Phase transition of tetragonal copper sulfide ${\mathrm{Cu}}_2\mathrm{S}$ at low temperatures

The low-temperature behavior of tetragonal copper sulfide, ${\mathrm{Cu}}_2\mathrm{S}$, was investigated by powder and single-crystal x-ray diffraction, calorimetry, electrical resistance measurements, and ambient temperature optical absorption spectroscopy. The experiments were complemented by density-functional-theory-based calculations. High-quality, polycrystalline samples and single crystals of tetragonal copper sulfide were synthesized at 5 GPa and 700 K in a large volume multianvil press. Tetragonal ${\mathrm{Cu}}_2\mathrm{S}$ undergoes a temperature-induced phase transition to an orthorhombic structure at around 202 K with a hysteresis of $\pm 21$ K, an enthalpy of reaction of 1.3(2) $\mathrm{kJ}{\mathrm{mol}}^{-1}$, and an entropy of reaction of 6.5(2) $\mathrm{J}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$. The temperature dependence of the heat capacity at the transition temperature indicates that the transition from the tetragonal to the low-temperature polymorph is not a single process. The structure of the low-temperature polymorph at 100 K was solved in space group $Pna{2}_1$. The structure is based on a slightly distorted cubic close packing of sulfur with copper in threefold coordination similar to the structure of tetragonal copper sulfide. The electrical resistance changes several orders of magnitude at the transition following the temperature hysteresis. The activation energy of the conductivity for the tetragonal phase and the low-temperature polymorph are 0.15(2) and 0.22(1) eV, respectively. The direct band gap of the tetragonal polymorph is found to be 1.04(2) eV with the absorption spectrum following Urbach's law. The activation energies and the band gaps of both phases are discussed with respect to the results of the calculated electronic band structures.

Figure 1

Crystal structure of tetragonal copper sulfide based on single-crystal x-ray diffraction data [19]. Yellow: sulfur; blue: copper.

Figure 2

Powder x-ray diffraction pattern of tetragonal copper sulfide, ${\mathrm{Cu}}_2\mathrm{S}$, at ambient pressure and temperature and Rietveld refinement based on structural data published by Janosi [20] ($\lambda =1.5406$ Å). Black crosses: measured data; red line: simulated pattern; green line: background; blue line: residuals; blue tick marks: calculated reflection positions.

Figure 3

(a) Fit of the excess heat capacity at the phase transition on cooling. $A_1,B_1,C_1$: center of Gauss functions; black line: measured data; red line: Gauss functions; green line: calculated model; blue line: residuals; $R^2=0.9949$. (b) Fit of the excess heat capacity at the phase transition on heating. $A_2,B_2,C_2$: center of Gauss functions; black line: measured data; red line: Gauss functions; green line: calculated model; blue line: residuals; $R^2=0.9986$.

Figure 4

(a) Heat capacity of tetragonal copper sulfide as a function of temperature on cooling. Blue squares: measured data; black line: fit, sixth-order polynomial function. Error bars smaller than symbol size. (b) Linear fit of the heat capacity based on Eq. (3). The Debye temperature ${\theta }_D=205\left(2\right)$ K. Blue squares: measured data; red line: calculated model. Error bars smaller than symbol size.

Figure 5

Reconstructed detector data: reflection overlap of the three twin domains ${\mathrm{D}}_1, {\mathrm{D}}_2$, and ${\mathrm{D}}_3$. Software: crysalis${}^{\text{Pro}}$ [31].

Figure 6

Orientation of the three twin domains ${\mathrm{D}}_1$ (blue), ${\mathrm{D}}_2$ (green), and ${\mathrm{D}}_3$ (red) in reciprocal space.

Figure 7

Crystal structure of the low-temperature polymorph of tetragonal copper sulfide. Black labels on sulfur atoms indicate the slightly distorted cubic close packing of sulfur.

Figure 8

(a) Blue: diffraction pattern of a sample at 100(10) K. The green stars mark the five reflections published by Stolen et al. [22] for the low-temperature polymorph of tetragonal copper sulfide; black: diffraction pattern of a recovered sample at ambient conditions; red: calculated powder diffraction pattern for tetragonal copper sulfide at ambient conditions; blue tick marks: calculated reflection positions for tetragonal copper sulfide at ambient conditions. (b) Powder x-ray diffraction pattern of the orthorhombic low-temperature polymorph at 100(10) K and Rietveld refinement based on structural data established by single-crystal diffraction experiments. Black crosses: measured data; red line: simulated pattern; green line: background; blue line: residuals; blue tick marks: calculated reflection positions; green stars: reflections that cannot be explained by the structural model of the orthorhombic low-temperature polymorph.

Figure 9

Logarithm of the electrical resistance of copper sulfide samples as a function of temperature. Red squares: measured data on heating; blue triangles: measured data on cooling; green dots: data from Pakeva and Germanova [21]. Black lines and labels correspond to the transition temperatures determined by differential scanning calorimetry. Error bars smaller than symbol size.

Figure 10

Linear fit of the natural logarithm of the reciprocal electrical resistance as a function of reciprocal temperature. Using the slope of the linear function of seven and four different datasets an activation energy of the conductivity of 0.15(2) and of 0.22(1) eV was calculated for tetragonal copper sulfide (a) and the low-temperature polymorph (b), respectively. Red open squares: measured data; blue line: linear fit. Error bars smaller than symbol size.

Figure 11

Electronic band structures and integrated partial density of states (PDOS) of the (a) tetragonal (space group $P{4}_3{2}_{1}2$) and (b) low-temperature orthorhombic (space group $Pna{2}_1$) polymorphs of ${\mathrm{Cu}}_2\mathrm{S}$. The continuous blue, red, green, and black lines in the DOS represent the contributions of the Cu $s$, S $p$, Cu $d$, and sum, respectively.

Figure 12

(a) Optical absorption spectrum (black) of tetragonal ${\mathrm{Cu}}_2\mathrm{S}$ at ambient temperature fitted to a direct-type Urbach law [44] (red). Linear range presented by the absorption edge of tetragonal ${\mathrm{Cu}}_2\mathrm{S}$ considering either (b) a direct-type Urbach tail [ln($\alpha$)] or (c) an indirect (${\alpha }^{1/2}$). The points are experimental data while the straight red lines are guides for the eye.