α-Ag2S: A Ductile Thermoelectric Material with High ZT.

Using first-principles calculation and Boltzmann electron/phonon transport theory, we present an accurate theoretical prediction of thermoelectric properties of the α-Ag2S crystal, a ductile inorganic semiconductor reported experimentally [Nat. Mater. 2018, 17, 421]. The semiconductor α-Ag2S has ultralow thermal conductivity associated with high anisotropy, which can be attributed to the complex crystalline structure and weak bonding. The optimal values of the Seebeck coefficient are 0.27 × 10-3 V/K for n-type and 0.21 × 10-3 V/K for p-type α-Ag2S, respectively, which are comparable to those of many promising thermoelectric materials. As a consequence, a maximum ZT value of 0.97/1.12 can be realized for p-type/n-type α-Ag2S at room temperature. More interestingly, the value of ZT can be further enhanced to 1.65 at room temperature by applying 5% compressive strain. Moreover, we find that the electronic thermal conductivity is a major factor limiting the ZT, which is several times the lattice thermal conductivity for n-type α-Ag2S. Our work demonstrates the great advantage of the α-Ag2S crystal as a ductile thermoelectric material and sparks new routes to improve its figure of merit.

Introduction

Thermoelectric (TE) materials, which enable the harvest of waste heat and direct conversion into electricity, provide an effective way for the solution of global energy crisis without producing new contaminants.[1−3] The energy conversion efficiency (η) of TE materials is evaluated by the ratio of output electrical power (P) to the input thermal power (Q) supplied[4]where TH and TC are temperatures of the hot end and cold end, respectively. The term ZT is a dimensionless figure of merit, which is defined as[4]where S, σ, T, κp, and κe are the Seebeck coefficient, electrical conductivity, absolute temperature, lattice thermal conductivity, and electronic thermal conductivity, respectively. Obviously, the materials with high ZT will lead to the high efficiency η. For practical application, a ZT of 1 at room temperature can have about 10% of efficiency with a temperature difference of 200 K, and an increased ZT can realize higher efficiency.[1−4]

Since 1950s, great research efforts have been made to improve the ZT of TE materials. In search of high-efficient TE materials, low thermal conductivity is essential and critical. Benefited from the development of nanofabrication technology[5] and phonon engineering,[6−8] lattice thermal conductivity can be significantly reduced, which makes some nanostructured semiconductors, such as silicon nanowires, promising in TE application, although their bulk counterparts are not considered as ideal candidates.[9−11] On the other hand, manipulating electronic band structures also promotes the power factor (PF = S2σ), partially because of the favorable effect of band convergence.[12−14] Moreover, novel theories emerged,[15,16] including topological surface states,[17,18] the wave nature of phonons,[19,20] nanophononic metamaterials,[21] rattling effect,[22] topological phonon states,[23−25] and magnetic molecular materials.[26] In recent years, “phonon-liquid electron-crystal” (PLEC) materials are suggested as promising TE materials because of their liquid-like ultralow thermal conductivity and crystal-like high electrical conductivity.[27−32] All of these efforts together consequently enhance ZT significantly since the beginning of this century.[33−36]

Despite past progress in TE materials, they are inorganic semiconductors, which are usually brittle. This hinders the application of inorganic TE materials in a flexible environment. Recently, the α-Ag2S crystal was reported to be a ductile inorganic semiconductor, with a large compression strain > 50%.[37] The ductility mechanism has been deeply discussed from chemical bonding analysis.[38] In addition, at present, most of the commercially available TE materials contain rare and toxic elements; therefore, Ag2S is very important because it is negligibly toxic and naturally abundant. Recently, Wang et al.[39] experimentally measured the TE properties of Ag2S and found that Ag2S has very low thermal conductivity. With the increase of temperature, a phase transition occurs, resulting in a significant improvement in TE properties. In addition, Wang et al.[40] also studied the TE properties of Ag4SSe experimentally and found that Ag4SSe is a promising TE material at the moderate temperature range. However, the systematic theoretical studies on TE properties of the α-Ag2S crystal have not been reported. Especially the effects of stress on TE properties of the α-Ag2S crystal are still open questions. In this work, using first-principles calculations and Boltzmann transport theory, we systematically investigated the TE transport properties of the α-Ag2S crystal. Ultralow lattice thermal conductivity is demonstrated. Combined with the extremely large power factor originating from the high Seebeck coefficient, high TE figure of merit ZT is realized, which can be further enhanced through compressive loading. The physical mechanism underlying the outstanding TE performance is discussed.

Theoretical Methods

All the first-principles calculations are performed using density functional theory as implemented in the widely used Vienna ab initio simulation package (VASP).[41] The Perdew–Burke–Ernzerhof form of the generalized gradient approximation is used for the exchange–correlation functional.[42] The crystal structure is relaxed with a total energy convergence criterion of 10–6 eV, and the force convergence criterion is 10–4 eV/Å. The kinetic energy cutoff for the plane-wave basis is chosen as 500 eV, and the Monkhorst–Pack k mesh is 5 × 5 × 5. The phonon spectrum is obtained by diagonalizing the dynamical matrix, which is constructed from the second-order interatomic force constants using the Phonopy code.[43] The second-order and third-order interatomic force constants are calculated using a 3 × 3 × 3 supercell with 2 × 2 × 2 Monkhorst–Pack k meshes and a 2 × 2 × 2 supercell with Γ point, respectively. The lattice thermal conductivity is calculated by solving the phonon Boltzmann transport equation (BTE) via the ShengBTE package,[44] and a fine 15 × 15 × 15 q-mesh is adopted to ensure the convergence. The selection of q-mesh is large enough to obtain converged values of thermal conductivity, as shown in Figure S1a in the Supporting Information.

The electronic transport coefficients are calculated by solving the BTEs with the relaxation time approximation (RTA) as implemented in the BoltzTraP code.[45] The electrical conductivity and Seebeck coefficient are calculated fromwhere f0 is the Fermi–Dirac distribution function and Ξ(ε) is the transport distribution function with Ξα,β(ε) = ∑δ(ε – ε)υαυβτ, in which υα is the αth component of the group velocity with wave vector k, τ is the relaxation time of the carrier with wave vector k, and ε is the energy of electrons or holes. For obtaining the accurate group velocities and the transport coefficient calculations to guarantee convergence, we use a more dense k-point Monkhorst–Pack mesh with 21 × 21 × 21. The relaxation time is calculated by deformation potential (DP) theory based on effective mass approximation.[46] For three-dimensional systems, the relaxation time is calculated as , where C3D is the elastic modulus and can be determined by , where E is the DP constant, which is calculated as , where ∂Eedge is the energy change at conduction band minimum (CBM) and valence band maximum (VBM) for electrons and holes, respectively, and m* is the effective mass tensor and can be calculated as , where ℏ is the reduced Planck constant and kB is the Boltzmann constant. C3D can be obtained through fitting the energy–strain curves, and the DP constant can be calculated by fitting the slope of the band edges as a function of strain. The calculated elastic modulus, DP constant, effective mass, and relaxation time are shown in Table . Once the lattice thermal conductivity, electrical conductivity, Seebeck coefficient, and electronic thermal conductivity are obtained, we can use eq to calculate ZT.

Table 1

Calculated Elastic Modulus, DP Constant, Effective Mass, and Relaxation Time of p- and n-Type Doped α-Ag2S along Different Directions at 300 K

		elastic modulus (GPa)	effective mass (me)	DP constant (eV)	relaxation time (ps)
xx	p	39.53	–1.19	5.21	11.7
	n		0.21	3.56	383.0
yy	p	27.41	–1.61	4.55	6.8
	n		0.20	6.25	91.4
zz	p	94.17	–1.48	4.90	22.8
	n		0.47	4.46	170.4

Results and Discussion

Ultralow Lattice Thermal Conductivity

The α-Ag2S crystal possesses a monoclinic structure (space group P21/c (14)) with eight Ag atoms and four sulfur atoms per unit cell at room temperature. The X–Y, X–Z, and Y–Z planes of the optimized α-Ag2S crystal structure are shown in Figure a–c, respectively. The optimized lattice constants a, b, and c are 4.40, 7.89, and 9.23 Å, and the angles between primitive vectors, α, β, and γ, are 90, 122.68, and 90°, respectively. Based on the optimized structure, we calculate the electronic band structure of α-Ag2S as shown in Figure d, which indicates that bulk α-Ag2S is an indirect band gap semiconductor. The band gap is 0.92 eV, in good agreement with the previously reported value 0.9 ± 0.1 eV.[37] In addition, the VBM is at the Brillouin zone center, and the CBM is very close to the Brillouin zone center. Therefore, α-Ag2S exhibits optical and electrical features similar to a direct band gap material. Similar calculation results have been found in previous studies.[47,48] With strain, the electronic band structure and total energy will change. The shifts of VBM, CBM, and total energy as a function of strain are shown in Figure , which can be used to calculate the elastic modulus and DP constant. The phonon spectrum of α-Ag2S is also calculated by first-principles calculation, as shown in Figure e. There is no imaginary frequency, confirming that the structure is dynamically stable. Moreover, it can be seen from the phonon spectrum that a lot of low-frequency optical branches mix with the acoustic branches along Γ–X, which implies that α-Ag2S may have a lower thermal conductivity along the X-direction (details will be discussed later). In contrast, the low-frequency optical branches along Γ–Y and Γ–Z directions have a large slope, which demonstrates that the optical branches may have important contribution to thermal conductivity in Y and Z directions.

Figure 1

(a) X–Y, (b) X–Z, and (c) Y–Z planes of the optimized α-Ag2S crystal structure. (d) Energy band structure and (e) phonon dispersion spectrum of α-Ag2S.

Figure 2

Shift of band edges as a function of strain of α-Ag2S along (a) X, (b) Y, and (c) Z directions, respectively. (d) Total energy–strain curves of α-Ag2S along different directions.

α-Ag2S is a layered structure, which is stacked by a zigzag-shaped Ag–S framework. The anisotropic nature of the structure will lead to direction-dependent physical properties, as observed in black phosphorene.[49] The lattice thermal conductivity of α-Ag2S along different directions is illustrated in Figure a. Because the α-Ag2S crystal is stable up to 451 K, above this temperature, the α-Ag2S crystal will be transformed into β-Ag2S, which has a body-centered cubic structure.[37] Therefore, we only show the thermal conductivity of α-Ag2S at temperatures below 400 K. With an increase in temperature, it is clear that the lattice thermal conductivity of α-Ag2S along all the three directions decreases following a T (α ≈ 1) dependence, which is due to the enhanced phonon–phonon anharmonic scattering. The fitting information is shown in Figure S2 of the Supporting Information. Moreover, the lattice thermal conductivity of α-Ag2S shows obvious anisotropic characteristic. The lattice thermal conductivity of α-Ag2S along three directions is in the order of κ > κ > κ within the considered range of temperature. At room temperature, κ is 0.13 W m–1 K–1, κ is 0.29 W m–1 K–1, and κ is 0.33 W m–1 K–1. α-Ag2S has ultralow thermal conductivity along the X-direction, less than half of those in Y and Z directions, indicating a large anisotropic thermal transport. This is because the zigzag-shaped Ag–S framework is closely connected with the Ag–S bonds in the Y–Z plane. However, in the X-direction, the zigzag-shaped Ag–S framework layers are stacked together by a relatively weak interaction, resulting in ultralow thermal conductivity. These results imply that α-Ag2S may have the potential for high TE efficiency.

Figure 3

(a) Lattice thermal conductivity of α-Ag2S along X, Y, and Z directions as a function of temperature. (b) Cumulative lattice thermal conductivity of α-Ag2S as a function of phonon frequency at room temperature.

In calculation of phonon thermal conductivity by solving BTE, RTA is usually adopted, in which each phonon mode relaxes independently to equilibrium. In low-dimensional materials, RTA of the BTE may underpredict the thermal conductivity[50,51] because the collective phonon excitations are starting to be responsible for heat transport.[52,53] Unlike RTA, the exact solution of the BTE with mode-dependent phonon–phonon interactions can provide the correct thermal conductivity in nanomaterials. We performed thermal conductivity calculation using both RTA and full iterative solution of the linearized BTE and found that the thermal conductivity predicted from these two methods is identical, as shown in Figure S1b. This reveals that the collective phonon excitation is trivial in the α-Ag2S crystal, and phonon–phonon Umklapp scattering is dominated in the scattering process, consistent with the T dependence as shown in Figure S2. In the following, the results of lattice thermal conductivity are obtained from RTA calculation.

In order to further understand the origin of the anisotropy in the thermal conductivity of α-Ag2S, we calculate the cumulative lattice thermal conductivity as a function of phonon frequency at room temperature, as shown in Figure b. For κ, we can find that the low-frequency acoustic phonons dominate heat transport. Moreover, from Figure e, the boundary frequency of acoustic phonons along Γ–X is 1.60 THz, which coincides with the cumulative lattice thermal conductivity of κ as a function of phonon frequency. For κ and κ, the boundary frequencies of acoustic phonons are 0.85 THz and 1.12 THz, respectively, as shown in Figure e. However, from Figure b, it can be seen that the phonons at frequencies ranging from 1.12 to 3.79 THz also contribute significantly to thermal conductivity. This means that the low-frequency optical phonons also play an important role in thermal transport along Y and Z directions because the low-frequency optical phonons along Γ–Y and Γ–Z have large group velocity, as shown in Figure e. Moreover, among these three different directions, the plateau region at frequencies ranging from 3.79 to 6.50 THz in Figure b corresponds to the band gap of the phonon spectrum in Figure e, and the high-frequency optical phonons at frequencies ranging from 6.50 to 9.38 THz also have non-negligible contributions.

High TE Figure of Merit ZT

We are now in a position to explore the TE transport properties of α-Ag2S by using the Boltzmann transport theory combined with DP theory. The electrical conductivity, Seebeck coefficient, power factor, electronic thermal conductivity, and ZT of α-Ag2S along different directions as a function of the chemical potential at room temperature are shown in Figure . From Figure a, we can see that the electrical conductivity increases with increasing chemical potential for both p- and n-type α-Ag2S. Here, the p- and n-type samples correspond to the negative and positive chemical potential, respectively. The electrical conductivity for n-type α-Ag2S is much larger than that for the p-type counterpart, which is originated from the longer relaxation time of electrons than holes, as shown in Table . In addition, the electrical conductivity of p-type α-Ag2S along the Z-direction is larger than that along the other two directions. However, for n-type α-Ag2S, the maximum electrical conductivity is arrived along the X-direction. These results are consistent with the calculated relaxation time of carriers, as shown in Table . Compared with the obvious orientation-dependent electrical conductivity, the Seebeck coefficient only has slight orientation dependence, as shown in Figure b. Moreover, the optimal values of the Seebeck coefficient are 0.27 × 10–3 V/K for n-type and 0.21 × 10–3 V/K for p-type samples, as shown in Table , respectively, which are comparable to those of many promising TE materials. For example, the theoretical optimal value of the Seebeck coefficient is about 0.2 × 10–3 V/K for p-type NbFeSb,[54] 0.6 × 10–3 V/K for bulk SnSe,[55] and 0.24 × 10–3 V/K for bulk Bi2Te3.[56] In order to offer the most fair comparison with other materials, here we only select theoretical values from calculations with the similar level of accuracy. Recently, Wang et al.[40] found that the theoretical optimal carrier concentration for the maximal ZT of Ag4SeS is about 2.0 × 1018 cm–3, and the corresponding Seebeck coefficient is about 0.2 × 10–3 V/K, which is consistent with our calculated value.

Figure 4

(a) Electrical conductivity, (b) Seebeck coefficient, (c) power factor, (d) electronic thermal conductivity, and (e) ZT of α-Ag2S as a function of the chemical potential at 300 K.

Table 2

Optimal Chemical Potential, Carrier Concentration, Electrical Conductivity, Seebeck Coefficient, Power Factor, Electronic Thermal Conductivity, Lattice Thermal Conductivity, and ZT of p- and n-Type α-Ag2S along the Z-Direction at 300 K

	μ (eV)	n (1019 cm–3)	σ (106 S/m)	S (10–3 V/K)	PF (10–3 W/m K2)	κe (W/m K)	κp (W/m K)	ZT
P	–0.476	2.07	0.042	0.207	1.796	0.227	0.328	0.97
N	0.429	0.77	0.065	0.273	4.826	0.963	0.328	1.12

The power factor, which is the product of the electrical conductivity and square of Seebeck coefficient, has a higher peak value for n-type than that for the p-type counterpart, as shown in Figure c. For n-type α-Ag2S, the maximum power factor appears along the X-direction, with a value of 18.13 × 10–3 W/m K2. On the other hand, for p-type α-Ag2S, the maximum power factor appears along the Z-direction, only about 1.95 × 10–3 W/m K2. For comparison, the maximum power factor (theoretical value) of p-type NbFeSb is about 11 × 10–3 W/m K2 at room temperature,[54] and the maximum power factor (theoretical value) of bulk Bi2Te3 is about 6 × 10–3 W/m K2 at room temperature;[56] both of them are regarded as a good TE material with high power factor.

As shown in Figure d, the trend of electronic thermal conductivity is similar to that of the electrical conductivity because the charge carriers are also heat carriers. Based on the obtained power factor, electronic thermal conductivity, and phonon thermal conductivity, we calculate the ZT of α-Ag2S along three directions as a function of the chemical potential at room temperature, as shown in Figure e. For p-type α-Ag2S, the maximum ZT along X, Y, and Z directions is 0.32, 0.11, and 0.97, respectively. The ZT along the Z-direction is much higher than that along the other two directions because of the much higher power factor. For n-type α-Ag2S, the maximum ZT can reach 1.12 along the Z-direction, which is also higher than that along X and Y directions. Although the power factor at the chemical potential corresponding to the maximum ZT along the Z-direction is lower than that along the X-direction (4.826 × 10–3 W/m K2 vs 6.417 × 10–3 W/m K2), the electronic thermal conductivity along the Z-direction (0.96 W/m K at the chemical potential corresponding to the maximum ZT) is much lower than that along the X-direction (2.46 W/m K at the chemical potential corresponding to the maximum ZT), as shown in Figure d and Table , which results in the higher ZT.

It is worth emphasizing that although the power factor of α-Ag2S is significantly high, the figure of merit ZT is not outstanding among the promising TE materials reported in recent years[3,4] because of the relatively high thermal conductivity. As shown in Figure , the lattice thermal conductivity of α-Ag2S is very low, benefited from the complex crystalline structure and weak bonding. Therefore, the major bottleneck in promoting its TE figure of merit is the high electronic thermal conductivity. For example, for n-type α-Ag2S, its X-direction electronic thermal conductivity is up to 2.46 W/m K at the chemical potential corresponding to the peak value of ZT. This is about 20 times the lattice thermal conductivity and is much higher than the electronic thermal conductivity of usually studied TE materials, for instance, 0.23 W/m K for Bi2Te3,[56] 1.5 W/m K for Mg2Si1–Sn solid solutions,[57] and 1 W/m K for the ZrSe2/HfSe2 superlattice monolayer.[58] This indicates that the dominated bottleneck that hinders the improvement of TE performance of α-Ag2S is the high electronic thermal conductivity.

It is noteworthy that the ZT value of α-Ag2S measured at room temperature by Wang et al.[39] is very low. This is due to the low carrier concentration in the experimentally studied samples, which is about 1.2 × 1014 cm–3 for n-type α-Ag2S samples. However, the optimal carrier concentration of n-type α-Ag2S for the maximal ZT from our theoretical calculation is about 0.77 × 1019 cm–3, as shown in Table . Therefore, higher ZT value can be expected in the experiment by increasing the carrier concentration of n-type α-Ag2S. Recent research shows that the maximal ZT value of Ag4SeS is about 0.6 at 373 K,[40] which agrees with the ZT value of n-type α-Ag2S along the Y-direction calculated at room temperature (ZT = 0.56).

Compressive Strain Effect on Lattice Thermal Conductivity

The experiment shows that α-Ag2S exhibits a metallic-like ductile behavior at room temperature and can also bear large compressive deformation along the X-direction. Next, we explore the structural stability of α-Ag2S under compressive strain along the X-direction. The compressive strain is defined as ε = (L0 – L)/L, where L is the strained lattice constant along the X-direction and L0 is the corresponding original lattice constant, respectively.

As shown in Figure , the structure is stable until ε = 5%. With further increase in strain, the out-of-plane acoustic (ZA) mode decreases, and slight imaginary frequency starts to appear along the Γ–X-direction when ε = 10%. It is worth noting that the previous experiment has shown that α-Ag2S exhibits an extraordinary metallic-like ductility with above 50% compressive plastic deformation, which comes from the slip between layers.[37,38] However, slip between layers is not considered in our study, causing the appearance of imaginary frequency with 10% compressive strain. Our results are meaningful under small deformation (ε ≤ 10%). In addition, we also find that α-Ag2S along the Y-direction has a negative Poisson’s ratio and that along the Z-direction has a positive Poisson’s ratio with the increasing strain along the X-direction. The changes of lattice constant with increasing strain are shown in Figure S3.

Figure 5

Phonon spectrum of α-Ag2S under compressive strains of (a) 0%, (b) 2%, (c) 5%, and (d) 10%, respectively.

Next, we studied the effect of X-direction compressive strain on the thermal transport properties of α-Ag2S, as strain is a widely adopted method to manipulate TE properties of materials.[59]Figure a shows the lattice thermal conductivity along different directions under compressive strains of 0%, 2%, and 5% at room temperature, respectively. It is clearly shown that the thermal conductivity of α-Ag2S decreases with the increase of compressive strain and then increases along three directions. α-Ag2S under 2% compressive strains achieves the lowest lattice thermal conductivity. In addition, the same phenomenon exists at other temperatures, as shown in Figure .

Figure 6

(a) Lattice thermal conductivity of α-Ag2S along X, Y, and Z directions under compressive strains of 0%, 2%, and 5% at 300 K, respectively. (b) Phonon scattering rate of α-Ag2S under different compressive strains at room temperature.

Figure 7

Lattice thermal conductivity of α-Ag2S along X, Y, and Z directions under compressive strains of 0%, 2%, and 5% at (a) 100 K, (b) 200 K, and (c) 400 K, respectively.

In order to explore the underlying physics, we calculate the phonon group velocity and phonon scattering rate under different compressive strains, as shown in Figures and 6b, respectively. As shown in Figure , under compressive strain from 2% to 5%, the group velocity of the three acoustic phonon branches changes slightly, showing that the harmonic effect does not have the dominant role. However, the compressive stress can significantly affect the phonon scattering rate of α-Ag2S, as shown in Figure b. With an increase in compressive strain from 0% to 2%, the scattering rate in the frequency range of 0–3 THz changes slightly. However, in the frequency range of 6.5–7.5 THz, the phonon scattering rate suddenly jumps, indicating a surprisingly giant anharmonic scattering, resulting in the reduction in thermal conductivity. With a further increase in compressive strain to 5%, phonon scattering rates in both high-frequency and low-frequency regions decrease, consequently increasing the lattice thermal conductivity. Therefore, the lattice thermal conductivity of α-Ag2S can be controlled by applying compressive stress through manipulating the phonon scattering rate.

Figure 8

Phonon group velocity of α-Ag2S under compressive strains of (a) 0%, (b) 2%, and (c) 5%, respectively.

Highly Tunable ZT Driven from Ductility

Finally, we study the influence of compressive strain on TE properties of α-Ag2S along the Z-direction because the ZT along the Z-direction is the largest compared with that along the other two directions. The TE transport coefficients with respect to the compressive strains of 0%, 2%, and 5% as a function of the chemical potential are shown in Figure . From Figure a, we can find that the electrical conductivity decreases with increasing compressive strain. In order to explain this phenomenon, the electronic band structures and electronic density of states (DOS) under different strains are provided in Figures and 11, respectively. We can find that the dispersion of the valence band decreases with the increase of stress, and at the same time, the band gap increases gradually with the increase of strain. Both of them reduce the electrical conductivity. Moreover, it is interesting to find that the degeneracy of the energy band valley can be increased by compressive strain, and the convergence of energy band appears under the compressive strain of 5%, which is a striking mechanism to increase the Seebeck coefficient,[4] as shown in Figure b.

Figure 9

TE properties of α-Ag2S as a function of chemical potential. The compressive strain is applied along the Z-direction, and temperature is at 300 K. (a) Electrical conductivity, (b) Seebeck coefficient, (c) power factor, (d) electronic thermal conductivity, and (e) ZT.

Figure 10

Electronic band structure of α-Ag2S under compressive strains of (a) 0%, (b) 2%, and (c) 5%, respectively.

Figure 11

Electronic DOS of α-Ag2S under compressive strains of (a) 0%, (b) 2%, and (c) 5, respectively.

From Figure , we also can find that the decrease of valence band dispersion and the degeneracy of energy band valley lead to the rapid increase of electron DOS near the valence band edge, which is beneficial to the improvement of Seebeck coefficient. Therefore, the power factor arrives at the maximum value at the compressive strain of 5% for p-type α-Ag2S. In addition, as the electrical conductivity decreases, the electronic thermal conductivity also decreases because of the Wiedemann–Franz law. Therefore, the remarkable increase of power factor combined with the decrease of electronic thermal conductivity leads to the remarkable increase of ZT (from 0.97 to 1.65) for p-type α-Ag2S at room temperature, as shown in Figure e. However, for n-type α-Ag2S, the compressive strain changes slightly the value of ZT, only inducing a blue shift in the chemical potential corresponding to the maximum ZT.

Conclusions

In the present work, the TE properties of α-Ag2S are investigated by using first-principles calculations combined with BTEs and DP theory. The results reveal a remarkable anisotropy for the lattice thermal conductivity in the α-Ag2S crystal. The thermal conductivity of α-Ag2S along the X-direction is 0.13 W/m K, and the ultralow thermal conductivity is due to the weak interlayer interaction. Moreover, the thermal conductivity of α-Ag2S can be controlled by applying compressive strain to control the phonon–phonon scattering rate. In addition, we also find that α-Ag2S has excellent TE properties, where the value of ZT can reach 0.97 for p-type and 1.12 for n-type α-Ag2S in the Z-direction at room temperature. Moreover, the value of ZT can be further enhanced to 1.65 at room temperature by applying 5% compressive strain. This is due to the increased Seebeck coefficient and reduced electronic thermal conductivity.