Hydrostatic Pressure Tuning of Thermal Conductivity for PbTe and PbSe Considering Pressure-Induced Phase Transitions.

Flexibly modulating thermal conductivity is of great significance to improve the application potential of materials. PbTe and PbSe are promising thermoelectric materials with pressure-induced phase transitions. Herein, the lattice thermal conductivities of PbTe and PbSe are investigated as a function of hydrostatic pressure by first-principles calculations. The thermal conductivities of both PbTe and PbSe in NaCl phase and Pnma phase exhibit complex pressure-dependence, which is mainly ascribed to the nonmonotonic variation of a phonon lifetime. In addition, the thermal transport properties of the Pnma phase behave anisotropically. The thermal conductivity of Pnma-PbTe is reduced below 1.1 W/m·K along the c-axis direction at 7-12 GPa. The mean free path for 50% cumulative thermal conductivity increases from 7 nm for NaCl-PbSe at 0 GPa to 47 nm for the Pnma-PbSe in the a-axis direction at 7 GPa, making it convenient for further thermal conductivity reduction by nanostructuring. The thermal conductivities of Pnma-PbTe in the c-axis direction and Pnma-PbSe in the a-axis direction are extremely low and hypersensitive to the nanostructure, showing important potential in thermoelectric applications. This work provides a comprehensive understanding of phonon behaviors to tune the thermal conductivity of PbTe and PbSe by hydrostatic pressure.

Introduction

Thermal conductivity is an essential thermophysical property of materials due to its significant effect on the performance of various devices from nanoelectronics to thermoelectrics.[1] In nanoelectronics, high thermal conductivity is required for efficient heat dissipation, while low thermal conductivity is urged in the applications of thermoelectrics.[2] Therefore, the flexible regulation of thermal conductivity is of utmost significance for broadening the application range of functional materials.[3,4] Hydrostatic pressure is a simple and effective method to modulate the thermal transport performance of materials,[5−7] which provides a promising approach to accomplish extraordinary thermoelectric performance at ambient temperature.

Thermoelectric materials can directly convert heat into electricity, which has attracted increasing attention.[8−10] Lead chalcogenides, such as PbTe and PbSe, are typical IV–VI narrow band gap semiconductors with outstanding thermoelectric performance.[11,12] The capability of the thermoelectric conversion is evaluated by the dimensionless figure of merit ZT = S2σT/(kl + ke), where S, σ, T, kl, and ke are Seebeck coefficient, electrical conductivity, absolute temperature, lattice thermal conductivity, and electronic thermal conductivity, respectively.[13] In general, phonons are the dominant heat carriers in semiconductors and insulators, and the lattice thermal conductivity is much greater than the electronic thermal conductivity.[14] The thermal conductivity mentioned in this work refers to the lattice thermal conductivity unless otherwise specified. PbTe is a promising thermoelectric material suitable for the temperature range of 400–800 K.[15] In recent decades, many efforts have been made to improve the ZT value of PbTe. The thermoelectric performance of n-PbTe is generally not as good as p-PbTe.[16] Yang et al.[17] reported an extraordinary ZT value of 1.35 in n-type Bi-doped PbTe at 675 K. Heremans et al.[18] achieved a high ZT value of 1.5 at 773 K in Tl-doped p-type PbTe by band structure engineering. Pei et al.[19] achieved a remarkable ZT value of 1.8 at about 850 K through producing high valley degeneracy in doped PbTe1–Se alloys. Zhang et al.[20] obtained an ultrahigh ZT value of 1.83 at 773 K in n-type PbTe-4%InSb composites by introducing the multiphase nanostructures. However, Te is quite scarce in the earth’s crust, which limits the practical applications of PbTe-based materials.[21] Compared with the Te element, Se is more earth-abundant and inexpensive.[22] As the sister compound of PbTe, PbSe shares similar crystal and band structures but exhibits superior chemical stability and mechanical properties.[23] Accordingly, PbSe is an ideal alternative to PbTe for large-scale applications. The ZT values of PbTe and PbSe were expected to reach 2.2 and 1.8 by band convergence, respectively.[24] Zhou et al.[25] achieved an extraordinary ZT value of 1.5 at 823 K in n-type PbSe by defect engineering. Wang et al.[26] observed ZT > 1.2 at 850 K in p-type Na-doped PbSe polycrystalline and demonstrated that PbSe is an excellent thermoelectric material for a mid-high temperature range.

As we can see, the existing studies mainly focus on the normal pressure phase. Nevertheless, PbTe and PbSe undergo a pressure-induced phase transition. At ambient conditions, lead chalcogenides PbX (X = Te and Se) are the NaCl-phase (B1).[27] Then, the NaCl phase transforms to an intermediate structure (Pnma phase) at 6.0 GPa[28] for PbTe and 6.01 GPa[29] for PbSe, respectively. The intermediate structure of PbTe and PbSe was unclear and it is identified as orthorhombic Pnma phase until the last decade.[27,28,30] A further structural phase transition occurs in PbTe and PbSe at 13 and 16 GPa,[31] respectively. The pressure-induced phase transitions of a crystal structure correspond to the electronic band structure transitions.[32] Herein, the NaCl phase and Pnma phase are semiconductor types and the high-pressure CsCl phase is a metal type. In thermoelectric conversion, the band gap is critical to the thermoelectric performance,[33] and narrow band gap semiconductors are generally preferred.[34] With regards to the aimed NaCl phase and Pnma phase, there are also some explorations. Ovsyannikov and Shchennikov[35] measured the power factor (PF) of PbTe and achieved a colossal improvement of PF in 2–3 and 6–6.5 GPa. Wang et al.[36] claimed that the ZT value of PbTe in the n-type Pnma phase at 6.5 GPa is about two times larger than that of the NaCl phase at ambient pressure, which is attributed to the high anisotropic electronic structure of the Pnma phase. Xu et al.[37] explored the thermoelectric performance of PbTe under pressure by first-principles calculations and demonstrated that the n-type Pnma phase at 6.7 GPa has a higher performance than the NaCl phase at atmospheric conditions. However, previous studies on the thermoelectric properties of PbTe and PbSe did not consider the effects of pressure and phase transition on thermal conductivity. Petersen et al.[38] calculated the structural, electronic, and mechanical properties of PbTe and PbSe in their three phases and indicated that thallium doping leads to a high PF value while iodine doping achieves a good mechanical performance. Recently, Chen et al.[11] obtained a striking ZT value of 1.7 at room temperature in Cr-doped PbSe by applying external pressure, which is ascribed to the pressure-driven topological phase transition. However, all such attention has been paid to the structural and electronic properties, while the thermal transport properties, particularly for the intermediate Pnma-phase, are not concerned. The structure of the Pnma phase is complex, and it may exhibit an anisotropic low thermal conductivity.[39] To date, the effect of pressure on the thermal conductivity of PbTe and PbSe is still unknown and the pressure-tuned phonon behaviors are not yet clear. Thus, it is of great significance to gain a thorough understanding of pressure-tuned thermal transport properties of PbTe and PbSe, which helps to expand their applications in thermoelectrics and other potential fields.

In this work, we systematically investigated the pressure-tuned thermal conductivity of PbTe and PbSe in NaCl phase and Pnma phase by first-principles calculations. The rest of the paper is organized as follows. Phase transitions and phonon dispersions are investigated in Section , and the effects of pressure and mean free path on the thermal conductivity of PbTe and PbSe are discussed in Section . A detailed phonon mode-level analysis is implemented in Section to unveil the underlying physics of the anomalous pressure-dependence of phonon transport behaviors. Then, the findings are summarized in Section . Finally, a brief description of the first-principles phonon Boltzmann transport equation (BTE) method and the computational details are introduced in Section .

Results and Discussion

Phase Transition and Phonon Dispersion

The enthalpy differences among NaCl phase, Pnma phase, and CsCl phase versus pressure are shown in Figure to identify the phase transition points of PbTe and PbSe. The enthalpy of the NaCl phase is set to zero for clear comparison. The phase transition appears at the intersection of the enthalpy difference curves, and the structure with lower enthalpy is more stable. There is a pressure-induced structural transition of NaCl phase → Pnma phase → CsCl phase in both PbTe and PbSe. The calculated phase transition pressures are also denoted in Figure . The experimental values of the phase transition pressure from NaCl phase to Pnma phase are 6.0 GPa[28] for PbTe and 6.01 GPa[29] for PbSe, and the phase transition pressures from Pnma phase to CsCl phase are 13.0 and 16.0 GPa[31] for PbTe and PbSe, respectively. The present results come considerably closer to available experimental data than previous calculations.[38,40,41] This is mainly ascribed to the fact that the PBEsol functional can accurately reproduce the lattice constants, which has been verified in many studies.[42−44] Lattice constants of PbTe and PbSe for NaCl phase, Pnma phase, and CsCl phase are presented in Table . The calculated results are compared with experimental data.

Figure 1

Enthalpy differences per formula unit for (a) PbTe and (b) PbSe as a function of pressure. The corresponding NaCl phase sets to zero. The insets are the crystal structures of three different phases for PbTe and PbSe. The gray, yellow, and purple spheres represent Pb, Te, and Se atoms, respectively.

Table 1

Lattice Constants of PbTe and PbSe for NaCl Phase, Pnma Phase, and CsCl Phase

		this work				experimental data
material	structure	pressure (GPa)	a (Å)	b (Å)	c (Å)	pressure (GPa)	a (Å)	b (Å)	c (Å)
PbTe	NaCl	0	6.45			0	6.46[45]
	Pnma	6.7	8.11	4.50	6.26	6.7	8.18[28]	4.50[28]	6.23[28]
	CsCl	24.7	3.57			>24.7	3.53[46]
PbSe	NaCl	0	6.11			0	6.12[45]
	Pnma	9.5	11.01	4.29	4.04	9.5	11.19[30]	4.17[30]	4.06[30]
	CsCl	16	3.56			>16	3.49[47]

Phonon dispersion relations reflect the harmonic properties of the system, and the slope of the dispersion curve represents the phonon group velocity. Figure shows the phonon dispersions along the high symmetry path in the first Brillouin zone. There are two atoms in the primitive cell of the NaCl phase and eight atoms in the primitive cell of the Pnma phase. No imaginary phonon frequencies appear in these systems, confirming the thermodynamic stability. The phonon dispersions of PbTe and PbSe with NaCl phase are presented in Figure a,b, respectively. Owing to the unusual large Born effective charges, the prominent splitting occurs between the transverse optical (TO) branch and the longitudinal optical (LO) branch at Γ point. There are six polarizations in the NaCl-phase phonon dispersions and all the branches stiffen with increasing pressure except for the lowest frequency acoustic branch. Figure c,d depicts the lowest frequency acoustic branch for PbTe and PbSe at various pressures, respectively. It can be seen that this branch stiffens with pressure in the Γ to L direction, but softens with pressure in other directions. For PbTe, the lowest frequency branch exhibits obvious softening with the pressure increasing from 0 to 2 GPa, while it shows a slight softening as the pressure increases from 2 to 4 GPa. As the pressure further increases, the change in the lowest acoustic branch is negligible. For PbSe, the lowest frequency acoustic branch displays a mild softening with increasing pressure. In addition, it can be noted that the interaction between LA and TO branch occurs in Γ → K and Γ → X directions at 0 GPa in both NaCl–PbTe and NaCl–PbSe, and the interaction is more notable in PbTe. Then, the interaction disappears with the applied pressure. The LA–TO coupling affects the heat-carrying LA phonons,[48] which may be one of the underlying causes of the extremely low thermal conductivity of these systems with a simple NaCl-phase crystal structure. The stiffening of acoustic branches indicates an increase in the phonon group velocity, while the softening of acoustic branches means a decrease in the phonon group velocity. Therefore, the pressure-dependence of NaCl-phase phonon dispersions is complex and the variation tendency of the phonon group velocity cannot be predicted clearly from the change of the phonon dispersions. Figure e,f shows the phonon dispersions of PbTe and PbSe in the Pnma phase under two typical pressures, respectively. The phonon dispersion of the Pnma phase contains 24 polarizations, three of which are acoustic branches. It can be seen that almost all the branches harden with the increasing pressure. The hardening of optical branches is more pronounced, while that of acoustic branches is slight. The LA–TO coupling is also observed in the Pnma phase, which indicates that the Pnma-phase PbTe and PbSe may also behave with low thermal conductivity. Furthermore, the optical branches of the NaCl phase and TO branches of the Pnma phase are not as flat as a general system. It means that they should also exhibit a fairly high phonon group velocity and play an important role in the phonon thermal transport. The frequency-dependent lattice thermal conductivities are presented in Figure S1. It can be observed that the contribution of optical branches to thermal conductivity is not negligible. In the following, we will specifically discuss the pressure-dependence and the phonon mean free path-dependence of thermal conductivity.

Figure 2

Phonon dispersions for NaCl phase of (a) PbTe and (b) PbSe at two typical pressures, and the lowest frequency acoustic branch for the NaCl phase of (c) PbTe and (d) PbSe at various pressures. Phonon dispersions for the Pnma phase of (e) PbTe and (f) PbSe at two typical pressures.

Lattice Thermal Conductivity

The calculated thermal conductivity of PbTe and PbSe with respect to pressure is summarized in Figure . We calculated the pressure-dependence of thermal conductivity of the NaCl phase and Pnma phase. For PbSe, the lattice thermal conductivity agrees well with both experimental[49] and theoretical[50] results at ambient pressure. However, for PbTe, the lattice thermal conductivity under ambient pressure is slightly lower than available experimental data,[49] which is attributed to that the contribution of electrons is not considered in the calculation. For the NaCl phase, the thermal conductivities of PbTe and PbSe are both isotropic. The thermal conductivity of NaCl-phase PbTe first decreases with pressure until it reaches the minimum value at around 2 GPa and then the thermal conductivity remains at a low plateau at 2–4 GPa. After that, the thermal conductivity continues to increase until the pressure reaches the phase transition point. The thermal conductivity of NaCl-phase PbTe achieves a minimum value at around 2–4 GPa, combined with the reported colossal (∼100 times) improvement of PF under 2–3 GPa,[35] which proves that the ZT value of PbTe can be greatly boosted under a pressure of ∼2–3 GPa. The thermal conductivity of NaCl-phase PbSe increases monotonously with pressure. This trend is very common in most bulk materials. However, the thermal conductivity of PbSe increases nonlinearly, and it appears to be an approximate plateau under a pressure of 2–4 GPa. A pressure-driven topological phase transition in PbSe is observed at 3 GPa, and the PF value increases significantly in the vicinity of 3 GPa.[11] Hence, the NaCl-phase PbSe also has excellent thermoelectric properties at 2–3 GPa. The thermal conductivity of the Pnma phase is anisotropic in general. For the Pnma-phase PbTe, the thermal conductivity first decreases with pressure until it attains a minimum at 10 GPa, and then it starts to increase until the pressure arrives at the phase transition point. The thermal conductivities in the a axis and b axis are similar but it is extremely low in the c axis. It has been predicted that the ZT value of Pnma-PbTe is greater than 0.4 at 10 GPa by setting the thermal conductivity to 2.0 W/m·K.[37] Combined with this work, if the lattice thermal conductivity along the a-axis and b-axis directions of 1.4 W/m·K is used, the ZT value could reach around 0.6. If using the lattice thermal conductivity along the c-axis direction of 0.7 W/m·K, an amazing ZT value of about 1.2 at room temperature can be achieved. For the Pnma-phase PbSe, the thermal conductivity shows an upward trend in the pressure range of 7–10 GPa, and then a sudden drop in 10–12 GPa. Finally, the thermal conductivity increases again with pressure in 12–16 GPa. The thermal conductivity of the Pnma phase is similar along the b axis and c axis, but the lowest along the a axis. The thermoelectric performance of Pnma-phase PbSe under high pressure is still unclear. This work provides valuable information for the investigation of the thermoelectric properties of Pnma-PbSe. The extremely low lattice thermal conductivity of Pnma-PbSe along the a-axis direction indicates that it is a promising thermoelectric material. Anisotropic thermal transport materials have widespread applications in many fields, such as thermal management, thermal insulations, and thermoelectrics.[5] The extremely low thermal conductivities in the c axis of PbTe and a axis of PbSe further clarify the great potential of Pnma-phase PbTe and PbSe in thermoelectric applications.

Figure 3

Lattice thermal conductivity against pressure in (a) PbTe and (b) PbSe with different phases at room temperature.

The thermal conductivity of the crystalline materials can be expressed as , where vg is the phonon group velocity, and Λ is the phonon mean free path. The thermal transport in nanostructures can be regulated by the phonon mean free path. Therefore, it is necessary to further study the contribution of the phonon mean free path to the thermal conductivity of PbTe and PbSe nanostructures. Figure shows the cumulative thermal conductivity of NaCl phase and Pnma phase versus phonon mean free path. It can be observed that for PbTe and PbSe in the NaCl phase, the phonon mean free paths that contribute to the thermal conductivity are concentrated in a narrow range, and the maximum value is lower than 130 nm. The mean free paths for 50% cumulative thermal conductivity are 8 nm for NaCl–PbTe and 7 nm for NaCl–PbSe, which proves that the thermal conductivities of NaCl-phase PbTe and PbSe are pretty hard to be further reduced by nanostructuring. For the Pnma phase, the mean free path of phonons that contribute to the thermal conductivity reaches 500 nm along the c-axis direction in PbTe and 3500 nm along the a-axis direction in PbSe. The mean free paths for 50% cumulative thermal conductivity are 20 nm for Pnma-PbTe in the c-axis direction and 47 nm for Pnma-PbSe in the a-axis direction. Half of the cumulative thermal conductivity of Pnma-PbTe in the c-axis direction is less than 0.5 W/m·K, which is pretty attractive for thermoelectric applications. Furthermore, the mean free path is highly efficient in manipulating the thermal conductivity of Pnma-PbSe in the a-axis direction, and a nanostructure of 47 nm can reduce its thermal conductivity by half. Compared with the NaCl phase, the thermal conductivities of Pnma-PbTe along the c-axis direction and Pnma-PbSe along the a-axis direction are hypersensitive to the nanostructure. Consequently, the thermal conductivities of PbTe in the c-axis direction and PbSe in the a-axis direction are not only low but also can be further reduced by nanostructuring easily.

Figure 4

Cumulative thermal conductivity calculated for (a) PbTe and (b) PbSe as a function of phonon mean free path. The NaCl phase and Pnma phase are at 0 and 7 GPa, respectively. The vertical dotted lines represent the mean free path of phonons that contribute half of the total thermal conductivity.

Phonon Spectral Analysis

In this section, a detailed phonon behavior analysis is conducted to unveil the underlying physics responsible for the novel pressure-dependence of the thermal conductivity of PbTe and PbSe. The thermal conductivity is directly relevant to the phonon volumetric specific heat, phonon group velocity, and phonon lifetime. Therefore, we compare these three phonon properties carefully to gain the leading factor for the abnormal pressure-dependence of the thermal conductivity. The average phonon group velocity and phonon lifetime[51] are calculated to obtain a quantitative understanding of their contributions to the pressure-dependent thermal conductivity. Figure presents the phonon volumetric specific heat for PbTe and PbSe under various pressures. It can be observed that the volumetric specific heat of PbTe and PbSe increases linearly with pressure in both the NaCl phase and Pnma phase. The phonon group velocities are presented in Figure . There is a blue shift of phonon frequency in both the NaCl phase and Pnma phase with applied pressure, which is in line with the change of phonon spectra. With the increase in pressure, the phonon group velocities of the NaCl phase and Pnma phase do not change apparently. It thus can be concluded that the anomalous pressure-dependence of the thermal conductivity is not originated from the phonon group velocity. In addition, the phonon group velocity of the Pnma phase is higher than that of the NaCl phase. As can be seen from Figure , the phonon group velocity component of Pnma-phase PbTe exhibits conspicuous anisotropy. The components of phonon group velocity along the a axis and b axis do not differ greatly, but the group velocity component along the c axis is significantly low, which coincides with the anisotropy of the thermal conductivity in Pnma-phase PbTe. Based on the analysis of the phonon volumetric specific heat and the phonon group velocity, it can be concluded that these two factors are not responsible for the anomalous pressure-dependent thermal conductivity in PbTe and PbSe, but the anisotropy of the thermal conductivity in the Pnma phase stems from the anisotropy of the phonon group velocity components in different dicrections.

Figure 5

Phonon volumetric specific heat of PbTe and PbSe in the NaCl phase and Pnma phase under various pressures.

Figure 6

Phonon group velocities for the NaCl phase of (a) PbTe and (b) PbSe and for the Pnma phase of (c) PbTe and (d) PbSe under two typical pressures. The insets show the average phonon group velocity as a function of pressure.

Figure 7

Frequency-dependent phonon group velocity components of Pnma-phase PbTe in the a-axis, b-axis, and c-axis directions.

Phonon lifetime is also one of the basic factors directly related to thermal conductivity. Figure shows the frequency-dependent phonon lifetime of PbTe and PbSe under various pressures. The phonon frequency range expands with pressure in all systems, which is in agreement with the variation of the phonon dispersions. The phonon lifetime distributions in Figure a,b are similar, and the phonon lifetime distributions in Figure c,d are also similar but completely different from those in Figure a,b. It can be seen that the crystal structure is vital to the distribution of the phonon lifetime. The average phonon lifetimes of all systems are extremely low and less than 10 ps, which is the root of their low thermal conductivity. Phonons in the low-frequency region possess larger lifetimes and contribute more to thermal conductivity. For PbTe in the NaCl phase (Figure a) and Pnma phase (Figure c), the average phonon lifetime first decreases and then increases with pressure. The average phonon lifetime of PbSe in the NaCl phase (Figure b) increases monotonically with pressure and an approximate plateau appears in 2–4 GPa. The average phonon lifetime of PbTe in the Pnma phase (Figure d) exhibits a complex fluctuating trend with pressure. Interestingly, it can be observed from Figure that the changing tendency of the average phonon lifetime with pressure is almost the same as that of thermal conductivity in Figure . Therefore, one can conclude that the anomalous pressure-dependent thermal conductivity in PbTe and PbSe is caused by the abnormal change of phonon lifetime with pressure.

Figure 8

Phonon lifetime for the NaCl phase of (a) PbTe and (b) PbSe and for the Pnma phase of (c) PbTe and (d) PbSe at two typical pressures. The insets depict the dependence of the average phonon lifetime on pressure.

To further explore the physical mechanism of the unusual change of phonon lifetime with pressure, the variations of the square of the Grüneisen parameter and three-phonon scattering phase space are depicted in Figure . The Grüneisen parameter represents the strength of every three-phonon scattering process, reflecting the anharmonicity of phonons. The three-phonon scattering phase space can evaluate the probability of occurrence of three-phonon scattering. PbSe has a higher phonon cutoff frequency under the same pressure, mainly because of the larger mass difference between Pb and Se atoms, and the high-frequency optical phonon mode is mainly contributed by Se atoms. For the NaCl-phase structure, it can be seen clearly that both PbTe and PbSe show strong phonon anharmonicity in the low-frequency region, and the higher proportion of high-frequency phonon modes in PbSe may result in weaker anharmonicity of the system. The phase space of PbTe is larger than that of PbSe almost throughout the entire frequency range, implying more accessible scattering channels in NaCl-phase PbTe. Figure e,f depicts the average of the square of the Grüneisen parameter and the three-phonon scattering phase space with respect to pressure, respectively. The phase space of all systems decreases with pressure, and the variations of the square of the Grüneisen parameter are more complicated. From the left part of Figure e,f, we can observe that the anharmonicity and the three-phonon scattering channels in NaCl–PbTe are larger than those in NaCl–PbSe, which can account for the lower thermal conductivity of NaCl-phase PbTe. The anharmonicity of NaCl-phase PbTe increases with pressure, while the anharmonicity of NaCl-phase PbSe first decreases and then increases with pressure. Therefore, the anomalous pressure-dependence of the phonon lifetime of NaCl-phase PbTe and PbSe originates from the synergistic effect of the lattice anharmonicity and the three-phonon scattering phase space. Compared with the NaCl-phase structure, the anharmonicity of the Pnma-phase structure is greatly reduced, but the phase space is significantly increased. The variation trends of the anharmonicity and the phonon lifetime of Pnma-PbTe and Pnma-PbSe with pressure are exactly opposite, which indicates that the lattice anharmonicity plays an important role in the nonmonotonic pressure-dependence of the phonon lifetime for the Pnma-phase structure. It can be concluded from the above analysis that the synergy between the lattice anharmonicity and three-phonon scattering phase space results in the anomalous pressure-dependence of the phonon lifetime in the NaCl-phase structure, while in the Pnma-phase structure, the lattice anharmonicity plays a dominant role.

Figure 9

Frequency dependence of (a, c) the square of the Grüneisen parameter and (b, d) three-phonon scattering phase space for (a, b) NaCl phase and (c, d) Pnma phase. The average of (e) square of the Grüneisen parameter and (f) three-phonon scattering phase space at various pressures.

Conclusions

In conclusion, the pressure-tuned phonon transport properties of PbTe and PbSe were investigated thoroughly by first-principles calculations. The thermal conductivity of PbTe and PbSe shows an anomalous pressure-dependence, which originates from the nonmonotonic pressure-dependent phonon lifetime. The thermal conductivity in the NaCl phase is isotropic, while remarkable anisotropy of the thermal conductivity is observed in the Pnma phase. The anisotropy of thermal conductivity in the Pnma phase is ascribed to the anisotropic phonon group velocity component in different directions. The thermal conductivity of Pnma-PbTe in the c-axis direction is as low as 0.7 W/m·K under a pressure of 10 GPa and the thermal conductivity of Pnma-PbSe in the a-axis direction is also lower than in the other two directions. Compared with the NaCl phase, the thermal conductivities of Pnma-PbTe in the c-axis direction and Pnma-PbSe in the a-axis direction are hypersensitive to the nanostructure, which can be further reduced by nanostructuring conveniently. Combined with available study on power factor, it can be inferred that the optimal thermoelectric performance of NaCl-phase PbTe and PbSe achieves around 2–3 GPa. For the Pnma phase, making full use of the anisotropy of thermal conductivity is the key to achieving excellent thermoelectric performance. A high ZT value of about 1.2 is expected to achieve in Pnma-PbTe at room temperature under a pressure of 10 GPa. This study offers a simple method to further improve the thermoelectric performance of existing thermoelectric materials at room temperature by applying hydrostatic pressure.

Methods and Simulation Details

All the presented first-principles calculations were implemented in Vienna Ab initio Simulation Package (VASP)[52] within the framework of density functional theory (DFT) using the projected augmented wave (PAW) pseudopotentials[53] with generalized gradient approximation (GGA) of PBEsol[54] for electronic exchange–correlation. The 5d orbital of lead was included in the valance bands in all the calculations. We adopted 10–6 eV as the energy convergence threshold, and the crystal structures were fully relaxed until the residual force was smaller than 0.01 eV Å–1. The computational details in geometric structure optimizations are presented in Table S1. Both the plane wave cutoff energy and Monkhorst-Pack k-mesh were strictly tested for convergence to ensure sufficient calculation accuracy. The enthalpy of systems was adopted to estimate the stability of the structures, which was calculated from H = U + PV by setting the temperature to zero, where H, U, P, and V are enthalpy, internal energy, pressure, and volume, respectively.

The lattice thermal conductivity was computed by iteratively solving the phonon Boltzmann transport equation in the ShengBTE package.[55] The harmonic (second-order) interatomic force constants (IFCs) were obtained based on the density functional perturbation theory (DFPT) calculations with the Phonopy[56] code. The anharmonic (third-order) IFCs were calculated by the finite-displacement method as performed in the thirdorder py[55] code. The harmonic and anharmonic IFCs are the main input parameters for the calculations of the lattice thermal conductivity, and the computational details, such as the cutoff energy, the size of the supercell, the k-mesh, the force cutoff distance for anharmonic IFCs calculation, and the q-grid, are detailed in Table S2. All these parameters have been carefully verified to ensure the convergence of present calculations. The lattice thermal conductivity kLαβ is expressed aswhere α and β are the Cartesian directions, kB is the Boltzmann constant, T is the absolute temperature, V is the volume of the unit cell, N is the number of discrete q sampling in the Brillouin zone, λ denotes the phonon mode with wave vector q and polarization v, fλ0 refers to the phonon distribution at equilibrium, and ℏ is the reduced Planck constant. CV, λ, wλ, vλ, and τλ are the phonon volumetric specific heat, phonon frequency, phonon group velocity, and phonon lifetime, respectively. Atomic displacements affect the macroscopic dipole moment and induce the electromagnetic field, which could alter the IFCs near Γ point.[57] This phenomenon leads to LO–TO splitting and can be taken into account through non-analytical term correction. In this work, Born effective charges and dielectric constants were obtained by the DFPT method to consider the LO–TO splitting. In addition, the spin–orbit interaction (SOI) was not treated in this study. For the NaCl phase, it has been demonstrated that the influence of SOI on lattice vibrations and thermodynamic properties can be ignored.[42,50,58] For the Pnma phase, we performed test calculations and found that the lattice constants with and without SOI differ less than 0.1%. It was confirmed that the SOI plays a non-negligible role in the thermodynamics properties of Bi, whose lattice constant changes by 0.93% with the influence of SOI.[59] Compared with Bi, the difference of 0.1% in the lattice constant of the present Pnma-phase structure is quite small. The SOI usually has no significant influence on the thermophysical properties when the lattice constants differ by ∼0.1%.[50,60] In addition, the calculated phase transition pressures are in good agreement with the experimental data (as described in Section ), which shows that the energy of the system can be accurately predicted without considering SOI. Therefore, the SOI is ignored in the present work considering its small impact but with a large computational cost.