, as observed in Eq. 2, the heat capacity C is a proportional quantity in kL. The heat capacity above the Debye temperature (TD(Cu) = 310 K, TD(S) = 527 K)36 can be described by Dulong-petit law. However, due to Cu fluidity, C cannot reach the Dulong-petit limit of 3NkB but approaches 2.5NkB2,4 Here, since the C is proportional to the mole number N, an increase in defect x in Cu2-xS would reduce kL. Mansour et al. 37 experimentally demonstrated that the heat capacity of Cu2-xS decreases with an increasing number of Cu defects.

However, according to refs. 2,5,6,7,8, in Cu2-x(S, Se) systems, the lattice thermal conductivity does not decrease with the number of Cu defects but shows an increasing trend.

Xiao et al.5 reported Cu2-xSe in a wide Cu-deficient range (0 ≤ x ≤ 0.25). Their kL values showed a clear increasing trend with Cu defect. Regarding this trend, they speculated this may be related to the strong phonon scattering induced by short-range disorders but did not provide any experimental evidence. Geng et al. 6 showed the lattice thermal conductivity of Cu2-xSe (0.01 ≤ x ≤ 0.03), despite their close chemical component, their kL values showed an increasing trend with Cu defects. Regarding this variation of kL, they noticed weak temperature dependence but did not pay attention to the defect dependence. Yang et al. 7 reported the lattice thermal conductivity of Cu2-xSe (0 ≤ x ≤ 0.05). They stated that the kL values did not make a significant difference, unlike the ke values. However, their Cu-deficient samples have higher lattice thermal conductivity. Moreover, ref. 8 and ref. 2 showed kL values of Cu2Se1+x (0 ≤ x ≤ 0.02) and Cu2-xS (0 ≤ x ≤ 0.03; see their Supplementary Data), respectively, showing the most Cu deficient sample recorded the highest kL values, but they did not provide any explanation. It should be noted that because of the detailed synthesis methods used in the refs. 2,5,6,7,8 are not identical, the grain sizes and grain boundary characteristics in their samples may differ. This is why the precise kL values are not the same across studies.

However, since each study followed its respective synthesis procedure, the microstructure of each Cu2-x(S, Se) sample in each report can be considered consistent within its context. Thus, the trend of kL according to Cu defect can be regarded as a common characteristic of Cu2-x(S, Se). This experimental finding is not fully understood, suggesting that there are some other mechanisms affecting the lattice thermal conductivity.

In this study, we propose that anharmonicity may explain the abnormal increase in lattice thermal conductivity. To the best of our knowledge, this work is the first to clarify this mechanism.

To this end, we performed high-temperature neutron powder diffraction (ND) on Cu defect-controlled Cu2-xS-powder samples and conducted a careful structural analysis using the Rietveld method and the maximum entropy method (MEM). This experiment allowed us to clarify the detailed structural changes caused by Cu defects. Additionally, Cu K-edge X-ray absorption spectroscopy (XAS) was conducted to investigate the changes in Cu valence under Cu defects. Moreover, we used the results obtained from the MEM analysis to calculate the one-particle potentials (OPPs) of Cu and S atoms based on the Boltzmann distribution. The calculated OPPs were fitted to a theoretical potential function assuming cubic symmetry, providing evidence of the dependence of anharmonicity on the number of Cu defects.

Results and discussion

Rietveld analysis

The ND data obtained from the SuperHRPD and SPICA diffractometers exhibit sharp and well-defined diffraction peaks (see Supplementary Note 2), indicating that Cu-vacancy defects are evenly distributed throughout the global structure. The ND data were analyzed using the Rietveld method based on the space group \({Fm}\bar{3}m\) (No. 225) with a lattice constant of a ≈ 5.65 Å. While the S-atom is located at 4a (0, 0, 0) site having \(m\bar{3}m\) point symmetry, various models of the atomic sites have been proposed for Cu atoms, solely at either 8c (1/4, 1/4, 1/4) with \(\bar{4}3m\) or 32f (x, x, x; x ≈ 0.34) with 0.3m or with an additional site at 4b (1/2, 1/2, 1/2) with \(m\bar{3}m\)38,39,40,41 Preliminarily, we attempted Rietveld refinement using these models. However, they did not yield satisfactory results. Finally, we located the Cu atoms at two split sites 8c (Cu1) and 32f (Cu2), which represent the tetrahedral-anisotropic thermal vibrations of the Cu atom (Fig. 1a). This model provided an excellent fit (Supplementary Fig. S2d) and satisfactory results. Notably, according to ref. 42, this structural model was identified as the most suitable for Cu2-xSe as well.

Fig. 1: The crystal structure and some of the analysis results of Cu2-xS.
figure 1

a Crystal structure of Cu2-xS, where the Cu1 and Cu2 sites are not fully occupied, as illustrated by white portions in the structure. The compositional and temperature dependence of b lattice constant, c the ratio of NTot(Cu1) and NTot(Cu2), and d the volume of the [Cu2]4-tetrahedra. Here, NTot(Cu1) and NTot(Cu2) imply the total number of Cu atoms per chemical formula at the 8c (Cu1) and the 32 f (Cu2) site, respectively (see text). The error bars represent standard deviations obtained from Rietveld refinement.

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In the analysis, full occupancy was assumed for the S site, g(S) = 1, while the occupancies of the Cu sites, g(Cu1) and g(Cu2), were refined using the constraint Eq. 2*g(Cu1) + 8*g(Cu2) = 2 – x in Cu2-xS. Additionally, isotropic atomic displacement parameters were applied to all sites. The refined structure parameters for the SuperHRPD and SPICA data are listed in Supplementary Tables S2 and S3, respectively.

Figure 1b–d illustrates the results of the structural analysis. Figure 1b presents the temperature dependence of lattice constant a, showing a good agreement with previously reported values43,44 The lattice constant a decreases as the number of Cu defects increases. We did not observe a significant compositional dependence of the linear thermal expansion coefficients. The total number of Cu atoms at each 8c (Cu1) and 32f (Cu2) site, NTot(Cu1) and NTot(Cu2), was calculated based on the site occupancies per chemical formula unit as follows: NTot(Cu1) = 2*g(Cu1), and NTot(Cu2) = 8*g(Cu2) and NTot(Cu1) + NTot(Cu2) = 2 – x in Cu2-xS.

Figure 1c illustrates the ratio of NTot(Cu1) and NTot(Cu2): NTot(Cu2)/NTot(Cu1). Across all temperatures as Cu defect increases, NTot(Cu2)/NTot(Cu1) decreases, indicating that the Cu atoms become more centralized at Cu1 site. Interestingly, with increasing Cu defects, not only does the number of atoms at the Cu2 site decreases, but also the number of Cu atoms at the Cu1 site increases. This detailed comparison can be found in Supplementary Fig. S2e. Moreover, the Cu distribution can be quantitatively discussed by considering the volume of [Cu2]4-tetrahedra. As Cu defect increases from x = 0 to 0.16, the volume decreases significantly (–17.4% at 750 K). In contrast, the effective ionic radius estimated from Cu valence remained almost unchanged (–0.98%; see Supplementary Table S4). Therefore, the contraction of Cu distribution is not due to the change in the size of the Cu cation, but rather to changes in the potential surrounding the Cu atom. The isotropic atomic displacement parameters B(Cu1) and B(Cu2) had larger values than B(S), indicating that the S atom formed a rigid framework, whereas the Cu atom was loosely bound with large mobility (Supplementary Fig. S2g). The atomic displacement parameter values were found to have nearly constant regardless of the amount of Cu defect.

Maximum entropy method analysis

In the Rietveld analysis, the atomic thermal distribution, represented by the atomic displacement parameter, is based on the simple harmonic Gaussian model. However, the actual atomic distribution would be anharmonic affected by the surrounding environment. The precise thermal atomic distribution can be discussed using the statistical method MEM45,46.

The neutron scattering amplitude density distribution was calculated using the MEM after the Rietveld analysis of high statistical ND data obtained from SuperHRPD (RE < 1%). Figure 2a illustrates a representative density distribution (MEM density) of the Cu1.84S sample measured at 750 K. Although the MEM density was in good agreement with the structural model used in the Rietveld analysis, the detailed one-dimensional (1D) distribution along the <111> direction presented in Fig. 2b provides further insights: the Cu1 and Cu2 atoms were distributed continuously rather than discretely. Namely, the distribution of Cu is not on separate two sites but has a continuous tetrahedral feature, implying the existence of a deformed anharmonic potential surrounding the Cu atom. Additionally, the density at the 4b site (1/2, 1/2, 1/2) was negligible, indicating the absence of a Cu atom at this site. This continuous distribution shows that the actual Cu atoms vibrate around a single site centered on Cu1. This is consistent with the Cu trajectory simulations on Cu2-x(S, Se) by ref. 47, which identified the most frequent trajectories as rattling motions centered at the Cu1 site. It is interesting that such dynamic features appear in our time-averaged density distribution.

Fig. 2: MEM density distributions for Cu2-xS at 750 K.
figure 2

a The iso-surface of the MEM density distribution (see text) for Cu1.84S at 750 K (iso-surface level: 3 fm Å−3), b One-dimensional MEM density along <111> for Cu1.84S at 750 K, c One-dimensional MEM density along <111> around Cu atoms for Cu2-xS (x = 0.16, 0.08 and 0) at 750 K, and (d) 1D MEM density along <111> around S atoms for Cu2-xS (x = 0.16, 0.08, and 0) at 750 K.

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Figure 2c illustrates the Cu-defect dependence of the 1D MEM density distribution of the Cu atom. As the number of Cu defects increased, the density at the Cu1 site (x = 0.25) increased, while the density at the Cu2 site (x ≈ 0.34) decreased, showing a trend similar to the Cu-defect dependence of NTot(Cu2) observed in the Rietveld analysis. Figure 2d illustrates the Cu-defect-dependent MEM density of the S atom. The dependence of the S-density distribution on Cu defects was minimal, with only minor variations detected.

One-particle potential analysis

Using the Boltzmann distribution in Eq. (3), the probability distribution Pi of state i in a system at specific temperature T is determined by the energy of the state Ei.

$${P}_{i}=\frac{\exp \left[-\frac{{E}_{i}}{{k}_{{{{\rm{B}}}}}T}\right]}{Z},Z=\iint \!\!\!\int \exp \left[-\frac{{E}_{i}}{{k}_{{{{\rm{B}}}}}T}\right]{dx\; dy\; dz}\,$$
(3)

The MEM density distribution PMEM(x) can be applied to the Boltzmann distribution as follows:

$${P}_{{{{\rm{MEM}}}}}({{{\bf{x}}}})=\frac{\exp \left[\frac{-V({{{\bf{x}}}})}{{k}_{{{{\rm{B}}}}}T}\right]}{Z}$$
(4)

The potential surrounding an atom V(x), known as the OPP, can be calculated as follows48:

$$V\left({{{\bf{x}}}}\right)=-{k}_{{{{\rm{B}}}}}T{{\mathrm{ln}}}\frac{{P}_{{{{\rm{MEM}}}}}\,\left({{{\bf{x}}}}\right)}{{P}_{{{{\rm{MEM}}}}}\,\left({x}_{0}\right)}$$
(5)

where PMEM(x0) is the MEM density at the equilibrium position of x0; the potential must be adjusted to 0 at the equilibrium position of the atoms.

Figure 3a, b present the calculated OPP of the Cu and S atoms of Cu1.92S at 750 K, respectively. The figures clearly show that the shape of the potential of the S atom is rather harmonic, whereas that of the Cu atom is anharmonic. Figure 3c, d illustrates the Cu-defect dependence of the 1D potential along <111> for the Cu and S atoms, respectively. As the number of Cu defects increased, the Cu potential became steeper for x > 0.25, yet remained unchanged for x < 0.25. Notably, no significant dependence of the S potential on the Cu defects was observed.

Fig. 3: One-particle potentials at 750 K.
figure 3

a The iso-surface of the one-particle potential of Cu for Cu1.92S at 750 K (iso-surface level: 0.024 eV)), b The iso-surface of the one-particle potential of S for Cu1.92S at 750 K (iso-surface level: 0.06 eV)), c One-dimensional one-particle potential of Cu on <111>, and d 1D one-particle potential of S on <111>.

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The calculated OPP was fitted with a theoretical potential function48 which was expanded under cubic symmetry.

$$V= \, {V}_{0}+\frac{1}{2}\alpha \left({x}^{2}+{y}^{2}+{z}^{2}\right)+\beta \left({xyz}\right)+\gamma {\left({x}^{2}+{y}^{2}+{z}^{2}\right)}^{2}\\ \quad +\delta \left({x}^{4}+{y}^{4}+{z}^{4}-\frac{3}{5}{r}^{4}\right)+\ldots$$
(6)

The shape made by each term of Eq. (6) is illustrated in Supplementary Fig. S4.

The first-order constant term V0 was fixed at 0. The second-order term with the coefficient α represents the harmonic potential, which is the simplest spring model of the lattice vibration kBT = ½αu2, where u is the displacement from the equilibrium point. The third-order term with the coefficient β represents the anisotropic anharmonic potential. The shape of the equipotential level is a tetrahedral shape, and the plus or minus sign before β determines the direction of the tetrahedron toward [1, 1, 1] or [\(\bar{1},\bar{1},\bar{1}\)], respectively. Because the S (\(m\bar{3}m\)) site is centrosymmetric, β = 0. The fourth-order terms contain an isotropic γ term and anisotropic δ term. The δ term affects the potential in the direction of the cubic vertex and makes the spherical harmonic-potential distribution cube-like (when δ is +) or octahedron-like (when δ is –). It does not fit the fourth-order term owing to the tetrahedral symmetry of the Cu (\(\bar{4}3m\)) site.

Figure 4a plots the harmonic coefficients α of Cu (open square) and S (open circle). With increasing Cu defects, both α(S) and α(Cu) slightly increased; however, no clear temperature dependence was observed. Notably, α(S) was approximately three times larger than α(Cu). This indicates that the S atom was rigidly bound, while the Cu atom was loosely bound at their equilibrium positions. In Fig. 4b, the anisotropic anharmonic coefficients, |β| for Cu (open square, right axis) and δ for S (open circle, left axis), are plotted together. As the temperature increased, both |β| and δ decreased, yet they did not exhibit clear trends with respect to Cu defect variations. Because δ(S) < 0, the potentials of the S had an octahedral shape, which was flat toward the Cu atom (<111> direction). The fitting results of γ for the S-atom potential are illustrated in Fig. 4c; γ <0 at all temperatures and compositions, and |γ| increased as the number of Cu defects increased.

Fig. 4: Fitting results of the one-particle potentials with a theoretical potential function.
figure 4

The obtained coefficients in the theoretical potential function (Eq. 6) are shown as follows: a harmonic coefficients α(S) and α(Cu); b anisotropic anharmonic coefficients, including the 3rd-order coefficient |β|(Cu) and the fourth-order coefficient δ(S); c the isotropic anharmonic fourth-order coefficient γ(S); and d anharmonicity ratios |β/α|(Cu) and |δ/α|(S). The error bars represent standard deviations estimated from least squares fitting in Igor-Pro 8 software.

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Degree of anharmonicity

In the simplest model of atomic vibrations in a lattice, known as the Einstein model, atoms oscillate harmonically at their lattice points. Heat is transferred through the restoring force of atomic vibrations, making a larger harmonic coefficient α more effective for heat transfer. In this harmonic model, phonons do not scatter and have infinite lifetimes; however, anharmonic vibrations cause phonon scattering, resulting in finite lifetimes. Since anharmonicity and harmonicity have opposite effects on lattice thermal conductivity, discussing them together as a ratio is useful. The ratios |β/α|(Cu) and |δ/α|(S), the degrees of anharmonicity, are shown in Fig. 4d. These ratios are largest in Cu2S and decrease as the number of Cu defects increases, implying that phonon scattering from anharmonic vibrations is largest in Cu2S and decreases with increasing Cu defects.

This result successfully explains the increase in lattice thermal conductivity due to the increase in Cu defects. Since the degree of anharmonicity alone provides a successful explanation that overcomes the opposite effects from heat capacity and defect-phonon scattering, we conclude that anharmonicity plays a crucial role in explaining the low thermal conductivity in this system.

Furthermore, it is worth mentioning that the Gram-Charlier expansion49,50 could provide an alternative approach to describing anharmonic atomic vibrations. Future studies may benefit from considering this expansion to gain deeper insights into the anharmonic behavior of similar systems.

Our results demonstrate that even a small number of Cu defects (≥4%) can induce an abrupt reduction of the Cu atomic distribution toward the center site (Cu1 site). This rapid contraction of the Cu distribution is attributed to the change in the potential around the Cu atom. Consequently, the potential becomes more harmonic, and the anharmonicity rate is reduced. This, in turn, decreases phonon scattering, increasing the lattice thermal conductivity of Cu-deficient Cu2S. The observed change in anharmonicity with Cu defects strongly suggests a novel way to tune thermoelectric properties without external doping, pressure, or heat. However, to fully optimize anharmonic potential engineering, further investigation into which phonon mode efficiently contributes among the various phonon branches would be necessary. Therefore, additional theoretical and experimental studies focusing on the role of the anharmonic potential should be conducted on binary chalcogenide compounds.

Conclusions

We performed ND on Cu2-xS, which exhibited an anomalous increase in lattice thermal conductivity with increasing Cu defects. Rietveld analysis was conducted assuming a single S site and two Cu sites. The structural analysis results showed that as Cu defects increased, Cu occupancy at the Cu1 site increased, while occupancy at the Cu2 site decreased. Moreover, the Cu distribution estimated by the volume of [Cu2]4-tetrahedra rapidly decreased. These results indicate that the presence of Cu defect alters the Cu distribution to centralize. The MEM density distribution exhibited a similar trend. Moreover, the subsequent OPP analysis revealed that the degree of anharmonicity is highest in Cu2S and decreases with increasing Cu defects. This provides a direct explanation for the anomalous increase in lattice thermal conductivity with increasing Cu-defect in Cu2-x(S, Se), as reported in previous papers. Furthermore, in this system, anharmonicity appears to be the primary factor governing lattice thermal conductivity. This is because its effect outweighs the increase in lattice thermal conductivity resulting from heat capacity reduction and enhanced point-defect scattering caused by Cu defects. Our findings demonstrate that anharmonicity can be tuned via Cu defects and establish a new fundamental mechanism underlying the low thermal conductivity of Cu2-x(S, Se).

Methods

Sample preparation and X-ray diffraction

Cu2-xS (0 ≤ x ≤ 0.2) powder samples were synthesized using the liquid–solid reaction method. The Cu (granules, 99.9999%, Sigma-Aldrich, USA) and S (powder, 99.998%, Sigma-Aldrich, USA) raw materials were sealed in each quartz tube under vacuum conditions (2 \(\times\) 10−3 Pa), heated to 400 °C at a rate of 2 °C·min−1 and maintained at that temperature for 24 h. The samples were further melted to ensure compositional homogeneity; they were heated to 1140 °C at a rate of 3 °C·min−1 and maintained at that temperature for 2 h, followed by cooling at a rate of 1 °C·min−1. X-ray diffraction (XRD) measurements were performed to determine the room temperature phase of the synthesized Cu2-xS. XRD data were collected using Cu Kα1 radiation (SmartLab, Rigaku Co., Japan). Supplementary Note 5 presents the result of room temperature XRD.

Neutron powder diffraction

Time-of-flight ND measurements were conducted at high temperatures using two high-resolution powder diffractometers, SuperHRPD and SPICA, at the Materials and Life Science Experimental Facility, Japan Proton Accelerator Research Complex (MLF, J-PARC). Cu1.84S, Cu1.92S, and Cu2S were measured at 500–1000 K using the SuperHRPD diffractometer, and Cu1.84S and Cu1.88S were measured at 400–700 K using the SPICA diffractometer. The best resolutions for SuperHRPD and SPICA were 0.04 and 0.09%, respectively51,52.

As reported in ref. 53, S evaporation was observed at high temperatures (see Supplementary Note 6). To maintain the stoichiometric composition of Cu2-xS due to S evaporation during high-temperature measurements, pelletized powder samples were sealed in quartz capillaries under vacuum (2 \(\times\) 10−3 Pa) environments.

Rietveld analyses and the following MEM analyses were conducted using Z-Rietveld software46,54 The crystal structures of Cu2-xS were drawn using VESTA visualization software. The fitting analysis of the calculated OPPs was conducted using Igor-Pro 8.

X-ray absorption spectroscopy

Cu K-edge XAS was conducted at room temperature using the transmission yield mode at the 8C beamline of the Pohang Light Source-II (PLS-II), Republic of Korea.

Supplementary Fig. S3 illustrates the Cu K-edge XAS spectra of Cu1.8S, Cu1.88S, Cu2S, and Cu1.8S annealed at 700 °C, along with their derivative spectra (dI/dE). The spectrum of Cu metal is included for comparison. The peak positions of the derivative spectra were determined using Gaussian fitting. The peak energies of the Cu cations in Cu2-xS were higher than that of the Cu metal, indicating positive valences. Based on Zhou’s study55 of Cu2O (Cu+1), the excitation energy of Cu+1 was determined to be 8980.5 eV. The average Cu valence for each Cu2-xS sample was estimated by assuming a linear relationship between the chemical shift and valence. The dI/dE-peak positions and estimated valences of the Cu cations are summarized in Supplementary Table S4. The Cu valences of all tested Cu2-xS samples were not larger than that of Cu2O (+1). The relatively small oxidation numbers for Cu2-xS reflect the covalent nature of the Cu-S bonds; if the Cu-S bonding in Cu2-xS was purely ionic (without a covalent nature), the Cu valences in Cu2S and Cu1.8S would be +1 and +1.11, respectively, as the S anions would possess a fixed valence of –2. Because S has a lower electronegativity than O, the Cu-S bonds tend to exhibit a more covalent character than the Cu-O bonds. Thus, the small overall oxidation numbers of Cu2-xS are reasonable. Additionally, Cu1.8S annealed at 700 °C has the same valence as Cu2S. This indicates that the composition may have changed to that of Cu2S during annealing.