We report the phonon thermal transport properties of two-dimensional (2D) and bulk boron chalcogenides using the density functional theory-driven Boltzmann transport equation approach by considering three- and four-phonon scatterings. The calculated thermal conductivities for 2D boron sulphide (BS), boron selenide (BSe), and boron telluride (BTe) at 300 K are 210, 57, and 125 W/m K and vary non-monotonically with the chalcogen mass. The effect of four-phonon scattering is significant in all materials and the obtained thermal conductivities are overpredicted by as much as 83% when only three-phonon scattering is included. For bulk BS, the four-phonon scattering is significant for the cross-plane direction (64% reduction in thermal conductivity with four-phonon scattering) while the basal-plane transport stays unaffected (∼10% change). The phonons contributing to the cross-plane thermal transport in bulk BS has similar mean free paths as those for basal-plane transport, despite the two orders of magnitude lower thermal conductivity in the cross-plane direction.
I. INTRODUCTION
Two-dimensional (2D) materials, such as graphene, MoS2, and phosphorene offer excellent physical properties with potential applications in electronics, energy conversion, and storage devices.1,2 In many of these applications of 2D materials, the thermal transport understanding is crucial in determining the design and efficiency of the underlying device. For instance, a low thermal conductivity material is required for efficient thermoelectric devices, while a high thermal conductivity is desired for heat dissipation in electronic devices. Consequently, several studies have focused on the understanding of the thermal transport properties of 2D materials starting with graphene with the highest room temperature thermal conductivity among all reported 2D materials of 3000–5000 W/m K.3 Although this remarkably high thermal conductivity of graphene is attributed to its planar, non-buckled structure with out-of-plane mirror symmetry, other planar allotropes of carbon, such as biphenylene,4 are reported to have thermal conductivities in order of magnitude lower than that of graphene despite having the similar out-of-plane mirror symmetry. Boron-based 2D materials, including borophene, boron nitride, and boron pnictides, generally have thermal conductivities in the range of ∼600–2000 W/m K.5 Transition metal dichalcogenides, such as MoS2 and WS2,6,7 have room temperature thermal conductivity in the range of 50–150 W/m K and feature a direct electronic bandgap opposed to the semi-metallic behavior of graphene. Other chalcogenides, based on In, Ga, and Tl, have low thermal conductivities and are explored for potential use in thermoelectric devices.8
A relatively new family of semiconducting 2D materials, boron chalcogenides, is reported recently with potential application in thermoelectrics.9 Among this boron chalcogenide family of materials, boron sulfide (BS) is experimentally synthesized via exfoliation, and its bandgap is tunable through variation in the number of layers.10 Zhang et al. revealed that BS exhibits a wide range of photoluminescence responses and suggested the potential of oxygen-functionalized BS to act as an anode in lithium-ion batteries.11 These materials are also reported to have bifunctional catalytic activity for oxygen evolution and oxygen reduction reactions.12 Further, Mishra et al. reported that boron chalcogenides have exceptional thermoelectric performance.9 However, the authors considered only electronic contribution to the total thermal conductivity and the understanding of phonon thermal transport conductivity is still lacking for these materials.
In this work, we explore the phonon thermal transport properties of 2D boron chalcogenides BX (X = S, Se, and Te) using the first-principles-driven Boltzmann transport equation approach. We find that the four-phonon (4ph) scattering is crucial for the correct description of the thermal transport physics in considered materials and the phonon thermal conductivity of 2D boron selenide (BSe) reduces by 83% with four-phonon scattering compared to its value using only three-phonon scattering. The final obtained thermal conductivity for 2D BSe is 57 W/m K at 300 K, which is smaller than that of 2D boron telluride (BTe) (125 W/m K), despite the heavy mass of chalcogen atoms in BTe. While this large reduction in thermal conductivity with four-phonon scattering is also observed for bulk BS, it is limited to only cross-plane direction and the basal-plane transport remains unaffected (the corresponding reduction is ∼10%) with four-phonon scattering.
II. METHODOLOGY
We obtain the ground state structure within the framework of density functional theory (DFT) employing the plane wave basis set as implemented in the Quantum Espresso simulation package.13 The total energy and forces are calculated by using projected augmented wave (PAW) pseudopotentials with generalized gradient approximation (GGA).14 The Brillouin zone (BZ) integrations are performed using the Monkhorst-Pack electronic wavevector grid of 14 × 14 × 1 and 12 × 12 × 2 for 2D systems and bulk BS, respectively. The plane wave kinetic energy and charge density cutoffs are set to 100 and 800 Ry, respectively, for all considered systems. We find that the total energy is converged [convergence is reported in Fig. S1 of the supplementary material] to within ∼10−3 Ry/atom with these choices of electronic wavevector grid and plane wave energy cutoff. The geometry optimization is performed using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method15 with the force and energy convergence criteria of 10−5 Ry/bohr and 10−8 Ry, respectively. The electronic energies are obtained with self-consistent functional evaluation using a total energy convergence threshold of 10−12 Ry. All calculations are performed by including Grimme's DFT-D2 van der Waals functionals.16
III. RESULTS AND DISCUSSION
The crystal structure of 2D BX is presented in Fig. 1(a). The structures of 2D BX are constructed according to the configuration of 2D MoS2 or WS2, where Mo or W is replaced by two boron atoms stacked in the vertical direction, resulting in a honeycomb structure with four sublayers having an order of X–B–B–X. The lattice parameters, bond lengths, and bond angles of considered boron chalcogenides are reported in Table I. For 2D BS, our obtained lattice constant is in close agreement with the experimentally measured value (within 1.2%).10,11,23 We noticed that, as expected, the B–X bond length and the lattice parameters increase monotonically in changing the chalcogen atom from S to Te. The bulk phase of BS has a rhombohedral structure composed of three layers of S–B–B–S along the stacking/z axis.
(a) The crystal structure of 2D BX (X = S, Se, and Te) and (b) and (c) bulk BS obtained using ABC, AAA stackings. (d)–(g) The phonon dispersions of considered boron chalcogenides. The phonon bandgaps open with the increase in chalcogen mass.24
(a) The crystal structure of 2D BX (X = S, Se, and Te) and (b) and (c) bulk BS obtained using ABC, AAA stackings. (d)–(g) The phonon dispersions of considered boron chalcogenides. The phonon bandgaps open with the increase in chalcogen mass.24
Lattice constants, boron–boron (B–B)/boron–chalcogen (B–X) bond lengths, and bond angles for all systems. The values reported in parenthesis are from experiments (lattice constants) and other theoretical reports (B–B and B–X bond lengths).
System . | Lattice parameters (Å) . | B–B (Å) . | B–X (Å) . | B–B–X (°), B–X–B (°) . |
---|---|---|---|---|
2D BS | 3.050 (3.014)11 | 1.702 | 1.952 | 115.6, 102.7 |
2D BSe | 3.262 | 1.684 | 2.101 | 116.3, 101.9 |
2D BTe | 3.588 | 1.677 | 2.312 | 116.4, 101.8 |
Bulk BS | 3.049 (3.050),10 20.440 (20.384)10 | 1.701 (1.703) | 1.951 (1.949) | 115.1, 102.8 |
We considered two different stacking configurations, ABC and AAA, for bulk BS. After geometry optimization, we find that the ABC stacking is energetically more stable and, therefore, we report thermal transport properties of bulk BS for ABC stacking of the rhombohedral phase. The obtained in/basal-plane and cross-plane lattice parameters for bulk BS are within 0.5% of the experimentally measured values (see Table I).10
The phonon dispersions of considered boron chalcogenides are reported in Figs. 1(d)–1(g). All considered chalcogenides have real dispersion branches, confirming the dynamic stability of structures. The acoustic phonon modes in these materials have a major contribution from heavy chalcogen atoms; with an increase in chalcogen mass, the acoustic phonon frequencies decrease in going from BS to BTe. The high-frequency optical modes have a dominant contribution from boron atoms and have similar frequencies in all considered boron chalcogenides. Further, all considered chalcogenides have a frequency gap between low-lying optical and mid-range optical phonons and this gap widens up with increasing mass of the chalcogen atom. The presence of this frequency gap has important consequences on the phonon thermal conductivity as it impedes the scattering of acoustic phonons by optical phonons. This has been reported earlier for BAs, where the thermal conductivity obtained using only three-phonon scattering was an over-prediction of experimentally measured value and the four-phonon scattering was needed for the correct description of the thermal transport physics.25,26
At 300 K, the obtained phonon thermal conductivity of 2D BS, BSe, and BTe are 421, 332, and 155 W/m K, respectively. This decrease in thermal conductivity with increasing chalcogen mass is expected and originates from the reduction in phonon group velocities. However, with the inclusion of four-phonon scattering, the obtained thermal conductivities are reduced by 49%, 83%, and 19% and vary non-monotonically with chalcogen mass. The obtained thermal conductivities with the inclusion of 4ph-scattering are 210, 57, and 125 W/m K, and the thermal conductivity of 2D BSe is lower than that of 2D BTe despite the heavy mass of the Te atom. To understand this, we plot phonon and accumulation against phonon frequency and mean free paths (MFP) [obtained by including both three- and four-phonon (3 + 4ph) scatterings] in Figs. 2(a) and 2(b).
The phonon thermal conductivity accumulation as a function of (a) phonon frequency and (b) phonon mean free path (MFP). With 4ph scattering, the average phonon mean free reduces by 82% in BSe.
The phonon thermal conductivity accumulation as a function of (a) phonon frequency and (b) phonon mean free path (MFP). With 4ph scattering, the average phonon mean free reduces by 82% in BSe.
We find that at 300 K, only low-frequency phonon modes [modes below frequency bandgap in Figs. 1(d)–1(g)] are active in all considered chalcogenides. Of these, the contribution of phonons with frequencies less than <5 THz is surprisingly large in 2D BTe compared to 2D BS and BSe. Beyond 5 THz, while there is a frequency bandgap in BTe (and consequently the saturation of thermal conductivity), the thermal conductivity continues to rise till 12.9 THz for BS. All of these modes below the frequency bandgap have mean free paths less than 300 nm. Interestingly, mean free paths in 2D BTe and BS are significantly higher compared to 2D BSe with 3 + 4ph scattering, the average mean free path obtained by considering gray approximation is 21 nm in 2D BSe compared to 46 and 60 nm in 2D BS and BTe, respectively. As shown in Fig. 2(b), these shorter mean free paths in 2D BSe are obtained by considering both 3ph and 4ph scatterings; with only 3ph scattering, the mean free paths are similar in all three considered 2D materials.
The effect of 4ph scattering on phonon relaxation times is reported in Figs. 3(a)–3(c). With only 3ph scattering, the phonon relaxation times have a similar range in considered 2D boron chalcogenides, thus, suggesting that their thermal conductivity trend is majorly decided by phonon group velocities at the 3ph-scattering level, i.e., for 2D BS > BSe > BTe. With the inclusion of 4ph scattering, the phonon relaxation times in 2D BSe reduce by more than an order of magnitude while the corresponding change is much less in 2D BS and BTe. As shown in Fig. 3(d), this large effect of 4ph scattering in BSe originates from phonon–phonon scattering phase space, which is greatly enhanced with the inclusion of 4ph scattering in BSe. The large frequency gap between low-lying optical modes and high-frequency optical modes in 2D BSe makes it difficult for the phonons to get scattered through three-phonon processes. With inclusion of four-phonon processes, the phonons which otherwise had low three-phonon scattering phase space, undergo scattering. This frequency gap in the considered boron chalcogenides is also responsible for the non-monotonous thermal conductivity behavior, i.e., 2D BTe ∼ BSe > BS [Figs. 1(f) and 1(g)].27
(a)–(c) The effect of 4ph scattering on the phonon relaxation times (denoted by 3 + 4ph) and (d) the ratio of 4ph- to 3ph-scattering phase space of considered 2D Boron chalcogenide materials.
(a)–(c) The effect of 4ph scattering on the phonon relaxation times (denoted by 3 + 4ph) and (d) the ratio of 4ph- to 3ph-scattering phase space of considered 2D Boron chalcogenide materials.
It is worth noting that the obtained thermal conductivity of 2D BTe is an order of magnitude larger than the other telluride materials such as MgTe,28 GeTe,29,30 SnTe,29 PbTe,31 and Bi2Te3 (Ref. 32) despite having similar Te-dominated acoustic phonon modes. Recently, telluride-based materials have been explored as electrode materials for batteries.11 However, the low of tellurides may restrict the thermal stability. We believe that 2D BTe with high thermal conductivity can be explored to replace low thermal conductivity telluride-based electrodes to improve the stability and performance of battery electrodes.
A. Cross-plane thermal transport in bulk BS
For the layered materials, the thermal transport is anisotropic, and thermal conductivity is different along the basal- and cross-plane directions. For BS, we considered the bulk phase and computed phonon thermal conductivity along the basal- and cross-plane directions. By considering both three- and four-phonon scatterings, the obtained basal-plane thermal conductivity for bulk BS is 205 W/m K, which compares well with that of 2D BS. However, with only three-phonon scattering, the obtained basal-plane thermal conductivity of bulk BS is 239 W/m K, compared to more than 400 W/m-K for 2D BS. This finding is similar to that reported for graphene where for 2D, the low-frequency flexural phonon modes were found to be non-scattered by three-phonon processes, and with the inclusion of four-phonon scattering, the obtained thermal conductivity was drastically reduced and was comparable to that of the bulk-phase.33
The obtained cross-plane thermal conductivity of bulk BS is 2.9 W/m K, which reduces to 1.1 W/m K with the inclusion of four-phonon scattering [Fig. 4(a)]. While this cross-plane thermal conductivity is two orders of magnitude lower than that of basal-plane direction, the phonons that contribute to heat transfer in the cross-plane direction have MFPs similar to those of phonons contributing to heat transfer in the basal-plane direction. This similar behavior of comparable basal- vs cross-plane MFPs has also been reported recently for several low materials such as MoO3 and KCuSe.6 Here, we find that such behavior is not limited to low materials, and high materials, such as BS, can also have similar MFPs of heat-carrying phonons in the basal- vs cross-plane directions.
The phonon thermal conductivity accumulation of bulk BS with (a) phonon frequency and (b) phonon mean free paths. The results are obtained by considering only 3ph scattering (dashed lines) and both, 3ph/4ph scatterings (solid lines). The phonon contributing to cross-plane and basal-plane directions have a similar mean free path despite two orders of magnitude difference in corresponding thermal conductivities.
The phonon thermal conductivity accumulation of bulk BS with (a) phonon frequency and (b) phonon mean free paths. The results are obtained by considering only 3ph scattering (dashed lines) and both, 3ph/4ph scatterings (solid lines). The phonon contributing to cross-plane and basal-plane directions have a similar mean free path despite two orders of magnitude difference in corresponding thermal conductivities.
IV. CONCLUSIONS
In summary, we investigated the thermal transport in 2D boron chalcogenides using a density functional theory-driven Boltzmann transport equation-based approach. We find that with only three-phonon scattering, the thermal conductivity follows a typical mass trend: for BS > BSe > BTe arising from the mass of the chalcogen atom. However, with four-phonon scattering, an anomalous trend of thermal conductivity is observed, where the thermal conductivity of 2D BSe is lower than that of 2D BTe. The obtained 300 K thermal conductivities of boron chalcogenides are higher than other 2D chalcogenides and, in particular, for BTe, the obtained thermal conductivity is more than an order of magnitude larger than other reported tellurides. The computed thermal conductivities of boron chalcogenides are high (>50 W/m K at 300 K), suggesting that these materials are not suitable for thermoelectric applications but rather could be explored for heat dissipation applications.
SUPPLEMENTARY MATERIAL
The supplementary material contains convergence of (i) total energy with plane wave energy and electronic wavevector grid, (ii) quartic force constants with displacement size, and (iii) thermal conductivity with the interaction cutoff, supercell size, and phonon wavevector grid.
ACKNOWLEDGMENTS
The authors acknowledge the financial support from IIT Bombay; Core Research Grant, Science & Engineering Research Board, India (Grant No. CRG/2021/000010) and Nano Mission, Government of India (Grant No. DST/NM/NS/2020/340). The calculations were carried out on the SpaceTime-II supercomputing facility of IIT Bombay and PARAM Porul (located at NIT Trichy, India) supercomputing facility provided by the Centre for Development of Advanced Computing (CDAC).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Hardik L. Kagdada: Conceptualization (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Ankit Jain: Conceptualization (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.