Introduction

Materials that convert heat directly into electricity, called thermoelectrics, have been used to construct reliable generators of electrical power for spacecraft [1] and, under an applied current, as solid-state cooling devices [2]. Thermoelectric generators and coolers have the advantages of being silent, having no moving parts, and being free from greenhouse and ozone-depleting gases. The maximum efficiency of a thermoelectric material is governed by the figure-of-merit, zT, which is equal to S2Tρ−1κ−1, where S is the Seebeck coefficient, T is the absolute temperature, ρ is the electrical resistivity, and κ is the thermal conductivity. Thermoelectric materials in general must have low electrical resistivity, low thermal conductivity, and a controllable carrier concentration.

A new class of high-performance thermoelectric materials is the superionic coinage-metal chalcogenides. Some examples are Cu2−xSe [3], AgCrSe2 [4], Ag2+xSe [5–8], and Ag2Te [8]. These materials all undergo a structural phase transition above which the metal ions become mobile, which leads to enhanced phonon scattering and consequently low thermal conductivity. This combined with excellent electrical conductivity makes this class of materials promising for use as thermoelectrics.

Cu2-xSe was found to have a zT of 1.5 at 1,000 K [3]. A related compound is Cu1.97Ag0.03Se1+y [9], which was studied from the late 1960s through the late 1970s as a candidate for use in radioisotope thermal generators (RTGs) [10]. The y = 0 compound (Cu1.97Ag0.03Se), synthesized by 3 M, reached a zT of 1.1 at 870 K [10], which is comparable to the zT of 1.0 at 870 K achieved in Cu1.97Ag0.03Se in this work. Recently, the “overstoichiometric” composition Cu1.98Ag0.2Se was studied and found to reach a maximum zT of 0.52 at 650 K before the onset of bipolar conduction [11]. While Cu1.97Ag0.03Se1+y was tested under RTG conditions in order to evaluate its performance as an energy converter, no attempt was made to analyze its thermoelectric properties in order to optimize its zT. Furthermore, no such analysis has been done for Cu2−xSe. Therefore, we seek to analyze these materials together to determine the suitability of a single model for describing them and whether they can achieve greater zT values.

The data will be analyzed using an effective mass approach. The premise of this approach is that at a particular temperature, the intricacies of the band structure are reflected by two parameters, the effective mass and the mobility parameter, with the carrier concentration and chemical potential determined from the Hall effect and the Seebeck effect, respectively.

Experimental

Ingots of Cu2Se, Cu1.98Se, Cu1.97Ag0.03Se, and Cu1.97Ag0.03Se1.009 were formed by melting Cu (shot, 99.9999 % pure, Alfa Aesar, Puratronic), Ag (shot, 99.9999 % pure, Alfa Aesar, Puratronic), and Se (shot, 99.999 % pure, Alfa Aesar, Puratronic) in quartz ampoules evacuated to less than 6 × 10−5 torr. The ampoules were ramped to 1,373 K at 100 K/h, held at that temperature for 12 h, and then quenched. The ingots were ball-milled to form powders, then re-sealed in quartz ampoules, heated at 1,273 K for 5 days, cooled to 973 K, held at that temperature for 3 days and then quenched. The ingots thus obtained were ball-milled again and then hot-pressed at 973 K and 40 MPa for 6 h [12]. The geometric densities of the hot-pressed pellets were greater than 95 % of their theoretical values.

The thermoelectric properties of the samples were measured with custom-built and commercial apparatus. The thermal diffusivity α was measured with a Netzsch LFA 457 laser flash analysis unit. The total thermal conductivity was calculated from κ = αdC p , where d is the geometric density, and C p is the heat capacity. The Seebeck coefficient S was measured with a custom-built device [13] and with an ULVAC ZEM-3. The resistivity ρ and Hall resistance were measured with a custom-built system [14] using van der Pauw geometry and a 2-T magnetic field to determine the Hall carrier concentration n H  = 1/eRH and the Hall mobility μ H  = RH/ρ. C p was measured on a Netzsch 200F3 differential scanning calorimeter from 317 to 913 K.

Powder X-ray diffraction (PXRD) patterns were collected using a Rigaku SmartLab diffractometer equipped with a Cu Kα source, parallel beam optics, and a Rigaku D/tex detector. A hot-pressed pellet of Cu1.97Ag0.03Se was placed under dynamic vacuum on an Anton-Paar DHS-1100 hot stage with an X-ray transparent graphite dome. PXRD patterns were collected at 300 and 420 K and used for phase identification and refinement of the high-temperature phase using the FullProf suite [15]. Diffractograms were recorded every 20 K from 300 to 500 K in the 2Θ range from 10° to 100°. Between these measurements, the sample was heated at 1 K/min, while fast diffractograms were recorded from 24.5° to 54.5° 2Θ and a scan speed of 12°/min.

Results and discussion

Powder X-ray diffraction shows that Cu1.97Ag0.03Se1+y is not single phase (Fig. 1); rather it is composed of Cu2-xAg x Se, CuAgSe [16], and at least one more unidentified impurity phase, i.e., some of the Ag enters the Cu2Se matrix and some forms CuAgSe (which has also been evaluated as a thermoelectric material and found to have low zT values) [17]. In Cu1.97Ag0.03Se, the CuAgSe phase is observed to dissolve at about 380 K, slightly before the superionic phase transition just above 400 K. At 420 K, all peaks can be indexed and refined in the high-temperature Cu2Se structure (antifluorite, space group \(Fm\bar{3}m\)) except for a few, low-intensity peaks from one or more unidentified impurities. The main phase peaks are satisfactorily described by the antifluorite structure when Cu interstitials are incorporated on the octahedral sites and at trigonal planar sites. The atomic positions were stable when refined. The room-temperature Cu2Se structure is not known but comparison with phase pure Cu2Se PXRD patterns [18] reveals another set of peaks not belonging to the impurity at high temperatures, CuAgSe, or pure Cu2Se. These peaks disappear at the phase transition and can hence either be an impurity, which dissolves, or belong to the main phase if this has a slightly different structure than pure Cu2Se.

Fig. 1
figure 1

Temperature resolved PXRD of Cu1.97Ag0.03Se from 300 to 500 K. On the bottom, arrows mark CuAgSe peaks, while hat symbol marks impurities that dissolve at the phase transition. Asterisk marks the stable impurity peaks. Unmarked peaks on the bottom have corresponding peaks in the low-temperature structure of Cu2Se

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In this study, the compositions Cu2Se, Cu1.98Se, Cu1.97Ag0.03Se, and Cu1.97Ag0.03Se1.009 were synthesized, and data on Cu1.98Ag0.2Se from a recent publication by Ballikaya et al. [11] are included for a more complete analysis.

The thermoelectric properties of a material depend strongly on the Hall carrier concentration, n H , whose magnitude can vary with temperature and via chemical doping. In the aforementioned materials, n H can be decreased by substituting Ag for Cu, or increased by adding Se to create Cu+ vacancies. Each additional Se2− ion is equivalent to 2 Cu+ vacancies; each Cu+ vacancy donates one hole to the valence band. Furthermore, previous work on the band gap of Cu2Se [19] shows that the valence and conduction bands are separated by a band gap much greater than k B T, so that only one type of carrier is present. The Hall carrier concentration for all samples is shown in Fig. 2a. Above the phase transition, the Hall carrier concentration of Cu1.97Ag0.03Se1.009 is greater than that of Cu1.97Ag0.03Se due to the greater deficiency of metal ions. Likewise, the Hall carrier concentration of Cu1.98Se is greater than that of Cu2Se. This could be because substitution of Ag for Cu alters the native vacancy concentration of Cu2Se. The Hall carrier concentration of Cu1.98Ag0.2Se is at least an order of magnitude less (~1019 cm−3) than the other compositions (~1020–1021 cm−3) due to the excess of metal ions [11]. Between 750 and 800 K, the Hall carrier concentrations of Cu2−xSe and Cu1.97Ag0.03Se1+y dramatically increase. The exponential character of this increase seems to indicate bipolar conduction. However, the concomitant decreases in Seebeck and resistivity are not observed. While Hall carrier concentration data for Cu1.98Ag0.2Se were not available above 575 K in Ballikaya et al. [11], the influence of a conduction band separated from the valence band by a band gap of order k B T is corroborated by the decrease in the Seebeck coefficient of that composition at 725 K, as shown in Fig. 2b.

Fig. 2
figure 2

Transport properties as functions of temperature of Cu2−xSe, Cu1.97Ag0.03Se1+y and literature data on Cu1.98Ag0.2Se from Ballikaya et al. [11]. Hall carrier concentration n H , Seebeck coefficient S, resistivity ρ, Hall mobility μ H , total thermal conductivity, and figure-of-merit zT are shown in a, b, c, d, e, and f, respectively. The resistivity of Cu1.98Ag0.2Se is not shown because it is several times greater than that of the other compositions. Above the phase transition and up to 780 K, Cu1.97Ag0.03Se achieves higher zT values (f) than do Cu2Se and Cu1.98Se because its Hall carrier concentration is closer to the optimum value (Fig. 3). Cu1.98Ag0.2Se is under-doped compared with both compositions of Cu1.97Ag0.03Se1+y, as indicated by its greater Seebeck coefficient (a) and lower zT values at all temperatures

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Cu2-xSe and Cu1.97Ag0.03Se1+y exhibit the steady increase with temperature of the Seebeck coefficient (Fig. 2b) and of the resistivity ρ (Fig. 2c) expected of a single-carrier semiconductor. Cu1.98Ag0.2Se has greater values of S and ρ than do the other samples in the entire temperature range due to its much lower Hall carrier concentration, and it shows a peak in S at about 725 K.

The Hall mobilities, μ H , of Cu2−xSe, Cu1.97Ag0.03Se1+y, and Cu1.98Ag0.2Se above the phase transition are low compared with those of other high-performance p-type thermoelectric materials, such as 40–6 cm2 V−1 s−1 in Na-doped PbTe between 600 and 750 K, depending on carrier concentration [20]. The Hall mobility scales with T−p [21]. The average value of p taken from the data shown in Fig. 2d and above the phase transition is about 3.1. Values of p between 1 and 1.5 usually indicate that acoustic phonons limit electron mobility in the material [21], while values greater than 1.5 indicate a temperature-dependent effective mass [22].

The thermal conductivity data are shown in Fig. 2e. The sudden increase in κ around the phase transition temperature is due to the sharp peak in C p [18]. Cu1.98Ag0.2Se has the lowest thermal conductivity values because it has the lowest lattice thermal conductivity, presumably due to disorder caused by the greater amount of Ag, and because it has the lowest carrier concentration of the compositions studied and therefore the lowest electronic thermal conductivity.

The zT data are shown in Fig. 2f. The zT values of Cu2−xSe and Cu1.97Ag0.03Se1+y all increase continuously from the phase transition temperature to the maximum temperature at which they were measured. The Cu1.97Ag0.03Se sample reaches a zT of 1.0 at 870 K. Cu2Se reaches a zT of 1.16 at 870 K, but between 450 and 780 K has an average zT of 0.59, whereas Cu1.97Ag0.03Se has an average zT of 0.66 in the same temperature range. Above 780 K, the increasing values of ρ in Cu1.97Ag0.03Se and the decreasing values of κ in Cu2Se mean that Cu2Se has a greater zT. Cu1.98Ag0.2Se reaches a peak zT of 0.52 at 650 K and then decreases due to bipolar conduction.

To understand why Cu1.97Ag0.03Se achieves a greater zT up to 780 K than do the other samples, we analyze n H and the effective mass m*. The carrier mobility in this model is limited by acoustic phonon scattering, and the effective mass is treated as a constant at each temperature. The results of our analysis are shown in Fig. 3 and Table 1.

Fig. 3
figure 3

Analysis of the effective mass and Hall carrier concentration explains and predicts the optimization of Cu1.97Ag0.03Se for thermoelectric use. a The Seebeck coefficient as a function of Hall carrier concentration with the effective mass as a fitting parameter. b The Hall mobility as a function of Hall carrier concentration with μ0 as a fitting parameter. The lattice thermal conductivity (c) was computed from the resistivity and the Lorenz number L. The optimum Hall carrier concentration (d) increases with temperature. The Hall carrier concentration of Cu1.97Ag0.03Se also increases with temperature, so it has a Hall carrier concentration close to the optimum value up to 750 K. The lines in c are average values of κ L at the indicated temperature

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Table 1 Thermoelectric material properties estimated from the effective mass model
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We estimated the effective mass m* at 575 K (the highest temperature for which R H data were available for Cu1.98Ag0.2Se) and 750 K (the lowest temperature at which none of the samples exhibit a sharp increase in Hall carrier concentration) by using m* as a fitting parameter to fit a theoretical curve to S versus n H data (Fig. 3a). The theoretical dependence of S on the dimensionless Fermi level η is given by Eq. (1). The following equations are valid if the electron mobility is limited by acoustic phonon scattering, as per the discussion of our Hall mobility data above. k B is Boltzmann’s constant, e is the elementary charge, and F j (η) is the Fermi integral of order j, given by Eq. (2). ε is the electronic energy level normalized by k B T.

$$S = \frac{{k_{B} }}{e}\left( {\frac{{2F_{1} \left( \eta \right)}}{{F_{0} \left( \eta \right)}} - \eta } \right)$$
(1)
$$F_{j} \left( \eta \right) = \int\limits_{0}^{\infty } {\frac{{\varepsilon^{j} d\varepsilon }}{{1 + \exp \left( {\varepsilon - \eta } \right)}}}$$
(2)

η is set by the Hall carrier concentration and the effective mass m*. The relationship between these variables is given by Eq. (3).

$$n_{H} = 4\pi \left( {\frac{{2m^*k_{B} T}}{{h^{2} }}} \right)^{3/2} F_{1/2} \left( \eta \right)$$
(3)

At a fixed temperature, we compute η from S for each sample and then use m* to compute a theoretical n H that we fit to the n H data. The effective mass increases with temperature (Table 1), which we predicted based on the Hall mobility data. The same trends of effective mass, Hall carrier concentration, resistivity, and Seebeck coefficient with temperature were observed by Voskanyan et al. [23], though they estimated different values of the effective mass, e.g., 2.2 m e at 750 K as opposed to 6.2 m e at 750 K because they used assumed values of n H instead of calculating them from R H . Voskanyan et al. proposed a second valence band as a possible cause of the increasing effective mass. While this may explain of the trend of m* with T, a two-band model is much more complex than a single-band model, requires more assumptions, and does not guarantee a unique solution. Here, we use a single band in this analysis in order to estimate the maximum achievable zT and optimum Hall carrier concentration in this material. We must emphasize that because the effective mass is not constant with temperature, our predictions are valid only at fixed temperatures as a function of Hall carrier concentration.

The estimated values of μ0 (Table 1) fit to the data shown in Fig. 3b, decrease with temperature, as expected from the raw Hall mobility measurements. The Hall mobility is given by Eq. (4). μ0 is the mobility of a single electron in the material. The Fermi integral term accounts for energy level degeneracy and scattering. The effective mass m* and the n H data are used to compute η, which is then used with the μ H data to fit μ0.

$$\mu_{H} = \mu_{0} \frac{{F_{ - 1/2} \left( \eta \right)}}{{2F_{0} \left( \eta \right)}}$$
(4)

Cu1.97Ag0.03Se1.009 has a greater Hall mobility than does Cu1.97Ag0.03Se, despite having a greater carrier concentration and more defects.

The thermal conductivity is made up of a lattice contribution κ L and an electronic contribution κ E . κ E is equal to LT/ρ, where L is the Lorenz number, given by Eq. (5). We estimated the Lorenz number of each sample at each temperature studied using the previously determined Fermi level η. The Lorenz numbers of the samples were between 1.5 × 10−8 and 1.9 × 10−8 V2 K−2 at 575 and 750 K, respectively.

$$L = \left( \frac{k}{e} \right)^{2} \frac{{3F_{0} \left( \eta \right)F_{2} \left( \eta \right) - 4F_{1} \left( \eta \right)^{2} }}{{F_{0} \left( \eta \right)^{2} }}$$
(5)

κ L of each composition is shown in Fig. 3c, along with the average κ L at each temperature, the values of which are shown in Table 1. κ L does not change significantly from 575 to 750 K; therefore, the optimization of zT in this materials system will hinge only on the electrical transport properties. The slight increase in κ L with temperature in Table 1 is due to uncertainty in the calculated κ E . Taking the estimates for m*, μ0, and κ L , we can calculate a zT versus Hall carrier concentration curve to determine the maximum zT at a given temperature and the optimum Hall carrier concentration (Fig. 3d). Looking at the zT versus Hall carrier concentration curve for 575 K, it is clear that Cu1.98Ag0.2Se is under-doped, leading to a decrease in zT above 650 K due to bipolar conduction. Cu1.97Ag0.03Se has the Hall carrier concentration closest to the optimum value at every temperature at which we calculated an effective mass, which explains why that composition has the greatest zT of all the compositions included in the analysis.

According to our model, a maximum zT of 1.0 at 750 K is possible in this material system. The dimensionless quality factor B [24–26] (Eq. 6) is a measure of the maximum zT at a given temperature and depends only on material properties and temperature.

$$B = \frac{{k_{B}^{7/2} }}{{2^{1/2} \pi^{3/2} \hbar^{3} e}}\frac{{\mu_{{\mathbf{0}}} \left( {m^{*} } \right)^{3/2} T^{5/2} }}{{\kappa_{L} }}$$
(6)

The quality factors calculated for this material system increase with temperature (Table 1), meaning the theoretical maximum zT also increases with temperature. According to Fig. 3d, the optimum Hall carrier concentration n H,opt increases with temperature as well, which combined with the increasing trend with temperature of n H in Cu1.97Ag0.03Se means that that composition has a Hall carrier concentration close to nH,opt at and below 750 K.

Conclusions

Our transport property measurements show that below 780 K Cu1.97Ag0.03Se1+y has superior zT values compared with Cu2−xSe because it has a Hall carrier concentration closer to the optimum value. We have analyzed the Hall carrier concentration and effective mass in these materials and in recently published data on Cu1.98Ag0.2Se for a more complete analysis, and found that these materials together follow the trends expected despite the complexity of the atomic structure and presumed complexity of the electronic structure. This model predicts that a maximum zT of 1.0 at 750 K is possible in this material system.

Each zT versus Hall carrier concentration curve in Fig. 3 indicates that zT of Cu2Se can be increased simply by reducing the Hall carrier concentration. This means that if Cu2Se is to be a commercially viable thermoelectric material, the addition of expensive Ag may be unnecessary. Any means of removing charge carriers from the material could improve its thermoelectric properties above the phase transition. Such means could include anion substitution [27, 28] or doping with divalent cations [17]. Furthermore, our analysis is based only on electronic parameters, so separate optimization of the lattice thermal conductivity may improve the zT of this material even further.