Monolayer SnI₂: An Excellent p-Type Thermoelectric Material with Ultralow Lattice Thermal Conductivity

Abstract

Using density functional theory and semiclassical Boltzmann transport equation, the lattice thermal conductivity and electronic transport performance of monolayer SnI₂ were systematically investigated. The results show that its room temperature lattice thermal conductivities along the zigzag and armchair directions are as low as 0.33 and 0.19 W/mK, respectively. This is attributed to the strong anharmonicity, softened acoustic modes, and weak bonding interactions. Such values of the lattice thermal conductivity are lower than those of other famous two-dimensional thermoelectric materials such as MoO₃, SnSe, and KAgSe. The two quasi-degenerate band valleys for the valence band maximum make it a p-type thermoelectric material. Due to its ultralow lattice thermal conductivities, coupled with an ultrahigh Seebeck coefficient, monolayer SnI₂ possesses an ultrahigh figure of merits at 800 K, approaching 4.01 and 3.34 along the armchair and zigzag directions, respectively. The results indicate that monolayer SnI₂ is a promising low-dimensional thermoelectric system, and would stimulate further theoretical and experimental investigations of metal halides as thermoelectric materials.

1. Introduction

With more than 60% of energy in the world lost in the form of waste heat, the thermoelectric system has attracted widespread attention, since it directly converts the waste heat to electric energy through the Seebeck effect. It has the advantages of small size, high reliability, no pollutants, and a feasibility in a wide temperature range. Such a system is widely used in aerospace exploration and industrial production, such as in space probes, thermoelectric generators, and precise temperature controls [1,2]. The converting efficiency of TE materials is ruled by the dimensionless figure of merit (ZT); ZT = S²σT/(κ_L+ κ_e), where S, σ, T, κ_L, and κ_e are the Seebeck coefficient, electrical conductivity, absolute temperature, lattice thermal conductivity, and electronic thermal conductivity, respectively. The electronic transport properties S, σ, and κ_e, have a complex coupling relationship, and are difficult to decouple, even though they can be modified via carrier concentration [3,4,5]. Historically, two aspects were established to enhance ZT: one aspect is to optimize carrier concentration by the band structure, engineered to enhance the power factor (σS²) [6,7,8,9], and the other aspect is to reduce the lattice thermal conductivity, via alloying and nanostructuring [10]. Alternatively, it is more attractive to seek materials with an intrinsic low lattice thermal conductivity (generally associated with complex crystal structures [11]), strong anharmonicity [12,13], lone pair electrons [14,15], and liquid-like behavior [16,17,18], etc.

The group IVA metal dihalides are candidates for semiconductor optical devices and perovskite solar cells, due to their excellent properties, such as the visible-range band gap and the thickness-dependent band structure [19,20,21,22]. However, the application of the Pb-based materials, such as the layered 2H-PbI₂, has been greatly limited by their toxicity and environmental unfriendliness [23,24,25]. In addition, the surface of the bulk SnI₂ is generally very rough and accompanied by many defects, which strongly scatters carriers [26]. Fortunately, its vdW monolayer has been experimentally realized, via molecular beam epitaxy [27]. The low dimensionality provides an effective conductive channel for carriers, and also suppresses phonon thermal transport [28]. Thus, monolayer SnI₂ is a preferable option over bulk SnI₂ and monolayer PbI₂ in use as a TE material, and deserves to be carefully explored in a systematic study.

In this work, combining first-principles calculations and the semiclassical Boltzmann transport equation, the TE properties of monolayer SnI₂ were systematically explored. Results show that the intrinsic ultralow lattice thermal conductivity originates from the strong anharmonicity, weak bonding, and softened acoustic modes. The Grüneisen parameter, phonons scattering phase space, and phonon relaxation time were calculated to understand the micro-mechanism of the phonon transports. The two quasi-degenerate band valleys for the valence band maximum (VBM) in its electronic band structure led to a p-type TE material. The maximum ZT value along the armchair and zigzag directions at 800 K reach 4.01 and 3.34, respectively, using optimal p-type doping. These results indicate that monolayer SnI₂ exhibits an extraordinary TE response, and is an ideal material for TE applications.

2. Computational Methods

The first-principles calculations were implemented in the Vienna ab initio simulation package (VASP, VASP.5.3, Wien, Austria) [29]. The generalized gradient approximation (GGA) [30,31] in the Perdew–Burke–Ernzerhof (PBE) [32] form was employed to deal with the exchange–correlation functional, with a cutoff of 300 eV on a 9 × 9 × 1 Monkhorst–Pack k-mesh. To screen the interactions between adjacent images, the length of the unit cell of 20 Å was used along the z direction. The geometry structure was fully relaxed, with a criterion of convergence for residual forces of 0.001 eV/Å, and the total energy difference converged to within 10⁻⁸ eV/Å. To obtain an accurate band gap and electronic transport performance, the Heyd–Suseria–Ernzerhof (HSE06) [33] method was employed.

The electronic transport properties of the monolayer SnI₂ were calculated by solving the semiclassical Boltzmann transport equation, utilizing the BoltzTraP code (Georg K. H. Madsen, Århus, Denmark) [34] with a dense 35 × 35 × 1 k-mesh. The constant relaxation time approach (CRTA) was used, since the relaxation time is not strongly dependent on the energy scale of k_BT, and has accurately predicted TE properties of multitudinous materials. In this work, the electrons relaxation time was calculated using the deformation potential (DP) theory [35], which considers the primarily acoustic phonon scatterings, but ignores effects of the optical phonons in the single parabolic band (SPB) model. Based on the rigid band approximation (RBA) [36] and CRTA, the transport coefficients S, σ, and κ_e can be obtained by:

$$S= \frac{e{k}_B}{\sigma }\int \Xi (\epsilon )(-\frac{\partial f_0}{\partial \epsilon })\frac{\epsilon -u}{k_{B}T}d\epsilon$$

(1)

$$\sigma =e^2\int \Xi (\epsilon )(-\frac{\partial f_0}{\partial \epsilon })\frac{\epsilon -\mu }{k_{B}T}d\epsilon$$

(2)

$${\kappa }_{\mathrm{e}}={\kappa }_0-\sigma S^{2}T=L\sigma T$$

(3)

$$\Xi =\sum {\nu }_k{\nu }_k{\tau }_k$$

(4)

where k_B, f₀, and Ξ(ε) are the Boltzmann constant, the Fermi–Dirac distribution function, and the transport distribution function, respectively. The calculation of the relaxation time τ was extremely difficult, due to the various complex scattering mechanism in the crystals. The relaxation time was calculated based on the DP theory in the SPB model, which considers the predominant scatterings between carriers and acoustic phonons in the low-energy region. In fact, the scattering matrix element (${\left|M(\overrightarrow{k},\overrightarrow{k’})\right|}^2$) of the acoustic phonons in the long-wavelength can be approximated as k_BTE₁²/C_ii, where C_ii and E₁ are the elastic and DP constants, respectively. Thus, for the 2D material, the carrier mobility and electrons relaxation time can be approached as [37,38]

$${\mu }_{2D}=\frac{e{h}^3{C}_{ii}}{8{\pi }^3{k}_{B}T{m}_{d}m*E_1{}^2}$$

(5)

$$\tau =\frac{um*}{e}$$

(6)

where h, m*, and m_d are the Planck constant, the effective mass along the transport direction, and the averaged effective mass, respectively.

The harmonic second-order interaction force constants (2nd IFCs) and phonon spectrum were calculated using the Phonopy package [39], using a 2 × 2 × 1 supercell with a 5 × 5 × 1 k-mesh using the finite-difference method [40]. The anharmonic third-order interaction force constants (3rd IFCs), which consider the interactions between the sixth-nearest-neighbor atoms, were obtained using the ShengBTE package (ShengBTE version 1.0.2, Wu Li, Grenoble, France; Phonopy version 2.11.0, Atsushi Togo, Sakyo, Japan) [41] with the same supercell. The lattice thermal conductivity and phonon transport properties were obtained using the self-consistent iterative solution of the Boltzmann transport equation, with a dense 36 × 36 × 1 mesh, which had a good convergence. Based on the Boltzmann transport equation, with the Fourier’s low of heat conduction, the matrix elements of the phonon thermal conductivity can be expressed by [42,43]

$${\kappa }_{\mathrm{p}.\alpha \beta }=\frac{1}{N_q}{\displaystyle \sum _{\lambda }{c}_{\lambda }{v}_{\lambda ,\alpha }{v}_{\lambda ,\beta }{\tau }_{\lambda }}$$

(7)

where α and β are the Cartesian indices, N_r is the total number of q-points sampled in the first Brillouin zone, and c_λ, v_λ, and τ_λ are the mode-specific heat capacities, phonon group velocity, and the relaxation time, respectively. Here, the phonon thermal properties were calculated based on the 3rd IFCs, ignoring the fourth- and higher-order terms. This strategy described the phonon behavior of the most anharmonic materials [44,45]. To define the effective thickness of two-dimensional (2D) materials, the summation of interlayer distance and the vdW radii of the outermost surface atoms was adopted [44].

3. Results and Discussion

3.1. Geometry and Electronic Structure

Monolayer SnI₂ crystallizes in a hexagonal lattice with space group P-3m1 (164), as shown in Figure 1. The optimized lattice parameter is 4.57 Å, which is in good agreement with the previous experimental data of 4.48 Å [27]. The structure is analogous to H-MoS₂ [46], consisting of three layers, with Sn atoms as the middle layer, and I atoms as the upper and lower atomic layers. The electron localization function (ELF) provides a deeper insight to characterize the nature of, and strengthen, the chemical band. Figure 1c shows the calculated three-dimensional (3D) ELF map (isosurface level of 0.97). The ELF around I is in the shape of a “mushroom”, suggesting the existence of the lone pair electrons. Moreover, the electron sharing is better visualized by the 2D ELF map in Figure 1d. The interstitial electrons between Sn and I are close to I atoms, and the value of the area between them is 0.5, which shows the characteristics of a free electron gas and indicates a weak bonding between Sn and I atoms. The electronic repulsion between the lone pair electrons and the Sn–I bonding electrons results in strong anharmonicity, such as in CuSbS₂ [14]. The complex structure and lone pair electrons are also beneficial to its low lattice thermal conductivity.

The electronic structure plays a crucial role in characterizing the electronic transport properties. As shown in Figure 1b, the electronic band structures obtained from the PBE and HSE06 hybrid function potentials are presented. They exhibit analogous band structures, except for a more accurate bandgap (2.71 eV) that approaches the experiment value (2.9 eV) [27] for the latter structure. The conduction band mainly originates from the p orbitals of the Sn atom and the p orbitals of the I atom, whereas the valence band consists of the s orbital of the Sn atom and the p orbitals of the I atom. Below the VBM, there are two quasi-degenerate band valleys along the Γ–M and Γ–K directions with an energy difference of ~0.03 eV, which is far less than those in SnTe (~0.35eV) [47] and PbTe (~0.15 eV) [1]. The multi-energy valley enhances the TE performance and is verified in many materials [48,49]. However, the behavior of band valleys degenerate do not exist for the conduction band minimum (CBM), which is located at the Γ point. Hence, it is expected that the TE performance of the p-type could be superior to that of the n-type.

3.2. Electronic Transport Properties

All the parameters for electronic transport, calculated according to the DP theory, are tabulated in

Table 1. The calculated DP constant E₁, elastic constant, effective mass, carrier mobility, and relaxation time along the zigzag and armchair directions in monolayer SnI₂ at 300 K.

Direction	Type	E1 (eV)	Cii (J/m2)	m* (m0)	u (cm−2V−1S−1))	τ (fs)
Zigzag	e	−4.44	17.92	0.59	42.52	17.41
	h	−4.43	17.92	0.73	40.16	16.72
Armchair	e	−4.25	17.42	0.84	30.48	15.17
	h	−4.49	17.42	0.68	35.14	13.60

. Based on the reasonable relaxation time, all the electronic transport coefficients, Seebeck coefficient S, electrical conductivity σ, and power factor (PF), as a function of temperature at the corresponding optimal carrier concentration for p-type SnI_2, range from 1 × 10¹³ to 1 × 10¹⁴ cm⁻², and are presented in Figure 2a–c. For comparison, those for n-type SnI₂ at the same condition are also plotted in Figure 2d–f. Here, the tiny anisotropy characteristics are ignored.

Increasing the carrier concentration, the chemical potential enters in the deeper energy levels for both p- and n-types of SnI₂, resulting in a decreased Seebeck coefficient at the same temperature. It is clear that the values under p-type doping are always far larger than those under n-type at the same condition, which can be attributed to the multi-band character near the VBM, as shown in Figure 1b. For example, the absolute values of the S under p- and n-types of doping at 700 K are 439 and 146 uV/K, respectively, for a doping concentration of 1 × 10¹³ cm⁻². In addition, the trend of S under p-type doping is opposite to that under n-type. The former decreases while the latter increases upon heating.

Compared with S, the electronic conductivity σ exhibits a different behavior, due to the complete relationship between them, as Equations (1) and (2) show. Interestingly, in all cases, the electrical conductivities for p-type SnI₂ are always far less than those for n-type. For example, the values under p- and n-type doping are 0.46 and 2.43 S/cm⁻², respectively, at 700 K for a doping concentration of 1 × 10¹³ cm⁻². At 800 K, the electrical conductivity under p-type (n-type) increases from ~0.67 S/cm⁻² (~2.24 S/cm⁻²) for 1 × 10¹³ cm⁻² to 6.35 S/cm⁻² (9.43 S/cm⁻²) for 1 × 10¹⁴ cm⁻².

Ultimately, the PF decouples the complete relationship between the Seebeck coefficient and electrical conductivity. It is clearly seen that the p-type SnI₂ possesses significantly higher PF values than those of the n-type system. This is in good agreement with the previous analysis for the electronic structure. At 800 K, the PF for p-type increases from ~0.27 mW/mK² for 1 × 10¹³ cm⁻² to 0.62 mW/mK² for 1 × 10¹⁴ cm⁻². For the n-type system, however, the PF decreases from ~0.15 mW/mK² for 1 × 10¹³ cm⁻² to 0.04 mW/mK² for 1 × 10¹⁴ cm⁻². Thus, it is expected that the p-type SnI₂ possesses a more excellent performance than that of the n-type. According to the Wiedemann–Franz law, κ_e = LσT, where L is the Lorenz number, the classical value L = (πk_B)²/3e² ≈ 2.44 × 10⁻⁸ WΩK⁻² is adopted. This value meets the result of κ₀-σS²T [50]. Thus, the electronic conductivity is proportional to the electronic thermal conductivity. The results in the present work also obey this rule.

3.3. Phonon Transport Properties

As shown in Figure 3a, the calculated total κ_L is very low in the wide temperature of 300–800 K. Remarkably, the low room temperature phonon thermal conductivities of 0.33 and 0.19 W/mK along the armchair and zigzag directions, respectively, are fundamentally lower than those previously reported values for 2D MO₃ (1.57 W/mk) [51], SnSe (2.77 W/mK) [13], and CaP₃ (0.65 W/mK) [52]. Through analyzing the contributions of the acoustic phonon branches along the in-plane (TA and LA) or out-of-plane (ZA), as well as the optical phonon modes to the total κ_L, results show that the main contribution to the κ_L is from the ZA mode. In the following, we reveal the origins of such ultralow phonon conductivity, and present a comprehensive analysis to support this result.

The phonon dispersion curves are presented in Figure 3b. With three atoms in its primitive cell, there are three acoustic and six optical phonon modes for monolayer SnI₂. It is dynamically stable, since no imaginary frequency is observed. Near the long wave limit, the TA and LA branches are in linear trend, whereas the ZA branch exhibits a quadratic trend. These features are typical for 2D materials, and can be explained with the elastic theory of thin plate [53]. It is clearly seen that a narrow phonon gap of about 0.1 THz separates the phonon modes into the acoustic phonon part (0~1.35 THz) and the optical phonon part (1.45~4.10 THz). The cutoff acoustic phonon frequency, which is as low as 1.35 THz, is lower than those for SnSe (1.6 THz) and SnS (1.9 THz) [13]. The low-lying acoustic modes, as well as the soft mode for TA near the M point, imply the weak bonding between Sn and I atoms, consistent with the previous analysis. From the corresponding phonon density of states (PhDOS), the contributions from the I atomic vibrations are apparently larger than those from Sn, within the region of 2.8 to 4.10 THz. Both Sn and I evidently contribute in a wide energy range, implying the nature of covalent bonding [12]. In addition, it is recalled that the decoupling of the in-plane and out-of-plane phonon modes results in ultrahigh phonon thermal conductivity for graphene, due to its one atom plane nature [53,54]. However, such full decoupling behavior should not be observed for a finite thickness 2D system, such as the case of the present studied monolayer SnI₂.

To explain the anomalous thermal transport behavior of monolayer SnI₂, the cumulative lattice conductivity, and their derivatives, with respect to frequency at 300 K are calculated and presented in Figure 3c. Clearly, the κ_L is mainly caused by phonons of the cutoff acoustic part (the shadow region < 1.35 THz). Specifically, the contributions from ZA, TA, LA, and the optical branches are 51.21%, 15.14%, 10%, and 23.54%, respectively. For graphene, the low frequency (0–5 THz) also dominates the κ_L and the ZA mode, and contributes 75% to κ_L [54]. Comparing with graphene, the structure of the monolayer SnI₂ lacks mirror symmetry. Thus, the mirror symmetry does not reflect whether the ZA mode dominates the thermal transport [55].

Generally, there are two factors that dominate the κ_L: (i) anharmonic interaction matrix elements, and (ii) the inverse of phonon phase space volume. The Grüneisen parameter γ is generally employed to quantify the strength of anharmonicity [41]. Figure 3d shows the calculated γ of the ZA, TA, LA, and optical modes of the SnI₂ monolayer. As a large |γ| implies strong anharmonicity, it can be seen that the ZA mode exhibits giant anharmonicity. The large |γ| of this out-of-plane mode means that the anharmonicity of the bonding between Sn and I atoms in the vertical direction of the monolayer plate is strong. The phonon scattering phase space reveals all available scattering processes, which are ruled by the energy and (quasi)momentum conservation. As shown in Figure 3e, the total scatting phase space of the ZA, TA, and LA modes are 1.55 × 10⁻³, 1.41 × 10⁻³, and 1.35 × 10⁻³, respectively, which confirms more abundant scattering channels of the ZA mode, than those of the TA and LA modes. The larger the phonon phase space, the greater the contribution to the κ_L, thus, validating the decreasing contributions to κ_L from the ZA, TA, and then LA modes. The phonon relaxation time provides a deeper microcosmic insight to understand the ultralow κ_L of SnI₂. Compared with the monolayer SnP₃ [38], monolayer SnI₂ possesses a shorter phonon relaxation time, implying an ultralow κ_L.

3.4. Thermoelectric Figure of Merit

By combining the phonon and electron transport coefficients, the ZT of SnI₂ under p- and n-types of doping as functions of temperature and carrier concentration are presented in Figure 4. Owing to the calculated thermal conductivity, along the zigzag direction is slightly larger than the armchair direction, and it is expected that the ZT along the armchair direction is higher than that along the zigzag direction. In addition, the p-type SnI₂ is obviously superior to the n-type, since the two quasi-degenerate band valleys in VBM leads to a larger Seebeck coefficient. Results show that monolayer SnI₂ is thermally stable up to 800 K, by performing ab initio molecular dynamic (AIMD) simulations (see Figure S1). The values of ZT are as high as 4.01 and 3.34 along the armchair and zigzag directions, respectively, around concentrations of 1.0 × 10¹³ and 1.2 × 10¹³ cm⁻², at 800 K. Such large values are better than the experimental value of 2.6 for the well-known TE material SnSe along the specific axis at 925 K [3]. In addition, the doping carrier concentrations at such levels have been realized experimentally in monolayer MoS₂ [56]. Therefore, such a high TE value for monolayer SnI₂ is possible, and indicates its excellent potential TE performance.

To clearly compare the TE performances of the monolayer SnI₂ with some famous TE materials, the κ_L and the max ZT are listed in

Table 2. κ_L at 300 K and the max ZT at corresponding maximum thermodynamic temperature for monolayer SnI_2, as well as some typical TE materials.

Material	κL (W/mK)	ZT
SnI2	0.26	4.01 (800 K)	This work
SnS	1.5	1.00 (750 K)	Ref. [13]
SnSe	0.6	1.50 (750 K)	Ref. [13]
SnP3	4.97	3.46 (500 K)	Ref. [38]
SnSe	1.12	0.85 (900 K)	Ref. [48]
LaCuOSe	0.84	2.71 (900 K)	Ref. [50]

. According to these results, the monolayer SnI₂ is an emerging candidate for TE devices, due to its ultrahigh ZT. As seen, the κ_L of monolayer SnI₂ is very low, almost a twentieth of the monolayer SnP₃ [38], and is greatly lower than those of tin selenide and tin sulfide [13,48]. Such low value of the κ_L for monolayer SnI₂ make it a typical 2D TE system. After all, the electronic transport properties can be easily modulated in experiments, while the lattice thermal conductivity is very difficult to change. In addition, the κ_L values at the level of 0.1-0.5 W/mK in experiments are very few. To the best of our knowledge, the complex systems of CsAg₅Te₃ (0.18 W/mK) [57] and CsCu₅Se₃ (0.4~0.8 W/mK) [58], as well as the bulk superlattice material Bi₄O₄SeCl₂ (0.1 W/mK) [59], are typical low κ_L TE materials. Overall, the low thermal conductivity, as well as the high ZT, make monolayer SnI₂ a good TE material for low-dimensional devices.

4. Conclusions

In summary, the TE properties of monolayer SnI₂ were studied using DFT and the semi-classical Boltzmann transport equation. The results indicate that this 2D material possesses intrinsically ultralow lattice thermal conductivity. Its strong anharmonicity, weak bonding, and softened acoustic branches greatly suppress the phonon transport, and result in an ultralow κ_L of 0.33 and 0.18 W/mK at 300 K along the zigzag and armchair directions, respectively. The p-type SnI₂ possesses a superior electric transport performance than the n-type one, due to the two quasi-degenerate band valleys in its VBM. The ZT at 800 K under p-type doping is as high as 4.01 along the armchair direction. Collectively, these results indicate the great advantages of monolayer SnI₂ for converting heat energy with high efficiency at high temperatures. Generally, when the ZT value of a material exceeds 1.0, it is considered as an ideal TE material. Therefore, the high ZT of monolayer SnI₂ demonstrates it as an emerging candidate for TE applications.