Introduction

As a typical two-dimensional (2D) layered material, transition metal dichalcogenides (TMDCs)1,2,3 are promising high-performance thermoelectric materials due to their high electrical conductivity and low thermal conductivity, which are mainly used in the fabrication of wearable heating/cooling devices and thermoelectric generators. Moreover, TMDCs have received extensive attention in recent decades4,5,6 due to semiconducting properties and possess excellent mechanical properties, physical properties and chemical stability.

Increasing the thermoelectric efficiency of TMDC materials has been an ambitious goal for decades. It is found that there are many factors affecting the properties of thermoelectric materials, such as the structure, number of layers and external stress. There are two main stable structures of TMDCs at room temperature. One is the 2H phase with a triangular prism coordination structure with the space group \(P\overline{6} m2\), and the other is the 1T phase with an octahedral coordination structure, with space group \(P\overline{3} m1\). The most common TMDC materials with the 2H phase structure are MoS2, MoSe2, WS2 and WSe2, and the materials with 1T phase structure are HfS2, HfSe2, ZrS2 and ZrSe2. By comparing the band structure of the 1T and the 2H phase structure, it can be found that the band gap of the 1T phase structure is generally smaller and the 2H phase is larger.7,8 It is well known that a large band gap is not conducive to optimizing the thermoelectric properties of materials, because a large amount of doping is required to achieve the best performance of the material, and a large band gap usually means a lower carrier concentration. Meanwhile, a larger band gap may mean a larger Seebeck coefficient, but also a smaller conductivity, which may ultimately lead to a smaller power factor and poorer thermoelectric performance. Yumnam et al. recently confirmed that the lattice thermal conductivity of Zr- and Hf-based bulk TMDCs is lower than that of Mo- and W-based bulk TMDCs due to the strong coupling of low-frequency optical and acoustic modes.9 For instance, the lattice thermal conductivity of ZrS2 monolayer is only 3.29 W/mK at 300 K.10 But the lattice thermal conductivities of MoS2 and WSe2 monolayers are as high as 100 W/mK and 40 W/mK.8,11 Likewise, the lattice thermal conductivities of ZrSe2 and HfSe2 monolayers are 1.2 W/mK and 1.8 W/mK at 300 K.7 This may indicate that the thermoelectric performance of the 1T phase structure is better than that of the 2H phase structure in TMDCs.

With the development of fabrication technology, monolayer, bilayer and few-layer 2D materials can be successfully prepared. Recent studies on TMDCs have shown that the thermoelectric properties of 2D materials with different layer thicknesses will be different.3,12 Wickramaratne et al. reported that the thermoelectric properties of most MoS2, MoSe2, WS2 and WSe2 with bilayer structures are better than other few-layer structures and bulk structures.12 The band structure and density of states of 2D TMDCs change when increasing from monolayer to multilayer due to the interlayer coupling effect. The Seebeck coefficient of monolayer MoS2 is 2055.4 μV/K, while the S of its bilayer structure increases to 2344.9 μV/K.13 Meanwhile, the work by Wickramaratne et al. also showed that the power factors of an n-type doped MoS2 monolayer and WS2 monolayer are 0.13 W/mK2 and 0.24 W/mK2 at 300 K, and the power factors increase to 0.14 W/mK2 and 0.28 W/mK2 as the monolayer increases to a bilayer.12 Finally, the maximum ZT value of MoS2 increased from 0.967 (monolayer) to 0.974 (bilayer),13 and the maximum ZT value of MoS2 increased from 1.52 (monolayer) to 1.98 (bilayer).12 Yun et al.14 and Goyal et al.15 demonstrate that the thermoelectric characteristics of Bi2Te3 and Bi2Se3 increase with decreasing thickness, which is due to an increase in ZT produced by the form of the density of states, which enhances the power factor. Moreover, the bilayer BP has substantially lower lattice thermal conductivity than the monolayer 300 W/mK, which is roughly 24 W/mK, as a result of the van der Waals (vdW) interaction. As a result, at 1200 K, the BP bilayer obtains an outstanding ZT of around 1.8.16 Hence, thermoelectric properties are significantly influenced by weak coupling contact. However, the current studies on the thermoelectric properties of TMDCs focus on the 2H phase structure and lack the research on the thermoelectric properties of the 1T phase structure.

Furthermore, tensile strain has been found to also affect the thermoelectric properties of 2D TMDCs. Tseng et al.17 found that applying tensile strain to TiSe2 monolayer and HfSe2 monolayer changed the band structure and tuned band degeneracy and induced changes in electron transport properties. As the tensile strain increased from 0% to 8%, the power factors17 of TiSe2 monolayer and HfSe2 monolayer increased from 2.92 W/mK2 and 1.45 W/mK2 to 3.40 W/mK2 and 2.80 W/mK2, respectively. Finally, the ZT values of TiSe2 monolayer and HfSe2 monolayer are increased from 0.15 and 1.19 to 1.64 and 1.9917 through the tensile strain. Similar results are obtained by Qin et al.18 by applying tensile strain to a ZrSe2 monolayer. As the tensile strain increased from 0% to 7.5%, the maximum ZT values for p-type and n-type doping increase from 2.13 and 4.26 to 3.84 and 4.58,18 respectively.

Hf-based materials of transition metal dichalcogenides generally have lower lattice thermal conductivity due to the heavier Hf atom and the coupling between the acoustic branch and the low-frequency optical branch,7,19 which is advantageous for obtaining high thermoelectric performance.20,21,22 Studies have demonstrated that HfS2 has a high carrier mobility compared to other two-dimensional semiconductor materials (such MX2 (M = Mo, W, Sn; X = S, Se) of other transition metal dichalcogenides).23 This suggests that it has outstanding electrical transport capabilities, which results in a high power factor. Secondly, HfS2 has low thermal conductivity. Owing to its excellent characteristics, it has good thermal conductivity among many materials of MX2 (M = Cr, Mo, W, Ti, Zr, Hf; X = O, S, Se, Te). In summary, as our research object, HfS2 has excellent thermoelectric properties among many materials.9,20,24

Recently, based on the successful preparation of few-layer HfS2 by Wang et al.2 and combined with the findings of Wickramaratne et al.,12 we chose to study the thermoelectric properties of HfS2 bilayers in this paper. In pursuit of higher thermoelectric properties, we subjected the HfS2 bilayer to biaxial tensile strain and calculated its ZT value.

Computational Methods

Based on density functional theory, we use the generalized gradient approximation (GGA) within Perdew–Burke–Ernzerhof (PBE) formulation exchange–correlation interaction to optimize the HfS2 bilayer in the Vienna Ab initio Simulation Package (VASP).25,26,27 In addition, to avoid the influence of interlayer force, Grimme’s DFT-D3 method is used to carry out van der Waals (vdW) correction for the calculation method.28 The plane-wave energy cutoff is chosen as 600 eV for the HfS2 bilayer, and the Monkhorst–Pack k mesh is 21 × 21 × 1. Geometrical structures were relaxed until the force on each atom was less than 0.001 eV Å−1. In order to avoid the interaction with periodic images, a 26 Å thickness of vacuum slab is added in the direction of the vertical 2D plane for HfS2 bilayer. In the subsequent calculation of the band structure and electron transport properties, the Heyd–Scuseria–Ernzerhof (HSE) screened nonlocal exchange functional of the generalized Kohn–Sham scheme was used to correct the results.29

Thermoelectric transport properties including Seebeck coefficients (S) and the electrical conductivity over relaxation time (σ/τ) are calculated by using the electron Boltzmann transport theory within relaxation time approximation (RTA) as implemented in the BoltzTrap software package.30 A large 51 × 51 × 1 k-point Monkhorst–Pack grid was used in the calculation of the band energies, resulting in the final transport properties. Finally, the relaxation time (τ) is estimated by using Bardeen–Shockley deformation potential theory based on the effective mass approximation:

$$ \tau = \frac{{\mu m^{*} }}{e}, $$
(1)
$$ \mu = \frac{{e\hbar^{3} C_{2d} }}{{k_{{\text{B}}} Tm^{*} m_{d} E_{{\text{l}}}^{2} }}, $$
(2)

where μ, C2D, and El are the carrier mobility, the effective elastic modulus and the deformation potential constant, respectively, and \(m^{*} = \hbar^{2} \left[ {\partial^{2} E/\partial k^{2} } \right]^{ - 1}\) is the effective mass of the carrier, and \(m_{d} = \sqrt {m_{x}^{*} m_{y}^{*} }\) is the average effective mass.31,32 The electronic thermal conductivity was calculated by the Wiedemann–Franz law:

$$ \kappa_{e} = L\sigma T, $$
(3)

where L is the Lorenz number, and we obtain a sufficiently accurate value through a simple equation:33

$$ L = { 1}.{5 } + {\text{ exp }}\left[ { - \left| S \right|/{116}} \right] $$
(4)

where S is the Seebeck coefficient in μV/K and L is in 10−8 WΩK−2.

The lattice thermal transport was obtained by solved the phonon Boltzmann transport equation as implemented in ShengBTE.34 The phonon dispersion and second-order interatomic force constants (IFCs) were obtained by using the finite displacement method as implemented in the PHONOPY code from a 3 × 3 × 1 supercell with 5 × 5 × 1 k-point Monkhorst–Pack grid based on the relaxed unit cell.35 The third-order force constants were calculated and extracted using the same supercell and k mesh as the second-order force constants. The formula of κl can be expressed as:

$$ \kappa_{\alpha \beta } = \frac{1}{{{\text{SH}}}}\mathop \sum \limits_{\lambda } C_{\lambda } \nu_{\lambda \alpha } \nu_{\lambda \beta } \tau_{\lambda } $$
(5)

in which S is the surface area, H is determined by \(H_{{{\text{HfS}}_{2} }} = 2(h_{{{\text{HfS}}_{2} }} + r_{{\text{S}}} ) + d \) for HfS2 bilayer, where \(h_{{HfS_{2} }}\) is the height difference of the top S and the bottom S atom and d, \(r_{{\text{S}}}\) is the distance between the two layers and van der Waals radii of the S atom.36,37

Results and Discussion

Lattice Parameters and Structural Stability

First, we obtained the geometry and lattice parameters of the HfS2 bilayer by lattice optimization. The top and side views of the HfS2 bilayer structure are shown in Fig. 1, where d means the distance between the two layers. As shown in Table I, the lattice constant of the HfS2 bilayer is 3.616 Å and the d is 3.006 Å. This lattice constant is in good agreement with the values reported in previous literature.4,38,39 As shown in Fig. S1 of the supplemental material, we ran ab initio molecular dynamics (AIMD) simulations of the bilayer HfS2. Given that it maintains its stability at high temperatures beyond 10 ps, bilayer HfS2 is expected to mature into a medium- to high-temperature thermoelectric material.

Fig. 1
figure 1

(a) Top and (b) side views of HfS2 bilayer.

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Table I Comparison of the lattice constants of the HfS2 bilayer calculated in this paper with the lattice constants (Å) of bulk and monolayer HfS2 in the literature
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Based on the above optimized structure, we applied a series of in-plane biaxial tensile strains to the HfS2 bilayer, defining the strain degree as

$$ \varepsilon = \left( {a - a_{0} } \right)/a_{0} \times 100\% $$
(6)

where a0 and a denote the unstrained cell parameter and the strained cell parameter.

Biaxial tensile strains from 0 to 7% are considered. We calculated the phonon dispersion curve to verify the structural stability of the HfS2 bilayer in the range of 0–7% strain. As shown in Fig. 2, the phonon dispersion curve has no imaginary frequency, which means the structure is stable.

Fig. 2
figure 2

The calculated phonon spectra of the HfS2 bilayer under different biaxial strains, (a) 0%, (b) 1%, (c) 2%, (d) 3%, (e) 4%, (f) 5%, (g) 6%, (h) 7%.

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Electronic Transport Properties

In order to analyze the effect of strain on the electronic band structure of the HfS2 bilayer, the electronic band structures, total density of states and partial density of states of the Brillouin zone along the K–Γ–M–K path are calculated based on the HSE06 method, as shown in Fig. 3, and the HfS2 bilayer at strains from 0% to 7% behaves as an indirect-band-gap semiconductor. The conduction band minimum (CBM) and valence band maximum (VBM) of the unstrained HfS2 bilayer are located at the high symmetry point M point and Γ point, respectively, while at the Γ point, an obvious valley degeneracy that does not exist in the HfS2 monolayer is found. In addition, splitting of the band is also found at the M point, and the energy difference between the two conduction band valleys is about 0.07 eV. According to the research, when the energy difference between the band valley is less than a certain value (compared with kBT, where kB is the Boltzmann constant), it can be considered to achieve an effective degeneracy,15,40,41,42 and the thermoelectric properties of the material are affected.15,41,43 Moreover, the band gap gradually increases with the increase of strain. The band gap is 1.92 eV at 0% strain and the band gap is 2.37 eV at 7% strain. At 3% strain, the VBM moves from the high symmetry point Γ to between K and Γ, and the valence band top shifts again to the left in the Brillouin zone as the strain increases to 5%. Finally, the band structures are calculated in order to reveal the effect of the energy difference between the band valleys on the thermoelectric performance, as shown in Fig. 3a, where I and II denote the conduction band valley near the Fermi level (0 eV), and III, IV and V represent the valley of the valence band near the Fermi level, respectively. Furthermore, the energy difference between band valleys I and II (III and IV) is expressed as ΔC (ΔV), namely ΔC = EI–EII, ΔV = EIV–EIII, and the relationship of ΔC and ΔV as a function of strain is shown in Fig. 4. ΔC decreases gradually with the increase of strain, while ΔV first decreases and then increases, and when the strain reaches 3%, ΔV reaches the minimum value of −0.017 eV. Therefore, the valleys of the valence band at III, IV and V are degenerate and the degeneracy is maximum at 3% strain. ΔV becomes larger as the strain increases from 3% to 5%, but as the strain continues to increase to 6% and 7%, ΔV tends to decrease again. But at the same time, it can be found that when the strain increases to 5%, the band structure at Γ begins to split, and as the strain continues to increase, the splitting degree increases. Therefore, when the strain increases from 5% to 7%, ΔV still decreases, but the band degeneracy decreases.

Fig. 3
figure 3

The band structure of HfS2 bilayer under different biaxial strain, (a) 0%, (b) 1%, (c) 2%, (d) 3%, (e) 4%, (f) 5%, (g) 6%, (h) 7%. The two conduction band valleys near the Fermi level (0 eV) are denoted by I and II, respectively. The three valence band valleys near the Fermi level are denoted by III, IV and V, respectively. ΔC (ΔV) represents the energy difference between I and II (III and IV).

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Fig. 4
figure 4

Band gap (Δ) and energy difference between the conduction and valance band valleys (ΔC and ΔV) as a function of the applied biaxial strain.

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Such a trend can also be seen from the variation function of the electronic density of states with strain, as shown in Fig. 5. We can find that the valence band of the HfS2 bilayer is mainly contributed by the d orbital of the Hf atom, while the conduction band is contributed by the p orbital of the S atom. When the strain reaches 3%, the slope of the electron total density of states (TDOS) at the VBM also reaches a maximum, which is due to the maximum degeneracy at the VBM. The slope of TDOS at the CBM increases gradually with the strain, and the slope of TDOS reaches the maximum at 7% strain. A larger slope of the density of states means a higher Seebeck coefficient (S).44

Fig. 5
figure 5

The density of states of the HfS2 bilayer at 0%, 3%, 4% and 7% strain.

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Next, the electrical transport properties of the HfS2 bilayer under different strains and obtained τ by deformation potential theory (DP) are calculated (as shown in Table II). Ultimately, the relationship between the S, electrical conductivity (σ), power factor (PF) and the electronic thermal conductivity (κe) with the carrier concentration under different strains are obtained as shown in Fig. 6. As shown in Fig. 6a, the bilayer HfS2 Seebeck coefficient increases as the degree of strain increases for p-type, and after peaking at 3% strain, the Seebeck coefficient begins to decrease with increasing strain. The difference is that the S of n-type doping increases gradually with strain, and the S is maximum at 7% strain. We can find that the change of the S is consistent with the change trend of the slope of the TDOS, which also shows that the improvement of the band degeneracy can effectively increase S. It can be seen from Fig. 6b that the σ of n-type doping is generally higher than that of p-type doping. However, the σ of n-type doping is smaller than that of p-type when the strain is below 3% at low doping concentrations. The smaller σ of n-type doping is due to the larger electron effective mass and lower carrier mobility.

Table II The effective mass (m*), average effective mass (md), elastic modulus (C2D), deformation potential constant (El), electron and hole mobility (μ) and relaxation time (τ) of HfS2 bilayer at 0%, 1%, 2%, 3%, 4%, 5%, 6% and 7% biaxial strain
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Fig. 6
figure 6

The calculated electronic transport coefficients ((a) S, (b) σ, (c) PF, and (d) κe) as a function of carrier concentration for both p-type and n-type HfS2 bilayers at different biaxial strain.

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An evaluation of the thermoelectric performance of a material does not unilaterally consider the S or the σ, but the PF obtained by coupling S and σ. The biaxial strain affects the Seebeck coefficient and conductivity of the bilayer HfS2, as shown in Fig. 6c, but it also significantly alters PF. For the p-type, the S of the p-type is higher than the n-type when the strain is more significant than 3%; however, the PF of the n-type is greater than the p-type due to the high σ. Moreover, for p-type HfS2 bilayers, taking high carrier concentration as an example, PF reaches the peak of roughly 40 mW/mK2 at 2%, while 3% PF is also relatively high. The pattern for PF is similar to that of S in that it increases initially before declining. For the n-type, although S gradually increases with strain, σ also presents a decreasing trend. The PF of the HfS2 bilayer increases first and then decreases and reaches the peak at 3%, about 130 mW/mK2, which is about 1.3 times higher than 0% (98 mW/mK2). In addition, it is worth mentioning that when the strain is applied beyond a particular limit, PF will be smaller than the unstrained case, which is related to the mutual balance between the increased S and the decreased σ. This shows that optimizing the power factor of the HfS2 bilayer does not require maximizing either S or σ, but rather both achieve appropriate values.

Thermal Transport Properties

In this section, we discuss the effect of strain on thermal conductivity (κ). First, in order to obtain a more accurate κe, we obtained L under different doping concentrations through Eq. 4 as shown in Tables III and IV. Then, the variation function of κe with doping concentration can be obtained according to Eq. 3, as show in Fig. 6d. We can find that the variation trend of κe under different strains is basically consistent with the variation trend of σ.

Table III Lorentz constants of the p-type HfS2 bilayer at different biaxial strain
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Table IV Lorentz constants of the n-type HfS2 bilayer at different biaxial strain
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Next, the phonon transport properties at different strains are calculated. The lattice thermal conductivity (κl) of HfS2 bilayer under different strains as a function of temperature is show in Fig. 7. The κl not only decreases with the increase of temperature but also decreases gradually as the increase of strain. Note that we investigated the convergence of lattice thermal conductivity and the nearest neighbor number and discovered that employing three nearest neighbors is enough to get the lattice thermal conductivity to converge, as shown in Supplementary Material Fig. S2, which shows the accuracy of the calculation findings. Figure 2 shows the phonon dispersion curves under different strains, and it can be found that the acoustic branch splits at the Γ point, which is a phenomenon that does not exist in the phonon dispersion curves of the HfS2 monolayer. The reason for this phenomenon is that the phonon scattering is affected by the interlayer force, which leads to a wider phonon scattering and reduces κl. Meanwhile, it can be found that the frequencies of both the acoustic branch and the optical branch gradually decrease with the strain. A lower phonon frequency often means a lower phonon group velocity, which may result in a smaller κl.44 The phonon group velocity and phonon lifetime as a function of phonon frequency are given in Figs. 8 and 9 at 0%, 3%, 4% and 7% strain, respectively. Figure 8 shows that the phonon group velocities of both the acoustic and optical branches gradually decrease with strain. At 7% strain, the phonon group velocity of the acoustic branch is significantly reduced in the low-frequency region, which indicates that the phonon transmission is suppressed. It can be seen from Fig. 9 that the phonon lifetime, especially the phonon lifetime of the acoustic branch phonon is gradually decreasing with the increasing strain. The progressive enhancement of phonon scattering and the shortening of phonon lifetime also tend to imply a smaller κl.

Fig. 7
figure 7

The calculated lattice thermal conductivity of HfS2 bilayer as a function of temperature at different biaxial strain.

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Fig. 8
figure 8

Magnitude of phonon group velocity as a function of frequency in the full BZ under 0%, 3%, 4% and 7% biaxial strain.

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Fig. 9
figure 9

Mode-dependent phonon relaxation times as a function of frequency in the full BZ under 0%, 3%, 4% and 7% biaxial strain.

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Combining the obtained electronic thermal conductivities and lattice thermal conductivities, we obtain κ as a function of doping concentration for different strains at 300 K, as shown in Fig. 10. We can see from Fig. 10 that κ is decreasing with strain.

Fig. 10
figure 10

The calculated total thermal conductivity of HfS2 bilayer as a function of carrier concentration at different biaxial strain.

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Figure of Merit ZT

Combining the obtained PF and κ, \(ZT = S^{2} \sigma T/\left( {\kappa_{{\text{e}}} + \kappa_{l} } \right)\) as a function of doping concentration under different strains at a temperature of 300 K is finally obtained, as shown in Fig. 11. For the p-type HfS2 bilayer, the PF is relatively high when the strain is small, but the ZT value is relatively high when the strain is large. This is because the PF changes very little under different strains, but at 6% and 7% strain, κ decreases greatly, resulting in an increase in ZT. At 6% strain, the maximum ZT value is 1.83, the PF is 13.81 mW/mK2 and κ is 2.27 W/mK at a hole concentration of 1 × 1013 cm−2. Conversely, for n-type doping, the PF is 69.87 mW/mK2, κ is 6.36 W/mK and the maximum ZT value is 3.29 at 3% strain and electron concentration of 6 × 1012 cm−2. Although the κ of n-type doping is much higher than that of p-type at the optimal doping concentration, the PF is much higher than that of p-type. Moreover, the PF of n-type doping at 3% strain is much higher than that under other strains, which means that the main factor affecting the change of ZT value is PF rather than κ. For p-type doping, the opposite is true. Because the PF is not much different, the main factor affecting the ZT value is κ.

Fig. 11
figure 11

The calculated ZT as a function of carrier concentration for both p-type and n-type bilayer HfS2 at different biaxial strain.

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The thermoelectric performance of the HfS2 bilayer is higher compared with the HfS2 monolayer. For example, Bera et al. calculated a maximum ZT value of 0.60 for the HfS2 monolayer and Özbal et al. reported a maximum ZT value of 0.67.26,38 According to the research in this paper, the maximum ZT value of the unstrained HfS2 bilayer is 1.76, which is larger than the ZT value of the HfS2 monolayer.

Conclusion

In this chapter, the band structure, electronic transport properties and thermal transport properties of HfS2 bilayer under different biaxial strains are calculated based on first principles combined with Boltzmann equation and the thermoelectric properties are obtained finally. For the unstrained HfS2 bilayer, the maximum ZT value of p-type doping is 0.85, while the maximum ZT value of n-type is 1.76, which indicates that the bilayer structure can effectively improve the thermoelectric properties of 2D HfS2. The degenerate bands and stair-like 2D DOS contribute to the high PF, while the interlayer force leads to enhanced phonon scattering and reduces the κl, which improves the ZT value of the HfS2 bilayer.

By applying biaxial tensile strain to the HfS2 bilayer, it is found that the strain induces a change in the band structure and increases the degeneracy of the band, thereby improving the PF of the material. Meanwhile, the κl of HfS2 bilayer can be greatly reduced with the strain due to the enhancement of phonon scattering and the decrease of phonon lifetime. The thermoelectric performance of HfS2 bilayer can be greatly improved under the dual effect of the improvement of PF and the reduction of κ. Ultimately, we further enhance the thermoelectric properties of HfS2 bilayer by strain engineering. For p-type doping, the ZT value of the HfS2 bilayer at 6% strain is improved to 1.83; while the ZT value of the HfS2 bilayer at 3% strain under the n-type doping is more higher, reaching 3.29.