The van der Waals Heterostructures of Blue Phosphorene with GaN/GeC for High-Performance Thermoelectric Applications

doi:10.21203/rs.3.rs-4662525/v1

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The van der Waals Heterostructures of Blue Phosphorene with GaN/GeC for High-Performance Thermoelectric Applications

https://doi.org/10.21203/rs.3.rs-4662525/v1

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Thermoelectric devices have substantial potential as an environmentally compatible and sustainable power source. In this article, the density functional theory and Boltzmann's transport equation are employed to investigate the thermoelectric characteristics of blue phosphorene (P)/GaN and P/GeC van der Waals heterostructures (vdWHs). Both vdWHs exhibit a small lattice thermal conductivity at 300 K because of the enhanced phonon boundary dispersion. The anisotropy in terms of longer relaxation time results in a high figure of merit (ZT) for n-type doping up to 5.07 and 4.67 for both P/GaN and P/GeC vdWHs at effective temperature. Meanwhile, they also exhibit the average ZT as high as 3.91 (P/GaN) and 3.44 (P/GaC). We also observed that these vdWHs exhibit an excellent conversion efficiency, up to 26% (P/GaN) and 25% (P/GeC) at 700 K. Our findings demonstrate that both vdWHs have practical potential for thermoelectric applications, representing significant progress in developing better thermoelectric devices.

Environmental contamination and energy crises are the most significant challenges in modern society. Waste heat accounts for over two-thirds of all energy consumed worldwide; hence, recycling waste heat is crucial. Thermoelectric (TE) technology offers a practical way to address the abovementioned issues by converting waste heat into electric energy without producing any pollutants. In recent times, TE materials have garnered a lot of interests because of their capacity to function as a solid-state heat pump or directly convert heat into electricity without employing any hazardous chemicals or complicated designs for thermoelectric devices. Solid-state devices are advantageous for many applications, including spread waste heat recovery and quick small-scale cooling, because they have no moving parts, vibration- and noise-free, very scalable, and incredibly reliable^1–4.

It is well known that the dimensionless parameter figure-of-merit (ZT) describes the capabilities of a material as a thermoelectric generator (TEG) and is defined as⁵:

$$ZT=\frac{{{S^2}\sigma }}{\kappa }T$$

Where S, σ, κ and T denote the Seebeck coefficient, electrical conductivity, thermal conductivity and temperature, respectively. κ is made up of two components: electronic thermal conductivity (κ_e) and lattice thermal conductivity (κ_l). The S²σ is also known as power factor. A brilliant TE material typically has a high Seebeck coefficient, as well as strong electrical and minimal thermal conductivities. These intrinsic parameters exhibit a robust interdependence and are subject to the impact of various factors, including the electronic and crystal structures, carrier mobility, relaxation time and carrier concentration⁶. Therefore, improving the ZT value by only modifying one of these parameters is challenging because of the complex connection between them. To date, numerous attempts have been made to improve the ZT value by either lowering κ through nanostructures designing^7–9, heterostructures/interface^10,11 and hierarchical architectures^12,13, or by boosting the S²σ by resonant state doping¹⁴, band convergence effects^15,16, superlattice structures^17,18, etc.

The van der Waals heterostructures (vdWHs) exhibit exceptional features over all components subsequent to the development of two-dimensional (2D) materials. Because they hold an unique interfacial morphology, as well as the ability of nanostructuring and numerous constituent elements^11,19,20. The κ_l of vdWHs is significantly decreased due to the strong surface phonon dispersion^9,21. Furthermore, numerous investigations have demonstrated that constructing a vdWH could modify the nanomaterial interface, which not only boosts S due to the energy filtering effect but also simultaneously reduces κ_l^22–25. Therefore, creating vdWHs is believed to offer additional improvements in thermoelectric properties. However, according to earlier research, the thermoelectric effectiveness of vdWHs is not as good as projected. Li et al.²⁴ synthesized the 2D Bi₂Te_2.7S_0.3/Bi₂Te₃ hetero-nanosheet with atomically thin heterojunction surfaces, resulting in a maximum ZT value of 1.17 at 475 K. He et al.²⁶ theoretically investigated the TE properties of AlN/GaN bilayer and reported the maximum peak of ZT up to 1.41 for p-type doping at 700 K. In contrast to the inferior TE performance of the two separate black-p and blue-p monolayers, Hu et al. ²⁷ observed that black phosphorus/blue phosphorus vdWHs demonstrate unexpectedly excellent TE capability with an elevated ZT up to 3.2 at 700 K. The transport and electrical characteristics of SnSe/SnS vdWHs were investigated theoretically by Zhang et al.²⁸. It was shown that vdWHs can greatly enhance the thermoelectric properties of materials since the ZT_ave of SnSe/SnS increased by 25% and 63% compared to bilayer SnS and SnSe, respectively. More recently, Shahid et al.¹⁰ reported SiPGaS/As vdWHs thermoelectric properties with high ZT of about 2.66(Model-I) and 3.75(Model-II) for n-type doping.

So far, the rapidly growing optical and electron device industries, along with their suitability for high-temperature power-generating applications, have led to a growing percentage of reports on the thermoelectric features of GaN-based materials. Because of their significant band gaps and excellent thermal stability, these materials have been proposed as a key components for high-temperature power production^29–32. Other noteworthy benefits of the III-nitride systems are the wide range of band gap engineering options, irradiation hardness, and non-toxic behavior^33,34. Meanwhile, monolayer GeC has opened the door to new possibilities in the field of thermal transport materials. Researchers have predicted that GeC is a stable 2D planar honeycomb structure with a broad indirect band gap and has been prepared via various experimental techniques^35–38. Numerous applications can be found for GeC films, including those in the fields of temperature sensors and infrared optics^39,40. Blue phosphorous (P) belongs to the group of phosphorene allotropes and was synthesized by employing molecular beam epitaxial growth on the surface of Au(111)⁴¹. It has gotten a lot of interest because of its unique properties, such as significant band gap, excellent mobility, superconductivity, and thermoelectricity^42–45. Many theoretical studies have also been conducted to study the electrical characteristics and stability of its one-dimensional forms, such as P nanotubes⁴⁶ and P nanoribbons⁴⁷. The studies mentioned above collectively demonstrate that GaN, GeC, and P are key 2D semiconductor materials with intriguing potential across a wide range of applications. Even more important, the lattices of GaN and GeC are almost identical to those of P, making them ideal materials for the fabrication of vdWHs to modify their electrical and thermoelectric properties via interlayer interactions.

In this work, we built the P/GaN and P/GeC vdWHs to broaden the understanding of the potential of vdWHs in thermoelectricity. We thoroughly investigated the electrical and transport properties of these vdWHs. Our computational results show that the thermoelectric characteristics of n-type conduction are superior to that of p-type conduction. More importantly, both vdWHs demonstrate excellent thermoelectric effectiveness, suggesting that nano-heterostructures can significantly improve the thermoelectric properties.

The cell structures of monolayer P, GaN and GeC are presented in Fig. 1(a-c). The optimized in-plane lattice constants of these monolayers are a = b = 3.27 Å, 3.24 Å and 3.25 Å respectively, which are closely aligned with earlier reports^26,35,41,60. Obviously, the lattice misfit between monolayer P and GaN or GeC is less than 1%, which is an ideal condition for the epitaxial formation of outstanding vdWHs. Besides the lattice mismatch, the local atomic stacking arrangement will also affect the interfacial characteristics. Here, we configured P/GaN and P/GeC vdWHs with six different stacking arrangements, as presented in the supplementary information (SI) Fig. S1. To find the most stable stacking arrangements, we evaluated the binding energy (E_b) which is defined as E_b = E_vdWH - E_P - E_GaN/GeC. Where E_vdWH, E_P, and E_GaN/GeC represent the total energies of the vdWHs, P, and GaN/GeC monolayers, respectively. As listed in table S1, the stacking-A of P/GaN and stacking-D of P/GeC vdWHs having the binding energy of -0.67 eV and − 0.64 eV are the most stable configurations, respectively. Therefore, in the subsequent study, we will focus on the most stable vdWHs, as depicted in Fig. 1(d, e) for P/GaN and P/GeC.

To verify the dynamic stabilities of the P/GaN and P/GeC vdWHs, we first calculated their phonon band spectra. As shown in Fig. 2(a, b), the vdWHs hold a good dynamic stability because they do not have any negative frequencies in their phonon band spectra. Both vdWHs unit cells are composed of four atoms, resulting in 12 distinct vibrational modes, including three acoustic modes and nine optical modes. In Fig. 2(a, b), we highlight the three acoustic modes, i.e. ZA (red), TA (blue), and LA (purple), which have a notable impact on thermal conductivity. The lowest-frequency acoustic mode (ZA mode) displays a parabolic shape in the phonon spectrum. Because the frequency ω of ZA mode is a quadratic function of the wave vector k, i.e. $\omega ={\left( {\frac{\kappa }{\rho }} \right)^{1/2}}{k^2}$. This type of acoustic branch is a prevalent characteristic of 2D systems⁶¹. It has been reported phonon frequencies are correlated with their atomic masses⁶². In the P/GaN and P/GeC vdWHs, the lightest N/C and heaviest Ga/Ge atoms primarily act at high and low frequencies. Whereas the intermediate atomic mass P atoms are involved throughout the phonon frequency range. In addition, to evaluate the thermal stabilities, we also performed AIMD simulations of P/GaN and P/GeC vdWHs at 700 K for 18 ps. As displayed in Fig. 2(c, d), both vdWHs are thermally stable at 700K. Because the energy fluctuations are minimal during the whole 18 ps-AIMD, which implies there are no notable structural deformation. Therefore, exploring the thermoelectric characteristics at such a high temperature is also justifiable.

The electronic structure can reflect the fundamental features of electrical transport. The band structure and density of states (DOS) of parent monolayers are shown in Fig. S2. The P and GaN monolayers possess indirect semiconductive band gaps of 1.94/2.75 eV and 1.99/3.26 eV using PBE/HSE06 functional, respectively. Whereas the GeC monolayer has a 2.06/2.81 eV direct band gap using PBE/HSE06 functional. These electronic properties are well consistent with earlier literatures^26,35,41,60. Figure 3(a, b) displays the band structures of P/GaN and P/GeC vdWHs attained by the PBE (red) and HSE (blue) functionals, respectively. The estimated energy gaps are presented in Table S2. Both vdWHs exhibit an indirect band gap, as evidenced by the presence of conduction band minima (CBM) located at the M-point and valence band minima situated at the K-point of the irreducible Brillouin zone. Further, we also investigated the DOS of these vdWHs, as illustrated in Fig. 3(c, d). The DOS shows that both vdWHs retain the Type-II band alignment. The VBM of vdWHs arises due to the main involvement of P atom orbitals, whereas the CBM appears due to the primary contribution of C/N atom orbitals. In both vdWHs, band degeneracy is observable in the lower conduction band (CB) region, suggesting that n-type doping exhibits enhanced transport capabilities compared to p-type doping¹⁵. Additionally, the observed differences in band gap between heterostructure and their respective monolayers indicate that vdWHs have the potential to modulate the transport capabilities.

The assessment of TE capability primarily relies on two variables: carrier mobility (µ) and carrier relaxation time (τ). Based on the deformation potential (DP)⁶³ theory, we calculated the relaxation time and carrier mobility for vdWHs^10,64. The formulation for the carrier mobility and relaxation time can be written as follows:

$${\mu _{2D}}=\frac{{2e{\hbar ^3}{C_{2D}}}}{{3{K_B}T|{m^*}{|^2}E_{{DP}}^{2}}}$$

$$\tau =\frac{{{m^*}\mu }}{e}$$

In the preceding expression, e stands for the elementary charge, ħ is the reduced Planck constant, m* is the effective mass, K_B is the Boltzmann constant and T designates the temperature. ${C_{2D}}={1 \mathord{\left/ {\vphantom {1 {{S_0}}}} \right. \kern-0pt} {{S_0}}}({\partial ^2}{E_{total}}/\partial {\varepsilon ^2})$ is the in-plane stiffness of the 2D system. Where S₀ is the unstrained system surface area. $E=\partial {E_{total}}/\partial \varepsilon$ is the DP constant, representing a shift of the band edge site upon uniaxial strain. This constant describes the influence of lattice potential field modulation on carriers around the Fermi level, and quantifies the intensity of electro-acoustic coupling. Table S3 illustrates the coefficients linked with the DP theory and the carrier mobility at 300 K for the P/GaN and P/GeC vdWHs. For both vdWHs, the electron mobility is marginally higher than that of hole. The carrier relaxation time at different temperatures is presented in Fig. 4(a). According to the Eqs. (4) and (5), the relaxation time monotonically decreases as temperature increases.

To judge the TE performance of these vdWHs, we will focus on their electrical conductivity (σ), Seebeck coefficient (S), and electronic thermal conductivity (κ_e). The rigid band approximation is employed to examine the impact of doping on electrical transport parameters. Negative and positive values of chemical potential (µ) are used to describe the p- and n-types doping, respectively. The S of P/GaN and P/GeC vdWHs as a function of µ at numerous temperatures are presented in Fig. 5(a, d). The p-type and n-type doping at both sides near the fermi level result in more pronounced and powerful peaks of the S at 300 K. The best Seebeck coefficients for P/GaN and P/GeC vdWHs at 300 K are 1793 µV/K and 1210 µV/K with n-type doping, respectively. As reported previously, the Seebeck coefficient for a talented thermoelectric device often exceeds 200 µV/K^65,66. We can observe a distinct response to temperature wherein the parameter S decreases when subjected to heat. This is a common characteristic of many good TE materials due to their energy-sensitive DOS, which exhibits significant changes (∂n(ε)/∂ε) near the Fermi level. As shown in Fig. 5(a, d), the S of both vdWHs display peaks in lower µ region. The Mott equation can be better to define S of thermoelectric materials⁶⁷:

$S=\frac{{{\pi ^2}k_{B}^{2}T}}{{3q}}{\left\{ {\frac{1}{n}\frac{{\partial n(E)}}{{\partial E}}+\frac{1}{\mu }\frac{{\partial \mu (E)}}{{\partial E}}} \right\}_{E={E_f}}}$ (6) In the given equation, q represents the charge of the carrier, E_f denotes the Fermi energy, µ(E) stands for mobility, and n(E) is associated with the DOS. S can be enhanced through two approaches: optimizing localized DOS and adjusting carrier concentration. A more energy-sensitive DOS distribution can lead to an increase in ∂n(ε)/∂ε. Figure 3 (c, d) illustrates comparable high rates of DOS for P/GaN and P/GeC vdWHs near the Fermi level, which aligns with the order of their S coefficients. The monolayer P and GaN/GeC interface may create an energy barrier that filters or scatters low-energy carriers, while high-energy electrons are allowed to pass²⁴. The energy barrier and its carriers scattering effect play a crucial role in the enhancement of S. Thus, vdWHs could increase S and provide a large S²σ, which further increases ZT as a result¹⁰. Figure 5(b, e) presents the σ/τ as a function of µ for P/GaN and P/GeC vdWHs. In contrast to S, the σ/τ exhibits a slight variation for both vdWHs and is less affected by the change in temperature. It is evident that the n-type doping of both vdWHs exhibits a higher slope of (σ/τ) vs. µ, especially for µ in the range of -2.0 to 2.0 eV. This observation aligns with the outcomes mentioned above. The conduction electrons adhere to Fermi-Dirac statistics. So, the slopes of σ/τ gradually become flattened in the low µ region, even when the temperature rises. In order to accurately predict the TE performance of the materials, it is essential to consider electron thermal conductivity (κ_e/𝜏). The κ_e/𝜏 of both vdWHs shown in Fig. 5(c, f) are acquired using the Wiedemann-Franz law⁵⁷. The κ_e/𝜏 curves are similar to those of σ/𝜏, but are highly responsive to changes in temperature. The κ_e gradually increases as the µ moves away from the Fermi level (µ = 0). The transport coefficients presented above demonstrate the good electronic transport characteristics of the P/GaN and P/GeC vdWHs with n-type doping.

Next, we investigated the lattice thermal conductivity (κ_l) of these vdWHs. The lattice thermal conductance is mainly achieved by phonons. A good thermoelectric material should have an ineffective phonon thermal transport in order to preserve a uniform temperature gradient. The group velocity (ν), Grüneisen parameter (γ), and relaxation time of phonons are essential variables for calculating κ_l. The phonon velocity is a key physical parameter for assessing thermal transport. We can determine the phonon group velocities by:

$${\nu _i}{\text{(q)}}=\frac{{\partial {\omega _i}{\text{(q)}}}}{{\partial {\text{q}}}}$$

The ∂ω_i(q) represents phonon frequency. Figure 6(a, d) shows the group velocities of P/GaN and P/GeC vdWHs. Regarding both vdWHs, distinct colors are used to represent the acoustic phonon group velocities for the ZA, TA, and LA modes. Due to the relatively lower acoustic group velocities, especially for the ZA and TA modes, κ_l will exhibit a declining tendency. According to Fig. 6(a, d), the velocities in the low-frequency region (about 2 ~ 7 THz) are significantly greater, while they are quite minimal in the high-frequency zone. Meanwhile, there is a slight difference in the group velocity between both vdWHs. Figure 6(b, e) displays the Grüneisen parameters (γ) and phonon relaxation time. The Grüneisen parameter's describes the anharmonic interactions among lattice atoms. The Grüneisen parameter can be written as follows:

$${\gamma _i}{\text{(q)}}= - \frac{V}{{{\omega _i}{\text{(q)}}}}\frac{{\partial {\omega _i}{\text{(q)}}}}{{\partial V}}$$

The V factor in the above equation represents the absolute volume of the crystal. In general, a large |γ| implies a significant phonon-phonon anharmonic scattering⁶⁸, which reduces the κ_l, especially at high temperatures. Here, both vdWHs hold a moderate value of γ, suggesting a notable anharmonicity. When two phonons collide, anharmonicity (a marker of an imbalance in phonon transport) occurs⁶⁹. In the low-frequency region, the large |γ| strongly suppresses heat transmission and results in low κ_l. At low temperatures, anharmonicity is more significant because the nonbonding electrons in the form of lone-pairs usually couple with the valence electrons of the adjacent atoms⁷⁰. Then the anharmonic third-order IFCs are used to compute the phonon relaxation time, which is a key factor in κ_l. Figure 6(c, e) shows that both vdWHs have significantly shorter phonon relaxation times, indicating sufficient phonon scattering, which also infers a low κ_l.

Based on the harmonic and anharmonic force constants, we can obtain the temperature dependent lattice thermal conductivity (κ_l) of P/GaN and P/GeC vdWHs by

$${\kappa _{l,i}}=\sum\limits_{v} {\sum\limits_{{\text{q}}} {{C_{ph}}v_{{g,i}}^{2}({\text{q,}}v)\tau ({\text{q}},v)} }$$

The variable C_ph represents the specific heat per mode, $\nu _{{g,i}}^{2}$ denotes the phonon group velocity and 𝜏(q,ν) represents the relaxation time of the phonon mode with wave vector q and dispersion branch ν. As illustrated in Fig. 4(b), at 300 K, the κ_l values for the P/GaN and P/GeC vdWHs are 6.18 and 8.47 WK⁻¹m⁻¹, which are significantly lower than those of GaSe/SnS₂ (14.61 WK⁻¹m⁻¹)⁶², WSe₂/SnS₂ (22.69 WK⁻¹m⁻¹)⁷¹ and MoS₂/WS₂ (68.44 WK⁻¹m⁻¹)⁷² vdWHs. The lattice thermal conductivity declines as the temperature increases, following the expression ${\kappa _l} \propto {1 \mathord{\left/ {\vphantom {1 T}} \right. \kern-0pt} T}$, due to the more intense phonon thermal movement that reduces phonon mean free path and remarkable phonon-phonon scattering upon high temperature⁷³. Acoustical modes play a significant role in controlling the behavior of monolayer and heterostructure systems⁷⁴. These low-frequency acoustic modes may contribute low phonon group velocities, resulting in low κ_l. Ultimately, creating a heterostructure has a notable impact on the transmission of phonon heat.

Finally, based on the electronic and lattice transport parameters above, we calculated the the ZT values of P/GaN and P/GeC vdWHs at different temperatures (300 K, 500 K, and 700 K). Figure 7(a, b) demonstrates a better TE performance of n-type doping compared with that of p-type doping. This anisotropic feature can be attributed to S and σ as shown in Fig. 5. For both vdWHs, they exhibit a larger ZT upon higher temperature. Meanwhile, the peak positions of ZT also shift towards µ = 0 as the temperature rises. More importantly, on the effective temperature scale, the ZT values for P/GaN and P/GeC vdWHs reached remarkable values of 5.07 and 4.67, respectively. The ZT of our designed vdWHs is superior to those of previously proposed vdWHs and single layers, as displayed in Fig. 8(a)^{10,26–28,45,62,75–77}. The thermoelectric conversion efficiency (η_TEG) of a TEG can be estimated using the following formula⁷⁸:

$${\eta _{TEG}}=\frac{{{T_h} - {T_c}}}{{{T_h}}}\frac{{\sqrt {1+Z{T_{avg}}} - 1}}{{\sqrt {1+Z{T_{avg}}} +{{{T_c}} \mathord{\left/ {\vphantom {{{T_c}} {{T_h}}}} \right. \kern-0pt} {{T_h}}}}}$$

Where ZT_avg represents the average ZT over the temperature range. T_h and Tc are the temperatures of the material on hot and cold surfaces, respectively. TE materials with ZT_avg > 1 have the potential to be practically used in a wide temperature range ⁷⁹. Remarkably, the ZT_avg for P/GaN and P/GeC vdWHs are 3.91 and 3.44, respectively, which are much larger compared with the well-known SiPGaS/As, SnSe/SnS, SnSe and SnS TE systems^10,28. In addition, based on the ZT_avg, we calculated the thermoelectric conversion efficiency (η_TEG) by Eq. (10). The P/GaN and P/GeC vdWHs have much better conversion efficiency, up to 26% and 25% at 700 K, respectively. Compared with P/GeC, the P/GaN vdWH has a larger band gap, resulting in a larger S and a lower κ_l. This makes P/GaN a potential superior thermoelectric material. Our analysis offers valuable insights into thermoelectric transport, which can be utilized to design devices using vdWHs, thus driving advancements in thermoelectric technology.

In summary, we studied the thermoelectric characteristics of P/GaN and P/GeC vdWHs. The results indicate that both vdWHs are energetically, dynamically and thermally stable. These vdWHs have an indirect band gap with the Type-II band alignment. It is shown that at 300 K, P/GaN and P/GeC vdWHs show a minimal lattice thermal conductivity due to intrinsic phonon transport characteristics, such as slow group velocities, significant Grüneisen parameters, and short phonon relaxation times. More importantly, in the range of effective temperatures, P/GaN and P/GeC vdWHs exhibit a maximum ZT (ZT_avg) up to 5.07 (3.91) and 4.67 (3.44) for n-type doping, respectively, due to their anisotropy in terms of longer relaxation times. We also found that these vdWHs have significantly higher conversion efficiencies, reaching up to 26% and 25%, respectively, at 700 K. All of the outstanding TE properties of P/GaN and P/GaC vdWHs indicate that these materials have potential applications in fabricating high performance TE devices across a broad temperature range. As a whole, our results provide significant advancements in terms of the development of superior thermoelectric devices.

The computations in this manuscript were performed using density functional theory (DFT), as implemented in plane wave Vienna Ab initio Simulation Package (VASP)⁴⁸. The generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE)⁴⁹ functional was utilized in the projected augmented wave technique to evaluate the exchange-correlation functional and optimize the shape of the designed vdWHs. In order to achieve convergence standards, a 30×30×1 k-mesh grid in the Brillouin zone was used, with a cutoff energy of 500 eV for the plane-wave expansion. A 20 Å-thick vacuum along Z-axis was used to avoid periodic interactions. The optimized atomic positions and lattice parameters of the vdWHs were achieved when the forces and total energy differences between two successive self-consistency steps were less than 0.001 eV/Å and 10^− 5 eV, respectively. Using the Grimme (DFT-D3) technique^50,51, the weak van der Waals (vdW) interactions at the heterostructure interface were taken into consideration. In order to compute the electronic characteristics of vdWHs with precision, the exchange–correlation interactions are modeled by the typical PBE and Heyd–Scuseria–Ernzerhof (HSE06)⁵² functionals with the 30×30×1 and 16×16×1 k-mesh grid, respectively. The phonon propagation characteristics are determined by the finite displacement approach employed in the Phonopy package⁵³. In addition, we also performed ab initio molecular dynamic (AIMD)⁵⁴ simulations to validate the stability of both vdWHs. The simulations were carried out in 4×4×1 supercells by maintaining for 18 ps at 700 K, using time steps of 3.0 fs. The transport coefficients of P/GaN and P/GeC vdWHs are predicted using semi-classical Boltzmann transport theory within constant relaxation time approximation, as implemented in the BoltzTrap code^55,56. The electronic thermal conductivity (κ_e) are determined by the Wiedemann-Franz law⁵⁷:

$${\kappa _e}=L\sigma T$$

here, L is the Lorenz number⁵⁸ that is represented as:

$$L=1.5+\exp \left[ { - \frac{{|S|}}{{116}}} \right]$$

Further examination of the lattice thermal conductivity capability was done by employing the ShengBTE code⁵⁹. The anharmonic interatomic force constants (IFCs) were also estimated using the THIRDORDER.py⁵⁹ script with the 4×4×1 supercell.

Author contributions

Iqtidar Ahmad: Writing-original draft, Conceptualization, Investigation, Methodology, Visualization. Xue-Peng Wang: Writing-review, and editing, Project administration, Data curation. Ismail Shahid: Writing-review, and editing, Formal Analysis. Anwar Ali: Validation. Syed Hatim Shah: Writing-review, and editing. Tu Lyu: Writing-review, and editing. En Li: Data curation. Keyuan Ding: Writing-review, and editing. Xin Chen: Writing-review, and editing. Feng Rao: Supervision, Project administration, Software, Resources,Funding acquisition.

Acknowledgments

F. Rao thanks the National Natural Science Foundation of China (52032006), the Basic and Applied Basic Research Foundation of Guangdong (2020B1515120008), the Science and Technology Foundation of Shenzhen (ZDSYS20210623091813040), and Shenzhen University 2035 Program for Excellent Research (00000203). K. Ding thanks the Project of Department of Education of Guangdong Province (2021ZDZX1006). The authors also acknowledge the computational resources provided by National Supercomputing Center in Shenzhen.

Competing interests

All authors declare no financial or non-financial competing interests.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

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The van der Waals Heterostructures of Blue Phosphorene with GaN/GeC for High-Performance Thermoelectric Applications

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