GaInX₃ (X = S, Se, Te): Ultra-low thermal conductivity and excellent thermoelectric performance

Thermoelectric (TE) materials have the potential to convert waste heat directly into electricity and can play an important role in solving global warming and the energy crises.^[1–4] Seeking for high performance TE materials has been a hotspot of research. The performance of TE materials is determined by the dimensionless figure of merit ZT because it reflects the TE conversion efficiency. ZT can be expressed as

where S is the Seebeck coefficient, σ is the electronic conductivity, S²σ is the power factor, T is the absolute temperature, and κ_e and κ_l are the electronic and lattice thermal conductivities, respectively.^[5,6] Currently, the achieved ZT values of TE materials are relatively low, which strictly restricts their widespread application.^[7,8] A lot of effort has been made to improve the ZT values of existing TE materials, including band engineering to improve the power factor^[9–11] and phonon engineering to suppress the thermal conductivity.^[12–18] However, due to the Wiedemann–Franz law,^[19] to enhance ZT through band engineering is a complex issue because the power factor and the electronic thermal conductivity are complexly connected. In addition, phonon engineering methods such as hierarchical architecting and introducing defects would inevitably disturb the electronic transport.^[20,21]

Searching for materials with intrinsically low lattice thermal conductivity is another avenue to obtain high TE performance materials^[22,23] because this approach helps to simplify complex TE parameters and optimize TE performance. Due to their unique physical structures, low-dimensional materials possess a variety of physical properties,^[24–30] which provides new possibilities for achieving low thermal conductivity and high TE performance. For instance, the material SnSe exhibits a high TE figure of merit as a result of its low thermal conductivity, while the half-Heusler compound PCdNa is noteworthy for its low thermal conductivity coupled with superior TE performance.^[31,32]

Previous research has demonstrated that reducing a material’s geometric symmetry can significantly reduce its thermal conductivity.^[33] Vertically asymmetric Janus materials naturally have lower symmetry than their symmetric counterparts, and thus have been potential high performance TE materials. For example, the thermal conductivity and TE performance of Janus WSX (X = Se; Te), SnSSe, and MXY (M = Pd, Pt; X, Y = S, Se, Te) have been investigated.^[34–37] However, the obtained ZT values in these materials are still relatively low, and thus searching for new Janus materials with a high ZT value is of significant importance.

In this work, based on the Boltzmann transport theory combined with first-principles calculations, the carrier transport and TE properties of monolayer Janus GaInX₃ (X = S, Se, Te) were investigated. The lattice thermal conductivity was first studied at room temperature, considering factors such as phonon group velocity, Grüneisen parameters, and scattering rates. The obtained lattice thermal conductivities for GaInS₃, GaInSe₃, and GaInTe₃ are 3.07 W⋅m⁻¹⋅K⁻¹, 1.16 W⋅m⁻¹⋅K⁻¹ and 0.57 W⋅m⁻¹⋅K⁻¹, respectively, which are comparable with that of the well-studied TE material SnSe (0.62 W⋅m⁻¹⋅K⁻¹).^[31] This is attributed to the presence of low-frequency optical modes, which can enhance the scattering between optical and acoustic phonons, resulting in a lower thermal conductivity. Then, the electric transport properties and the figure of merit ZT were investigated. It is found that for GaInS₃, GaInSe₃, and GaInTe₃, the figure of merit ZT can reach maximum values as high as 0.95, 2.37, and 3.00, respectively.

The quantities involved in the calculations of lattice thermal conductivity and electronic transport calculations were obtained through first-principles calculations based on density functional theory (DFT) using the Vienna ab initio simulation package (VASP).^[38,39] The electron exchange and correlation interactions were treated with generalized gradient approximation (GGA) using the Perdew–Burke–Ernzerhof (PBE).^[40] To prevent spurious interactions between the two-dimensional (2D) sheet and its periodic images along the z-direction, a vacuum space of 25 Å was introduced. The structural relaxation was performed using the conjugate gradient method until the absolute value of the Hellman–Feynman force converged to less than 0.0001 eV/Å. The integration over the Brillouin zone was conducted using a 17 × 17 × 1 Monkhorst–Pack k-grid,^[41] with an energy convergence criterion set to 1 × 10⁻⁸ eV. The ion–electron interaction was carried out by using the projector augmented wave method with a kinetic energy cutoff of 400 eV.^[42] The HSE06 functional^[43] was employed to obtain more accurate electronic transport properties. Ab-initio molecular dynamics (AIMD) simulations were conducted for the Janus GaInX₃ (X = S, Se, Te) monolayers. The simulations employed a 2 fs time step and a total time of 10 ps in the canonical ensemble at the temperature of 900 K to assess their thermodynamic stabilities.^[44]

The calculation of second-order and third-order interatomic force constants (IFCs) was conducted using 5 × 5 × 1 and 4 × 4 × 1 supercells utilizing both the PHONOPY and Thirdorder packages, respectively.^[45,46] A cut-off radius up to the 12th nearest neighbors was chosen to obtain the third-order IFCs. The calculation of thermal conductivity convergence in 2D materials is contingent upon the quadratic characteristic of the ZA branch near the Γ point.^[47] To achieve a physically accurate quadratic phonon dispersion for low-frequency ZA phonons, we utilized the Hiphive program to manipulate second-order IFCs that were initially acquired through PHONOPY.^[48–50] Phonon transport properties are obtained using the phonon Boltzmann transport eqnarray as implemented in ShengBTE code,^[51] where a denser q-point mesh of 50 × 50 × 1 is used to ensure convergence (see the supporting information for details). The κ_l is given by^[51]

where k_B is the Boltzmann constant, T is the temperature, V is the volume of the primary cell, ħ is the reduced Planck constant, ω_λ is the angular frequency of the phonon mode λ, N is the total number of q sampling points, and v is the group velocity of phonons. Fλy is the projection of the mean free path along the y direction. The electronic transport properties are obtained by solving the semiclassical Boltzmann transport eqnarray with a denser k-mesh of 35 × 35 × 1 and the constant relaxation time approximation using the BoltzTraP2 software package.^[52] Here, S and σ can be expressed as

where n and k are the band index and wave vectors of the electronic states, respectively. E_nk is the electron energy, N is the total number of k points, and ν_nk is the group velocity. τ_e is the electron relaxation time, calculated through deformation potential theory within the effective mass approximation framework,^[53,54] which can be determined by the subsequent eqnarray

where C, m*, E₁ and k_B are the elastic modulus, the effective mass, the DP constant, and the Boltzmann constant, respectively. According to the Wiedemann–Franz law, κ_e is proportional to σ and temperature and can be expressed as

where L is the Lorenz number. To obtain more reliable results for κ_l, σ, and κ_e in 2D materials, a normalization process is conducted to compensate for the additional length introduced by the vacuum space applied along the non-periodic direction.^[12,55] This normalization is achieved by the ratio of L/t, where L represents the cell length along the direction with the added vacuum, and t stands for the effective thickness of the 2D materials (see the supporting information for details).

Monolayer Janus GaInX₃ (X = S, Se, Te) belongs to the R3m space group, as shown in Figs. 1(a) and 1(b). The optimized lattice constants for GaInS₃, GaInSe₃, and GaInTe₃ are 3.79 Å, 3.98 Å, and 4.24 Å, respectively, which align with the results reported in the literature.^[56] They are dynamically stable because there are no imaginary frequencies in their phonon spectra, as shown in Figs. 1(c)–1(e). In addition, AIMD simulations indicate that these materials remain stable at 900 K (see the supporting information for details). As shown in Figs. 1(f)–1(h), the HSE06 functional gives similar band dispersions as the PBE functional but with an enlarged bandgap. GaInS₃ is a direct bandgap semiconductor with the valence band maximum (VBM) and conduction band minimum (CBM) located at the Γ point. Meanwhile, GaInTe₃ and GaInSe₃ are indirect bandgap semiconductors with their CBM and VBM located at the Γ point and along the Γ–M line. The obtained bandgap energies of GaInS₃, GaInSe₃, and GaInTe₃ are 1.793 eV (0.935 eV), 1.130 eV (0.415 eV), and 0.68 eV (0.147 eV), respectively by using the HSE06 (PBE) functional. The gradual decrease of the bandgaps in this series mainly comes from the decrease of the X p orbital ionization energies as X becomes heavier^[57–60] (details are shown in Fig. S1 in the supporting information).

Figures 1(c)–1(e) show that low-energy optical phonon modes emerge in these materials, indicating strong scattering between optical and acoustic phonons.^[58,61–63] The lattice thermal conductivities (κ_l) of these materials are then investigated and the results are shown in Fig. 2(a). At 300 K, κ_l reaches values as low as 3.07 W⋅m⁻¹⋅K⁻¹, 1.16 W⋅m⁻¹⋅K⁻¹, and 0.57 W⋅m⁻¹⋅K⁻¹ for GaInS₃, GaInSe₃, and GaInTe₃, respectively, which are much lower than that of the well-studied monolayer MoS₂ (116.8 W⋅m⁻¹⋅K⁻¹)^[64] and are comparable to those of 2D group IV–VI semiconductors, such as SnSe, SnS, GeSe, and GeS (2.2–10.5 W⋅m⁻¹⋅K⁻¹).^[65] In addition, κ_l decreases with the increase of temperature. To understand the physical mechanism behind the low κ_l value obtained, the frequency-resolved phonon contributions to κ_l are investigated. Taking that obtained at 300 K as an example, it is shown that the low-frequency phonon contribution to κ_l is dominant and the accumulative κ_l achieves 80% of the total κ_l below 2.87 THz, 2.42 THz, and 2.17 THz for GaInS₃, GaInSe₃, and GaInTe₃, respectively, as depicted in Figs. 2(b)–2(d). Simultaneously, there is a pronounced decrease in the contribution of phonons to the thermal conductivity at frequencies of 2.24 THz, 1.1 THz, and 0.86 THz, in GaInS₃, GaInSe₃, and GaInTe₃, respectively. This phenomenon comes from the high phonon–phonon scattering rate, especially between TA, LA, and low frequency optical phonon branches as shown in Figs. 3(a)–3(c). This enhanced scattering effect consequently leads to a diminished contribution of acoustic phonons to the total κ_l, thereby reducing the overall thermal conductivity of the material.

From a general point of view, κ_l is primarily determined by the interactions among the harmonic and anharmonic parameters.^[66–68] The former include second-order force constants, phonon group velocities, and atomic masses, and the latter mainly involve third-order force constants and three-phonon scattering processes. This makes it very complicated to reveal the physics behind the low κ_l.^[69] However, it is found that κ_l is largely determined by the phonon group velocity v and the Grüneisen parameter γ (κ_l ∝ v³/γ²).^[70–73]v can be obtained from the phonon spectra via

As depicted in Figs. 3(d)–3(f), the calculated maximum values of v are 4.62 km/s, 4.11 km/s, and 2.67 km/s for GaInS₃, GaInSe₃, and GaInTe₃, respectively, which are comparable to the well-known 2D TE materials SnSe (2.7 km/s), PtSe (3.02 km/s), and SnTe (4.0 km/s).^[31,74] As X becomes heavier (from S to Te), the bond length increases and the bond strength weakens in the GaInX₃ series, which leads to a gradual softening of the phonon modes (Figs. 1(c)–1(e)). This gives a gradual reduction in the group velocity.^[75] (see the supporting information for more details.) γ describes the anharmonic interactions of a crystal, which can be expressed as

In general, systems with larger |γ| exhibit stronger anharmonicity, resulting in higher phonon-phonon scattering rates and lower κ_l.^[73] As depicted in Figs. 3(g)–3(i), the magnitude of |γ| is considerably large in low energy phonon regions, which is comparable to that of the excellent TE materials SnSe (|γ| = 2.83) and PbTe (|γ| = 1.45).^[31,61] To understand the strong anharmonicity shown in these materials, the bonding features are investigated. It is shown that nonbonding electron lone pairs^[12,76,77] emerge in these materials (details are shown in the supporting information). These electron lone pairs will interact with the bonding electrons of neighboring atoms, which creates nonlinear electrostatic repulsion. This nonlinear electrostatic repulsion is the origin of the observed strong lattice anharmonicity.^[21,71,78] A small v combined with a large |γ| results in a low κ_l in these materials. In particular, v decreases from GaInS₃ to GaInSe₃ and GaInTe₃, while |γ| is comparable in these materials in the low energy phonon region. This is the reason why the value of κ_l in GaInS₃ is the largest and in GaInTe₃ is the smallest.

According to Eqs. (4) and (6), the relaxation time τ is required to obtain the electrical conductivity (σ) and the electronic thermal conductivity (κ_e). Therefore, τ is first obtained based on Eq. (5) and the corresponding data are shown Table 1. It is noted that, because the band dispersions obtained using the HSE06 and the PBE functionals are similar, τ is obtained using the PBE functional to save computational efforts. It is shown that the electron has a much longer relaxation time than the hole in the three studied materials. This is mainly because the electron has a much smaller effective mass. Then, the three TE parameters S, σ and κ_e are investigated using the HSE06 functional to obtain more accurate results.

As shown in Figs. 4(a)–4(c), since the valence bands near the Fermi level have a flatter dispersion compared to that of the conduction bands, the density of states is larger for the valence bands.^[79] Consequently, the absolute values of S (|S|) for p-type doping are much larger than that for n-type doping in the three studied materials within the experimentally accessible doping regime. Meanwhile, σ and κ_e have the opposite trends, as shown in Figs. 4(d)–4(i), because they are proportional to the carrier relaxation time. Specifically, for p-type doping, |S| decreases with increasing carrier concentration in GaInS₃ and GaInSe₃. However, for GaInTe₃, |S| decreases with increasing carrier concentration at 300 K and 600 K. Meanwhile, it first increases and then decreases at 900 K. This discrepancy comes from the bipolar conduction effect that the thermally excited electrons and holes in low carrier concentration system at high temperature will have a high proportion in narrow bandgap semiconductors. Since the Seebeck effects from electrons and holes have opposite signs, they cancel each other to some extent, which results in a reduced |S| at elevated temperatures.^[80,81] For n-type doping, the variation of S is more complicated and it tends to zero eventually as a function of carrier concentration. Meanwhile, the electrical conductivities from the thermally excited electrons and holes have the same signs, leading to an enhanced σ at low carrier concentration in p-type GaInTe₃ at 900 K.

Finally, the TE performance of the three studied materials is investigated. As depicted in Fig. 5, the ZT value increases with temperature in GaInS₃ and GaInSe₃ for both n-type and p-type dopings. However, this trend breaks down in GaInTe₃ due to the bipolar effect. The optimal ZT values of the three materials within the experimentally accessible doping regime at 300 K, 600 K, and 900 K are shown in Table 2.

The TE performance of n-type doping is clearly superior to that of p-type doping for GaInS₃ and GaInSe₃ at 300 K, 600 K, and 900 K. Meanwhile, for GaInTe₃ n-type doping is still superior at 300 K and 600 K, but the maximum ZT value for p-type doping is larger than that of n-type doping at 900 K. The maximum ZT values for GaInSe₃, GaInSe₃, and GaInTe₃ are 0.95 (900 K), 2.37 (900 K), and 3.00 (900 K), respectively, which indicates that they are potential high-temperature TE materials. In addition, at 300 K, the maximum ZT value for GaInTe₃ is 0.85, which is notably larger than that of the well-studied SnSe (0.7).^[82] This demonstrates its great potential in room-temperature TE device applications.

In conclusion, we have studied the thermal transport and TE properties of Janus GaInX₃ (X = S, Se, Te). The presence of low-frequency optical phonon modes leads to a high scattering rate and a large Grüneisen parameter in the low-frequency phonon region, resulting in an ultra-low lattice thermal conductivity in these materials. Furthermore, within the experimentally accessible doping regime, the electronic properties of narrow-gap semiconductors GaInTe₃ are significantly influenced by the bipolar effect at high temperatures, which reduces their TE performance in such conditions. At room temperature, the obtained TE figure of merit ZT can reach values as high as 0.85 in GaInTe₃. In addition, the maximum values of ZT are 0.95 (900 K), 2.37 (900 K), and 3.00 (900 K) for GaInS₃, GaInSe₃, and GaInTe₃, respectively. Our research elucidates the mechanisms behind the low thermal conductivity of GaInX₃ (X = S, Se, Te) materials and highlights their potential in TE and heat management applications.