Thermal transport properties of monolayer phosphorene: a mini-review of theoretical studies

Abstract

Phosphorene, a two-dimensional (2D) elemental semiconductor with a high carrier mobility and intrinsic direct band gap, possesses fascinating chemical and physical properties distinctively different from other 2D materials. Its rapidly growing applications in nano-/opto-electronics and thermoelectrics call for fundamental understanding of the thermal transport properties. Considering the fact that there have been so many studies on the thermal transport in phosphorene, it is on emerging demand to have a review on the progress of previous studies and give an outlook on future work. In this mini-review, the unique thermal transport properties of phosphorene induced by the hinge-like structure are examined. There exists a huge deviation in the reported thermal conductivity of phosphorene in literature. Besides, the mechanism underlying the deviation is discussed by reviewing the effect of different functionals and cutoff distance in calculating the thermal transport properties of phosphorene. It is found that the van der Waals (vdW) interactions play a key role in the formation of resonant bonding, which leads to long-ranged interactions. Taking into account of the vdW interactions and including the long-ranged interactions caused by the resonant bonding with large cutoff distance are important for getting the accurate and converged thermal conductivity of phosphorene. Moreover, a fundamental insight into the thermal transport is provided based on the review of resonant bonding in phosphorene. This mini-review summarizes the progress of the thermal transport in phosphorene and gives an outlook on future horizons, which would benefit the design of phosphorene based nano-electronics.

Keywords

thermal transport / phosphorene / resonant bonding

Introduction

Two-dimensional (2D) materials represented by graphene have been intensively studied for many years for the promising applications in lots of fields [¹–⁴]. Phosphorene, the single-layer counterpart of bulk black phosphorus (BP), is a novel elemental 2D semiconductor with a high carrier mobility proved by experiments [⁵–⁷] and an intrinsically large direct band gap (~1.5 eV) [⁸], which makes bulk BP promising for lots of nano-electronic applications [⁹]. Recently, owing to the interlayer van der Waals (vdW) interactions, few-layer phosphorene has been successfully fabricated by exfoliating from the bulk BP [⁵–⁷,¹⁰]. Most interestingly, the properties of phosphorene exhibit distinct anisotropy arising from the specific anisotropic hinge-like (puckered) structure [⁶,¹¹], which is prior to other 2D materials such as graphene, silicene, and transition metal dichalcogenides (TMDCs) [⁵–²⁰]. The unique properties of phosphorene give rise to a great prospective for its applications as the active layer in nano-electronic devices, such as field-effect transistors and photo-transistors [⁵–⁷,¹⁰,¹⁷–¹⁹,²¹]. Besides the extensive studies related to its electrical properties, there are also a lot of explorations on its potential applications in thermoelectric [¹⁷,¹⁹,²¹].

Since high-performance thermal management plays a critical role in all these applications, such as minimized thermal conductivity for thermoelectrics and efficient heat dissipation for nano-/opto-electronic devices, tremendous amount of work have been dedicated to exploring the thermal transport properties of phosphorene [²¹–³²]. The thermal conductivity of phosphorene has been independently investigated by different theory groups using various methods [²¹,²²,²⁶–³²]. It is interesting to find that the preferred thermal transport direction of phosphorene is along the zigzag direction, which is orthogonal to the preferred electric transport (armchair) direction, showing an opposite anisotropy in thermal and electrical conductivities [²¹]. Considering the fact that there have been so many studies on the thermal transport properties of phosphorene, it is on emerging demand to have a review on the progress and give an outlook on future horizons, which would benefit the design of phosphorene based nano-electronics.

In this mini-review, the unique thermal transport properties of phosphorene caused by the hinge-like structure are examined. After that, the effect of different functionals and cutoff distances on the convergence behavior of the thermal conductivity of phosphorene is discussed. Then, a fundamental insight into the thermal transport based on the review of resonant bonding in phosphorene is provided. And finally, summary and perspectives are presented at the end of the paper.

Hinge-like structure of phosphorene

Bulk BP is a layered material in which individual atomic layers are stacked together by vdW interactions like bulk graphite. Thus, it is possible to fabricate 2D phosphorene by exfoliating from the bulk BP. In experiments, by heating red phosphorus to 1000°C and then slowly cooling to 600°C at a cooling rate of 100°C per hour, Li et al. [⁶] synthesized bulk BP under a constant pressure of 10 kbar. And Lee et al. [²⁴] synthesized bulk BP from red phosphorus powders with SnI4 (American Elements, electronic grade 99.995%) and Sn ingot (Sigma Aldrich) promoters, while Liu et al. [⁵] and Xia et al. [⁷] achieved atomically thin single-layer or few-layer phosphorene via mechanical exfoliation of commercially available (Smart-elements) bulk BP. In the work by Xia et al. [⁷], flakes with the lateral size in tens of micrometre scale could be isolated for BP films thicker than 8 nm. The thickness of the crystals could be determined by a Veeco Dimension 3100 atomic force microscope (AFM). Although BP films with a thickness down to around 1.2 nm (corresponding to 2 atomic layers) can be clearly identified, these bilayer BP flakes of few-micrometre scale are usually too small for experimental measurements of optical and electronic properties. Generally, measurements were performed on BP flakes with thicknesses ranging from 2 to>30 nanometres (corresponding to 4–60 layers).

2D phosphorene possesses a hinge-like (puckered) structure along the armchair direction (Fig. 1), which is distinctly different from the flat graphene and buckled silicone [¹⁷,³³–³⁶]. The lattice constants of the unit cell along the zigzag and armchair directions are 3.30 Å and 4.60 Å (the results from Perdew-Burke-Ernzerhof (PBE) functional), respectively [³⁰]. Similar to the case of bulk BP [¹³,³⁷], the obtained lattice constants are slightly different from each other with different functionals employed during the structure optimization. For example, if the local density approximation (LDA) functional is employed, the lattice constants along the zigzag and armchair directions are 3.27 Å and 4.37 Å, respectively. Moreover, if the optB88 functional is employed to include vdW interactions, the lattice constants along the zigzag and armchair directions are 3.32 Å and 4.58 Å, respectively [³⁸]. The different lattice constants would have a significant effect on the thermal transport properties (including the absolute values of thermal conductivity and the relative anisotropy), due to the change of the bonding distance between P atoms, especially the distance between non-covalently bonded P atoms [³⁸]. Considering the fact that the anisotropy in the thermal transport of group IV-VI compounds with the similar hinge-like structure as phosphorene [³⁹,⁴⁰], it can be concluded that the hinge-like structure could generally lead to anisotropy in carrier transport. In fact, it is found that the anisotropies of the electrical and thermal transport in phosphorene are orthogonal to each other [²¹]. Thermoelectrics can perform direct solid-state conversion from thermal to electrical energy or vice versa, which have a number of valuable applications and thus may make crucial contributions to the crisis of energy and environment. In general, the thermoelectric performance and efficiency are characterized by the dimensionless figure of merit ZT = S²sT/k, where S, s, T, and k are Seebeck coeffcient (thermopower), electrical conductivity, absolute temperature, and thermal conductivity, respectively. With phosphorene as the thermoelectric material, if the temperature gradient is applied along the armchair direction, the maximum electrical conductivity and minimized thermal conductivity can be obtained simultaneously, which would lead to the largest ZT value. Therefore, phosphorene is surmised to be an extremely valuable thermoelectric material.

Thermal conductivity

As a hot interesting topic, the thermal conductivity (k) of phosphorene was first predicted in May 2014 by Fei et al. [²¹] based on analytical estimation. Then, based on first-principles calculations, the thermal conductivity/conductance of phosphorene was almost simultaneously reported at the beginning of Sept. 2014 by three independent groups at arXiv [³⁰,³²,⁴¹]. Their results largely differ from each other, although the same method of the single relaxation time approximation (RTA) for solving Boltzmann transport equation (BTE) is employed [³⁰,³²]. As mentioned below, the thermal conductivity of phosphorene was reported theoretically by lots of groups using various methods, such as analytical estimation [²¹,²⁶], classical MD simulation with optimized Stillinger-Weber (SW) potential [²⁷–²⁹], RTA [²²,³⁰–³²], and iterative method for solving BTE [²²,³¹,⁴²]. However, the exact value of the thermal conductivity of phosphorene still remains unclear, since these results are one order of magnitude different from each other as shown in Fig. 2. For instance, the thermal conductivity of phosphorene along the zigzag direction ranges from 30 to 152.7 W·(m·K)⁻¹, while those along the armchair direction ranges from 9.9 to 63.9 W·(m·K)⁻¹ [²¹,²⁷–²⁹]. The huge deviation could be attributed to the different parameters or the different calculation methods, but the underlying mechanism remains largely unclear.

Recently, Qin et al. [³⁸] reported a low thermal conductivity of phosphorene (15.33 W·(m·K)⁻¹ (zigzag) and 4.59 W·(m·K)⁻¹ (armchair) at 300 K) by using the iterative method to solve phonon BTE based on first-principles anharmonic lattice dynamics (ALD). In addition to the results obtained from ALD/BTE, the thermal conductivity of phosphorene was also obtained based on ab-initio molecular dynamics (AIMD) simulations. Based on the phonon spectral energy density (SED) technique [⁴³–⁴⁵] coupled with the equilibrium AIMD simulations, the thermal conductivity of phosphorene was obtained to be 13.46 and 5.46 W·(m·K)^-1 for zigzag and armchair directions, respectively, which agreed very well with the results from ALD/BTE. Direct non-equilibrium AIMD (NEAIMD) simulations with heat flux along the armchair direction were also performed for evaluating the thermal conductivity of phosphorene. The size dependent thermal conductivity of phosphorene obtained by NEAIMD agreed very well with that estimated using the ALD/BTE method considering the boundary scattering (Fig. 3). The consistent thermal conductivities reported by Qin et al. were obtained based on three independent ab-initio methods based on first-principles calculations. The thermal conductivity is so low and is even below the lower bound of the range reported previously [²¹,²⁹,³⁸]. It was claimed that there existed a giant phonon anharmonicity in phosphorene, which was associated with the soft transverse optical (TO) phonon modes and arose from long-ranged interactions driven by the resonant bonding [³⁸]. A microscopic picture was provided by Qin et al. [³⁸] to connect the anisotropic and low thermal conductivity to the giant phonon anharmonicity and long-ranged interactions in phosphorene, which were further attributed to the resonant orbital occupations of electrons and characteristics of the intrinsic hinge-like structure.

Note that due to the limitations of the synthesis technique, only the thermal conductivity of phosphorene films with thicknesses ranging from 9.5 to 552 nm are available [²²–²⁵,⁴²,⁴⁶]. Currently, there is still no experimental report on the thermal conductivity of single-layer phosphorene (thickness of ~0.5 nm). Considering the huge discrepancy in the theoretical reports on the thermal conductivity of single-layer phosphorene, future experimental measurements of the thermal conductivity of single-layer phosphorene are expected, which would be helpful to verify the theoretical calculations.

Despite the discrepancy in the thermal conductivity values of phosphorene, there are some common conclusions on the thermal properties of phosphorene, such as the distinct anisotropy. All of the previous calculations show that the thermal conductivity along the zigzag direction is much larger than that along the armchair direction, with an anisotropy ratio ranging from 2.2 to 5.5 [²²,³⁸]. The anisotropy in thermal conductivity is attributed to the anisotropy in phonon group velocity determined by the phonon dispersion, rather than phonon scattering [²²,³¹,⁴²]. For example, the group velocity of the longitudinal acoustic (LA) phonon modes is much higher in the zigzag direction than that in the armchair direction, contributing to the distinct anisotropy of thermal transport in phosphorene. Moreover, it is interesting to find that the preferred thermal transport direction of phosphorene is along the zigzag direction, which is orthogonal to the preferred electric transport (armchair) direction, showing an opposite anisotropy in thermal and electrical conductivities [²¹].

Effect of functional and cutoff distance

There exists a huge discrepancy in the reported thermal conductivity of phosphorene as reviewed in Section 3 based on Fig. 2. The thermal conductivity of phosphorene along the zigzag direction ranges from 15.33 to 152.7 W·(m·K)⁻¹, while the thermal conductivity along the armchair direction ranges from 4.59 to 63.9 W·(m·K)⁻¹ [²¹,²⁷–²⁹]. In this section, the mechanism underlying the deviation is discussed by reviewing the effect of different functionals and cutoff distance on the thermal transport properties of phosphorene.

Over the past few years, the method of solving BTE based on first-principles calculations have well reproduced experimental thermal conductivities of various materials [⁴⁷–⁵²], and the prediction power has been widely acknowledged. Despite the powerful prediction of the BTE method, there are lots of parameters involved in the first-principles calculations and the procedure of applying the BTE method, that would have a significant effect on the predicted thermal conductivity, such as the selection of functional, including vdW interactions or not; the electronic wavevector grid; the planewave energy cutoff; the Fermi-Dirac function smearing; the amount of vacuum (to remove inter-layer interactions in 2D materials); the supercell size for harmonic and cubic force constant extraction; the displacement amount for generating displaced supercells for force constant extraction; the cutoff distance for evaluating cubic interatomic force constants (IFCs); the invariance constraints (translational and rotational invariance) for renormalizing the IFCs; the Q-grid size when calculating thermal conductivity; and the broadening parameter for phonon-phonon scattering delta function.

The non-exhausted list as presented above is so long that it is hard to fully conduct all the tests. In fact, some parameters can be chosen based on personal experience or taken from literature [⁵³]. However, some critical parameters should be carefully tested for the convergence, such as the cutoff distance when evaluating the cubic IFCs and the Q-grid size when calculating the thermal conductivity. Unfortunately, limited work thoroughly did such tests due to the large computational cost. As for the studies reporting the thermal conductivity of phosphorene, only two works by Qin et al. [³⁸] and Jain and McGaughey [³¹] performed the convergence test of thermal conductivity with respect to the cutoff distance, which are reproduced in Fig. 4.

Qin et al. [³⁸] performed the convergence test of thermal conductivity with respect to the cutoff distance (Fig. 4(a)). For first-principles calculations, the PBE of generalized gradient approximation (GGA) was chosen as the exchange-correlation functional and the vdW interactions were taken into account at the vdW-DF level with the optB88 used as exchange functional. With cutoff distance increasing, more interactions are included, leading to a stronger phonon anharmonicity and phononphonon scattering. Thus the thermal conductivity decreases with the increasing cutoff distance, and finally converges at a large enough cutoff distance. The thermal conductivities of phosphorene along both zigzag and armchair directions decrease with cutoff distance increasing, showing a distinct exponential-like decreasing trend and toward the converged thermal conductivity of phosphorene. A large decrease of thermal conductivity at the cutoff distance of ~6 Å is observed, which is due to the long-ranged interactions caused by resonant bonding in phosphorene [⁵⁴].

Another work that reported the convergence test of thermal conductivity of phosphorene with respect to the cutoff distance was presented by Jain and McGaughey [³¹]. For first-principles calculations, the scalar relativistic pseudopotential was used to describe interatomic interactions, which did not include the vdW interactions. The thermal conductivity was calculated using similar iterative method based on ALD/BTE as employed in the work by Qin et al. [³⁸]. The convergence results are shown in Fig. 4(b). However, the thermal conductivity seems not decreasing with the increasing cutoff distance, which does not show a distinct convergence trend while jumps in a quite wide range. Besides, it is a little bit confusing that the values of thermal conductivity as reported in the abstract and main text (zigzag: 110 W·(m·K)⁻¹, armchair: 36 W·(m·K)⁻¹) [³¹] are not consistent with the convergence results as shown in Fig. 4(b). Based on the convergence results of thermal conductivity with respect to cutoff distance in their Supplemental Materials [³¹] (reproduced here as Fig. 4(b)), the thermal conductivity along zigzag direction should be less than 100 W·(m·K)⁻¹.

To understand the difference between the convergence results reported by Qin et al. [³⁸] and Jain and McGaughey [³¹], further studies were performed by Qin et al. They performed additional test with the PBE of GGA chosen as the exchange-correlation functional NOT including vdW interactions for the first-principles calculations [³⁸]. The thermal conductivity was obtained for the same interactions cutoff radius (7.88 Å, 20th nearest neighbors) in case of not including vdW interactions (zigzag: 129.47 W·(m·K)⁻¹, armchair: 23.04 W·(m·K)⁻¹) (Fig. 5(a)), which were quite large compared with the results obtained in case of including vdW interactions (Fig. 4(a)). The large thermal conductivity was found primarily due to the enhanced phonon lifetime, which lay in the effect of vdW interactions on the formation of resonant bonding.

The vdW interactions were proved in previous studies to have a significant effect on the properties of bulk BP and phosphorene [¹³,⁵⁵]. Excluding vdW interactions will break the formation of resonant bonding by changing lattice constants and the distance between P atoms, leading to weakened long-ranged interactions, and then the much higher thermal conductivity with the same interactions cutoff radius (7.88 Å, 20th nearest neighbors) [³¹,³⁸]. The vdW interactions describe the long-ranged non-covalent-bonding interactions between atoms, which are usually used to describe the interlayer interactions in systems such as bilayer grapheme [⁵⁶–⁵⁹]. In the special single layer systems of phosphorene, there are some P atoms close to each other due to the unique hinge-like structure [¹⁷]. Although there is no covalent bonding between these atoms, there indeed exist some interactions (wave function overlap), which can be captured by the vdW functional [¹³]. The vdW interactions between these non-covalent-bonding atoms have a remarkable effect on the thermal transport properties of phosphorene, while the PBE functional fails to precisely describe the non-covalent-bonding interactions. In addition, the vdW interactions make the distance between the non-covalent bonding atoms closer and thus benefit the formation of resonant bonding [¹³,³⁰,³⁸]. Consequently, including the vdW interactions in phosphorene leads to the resonant bonding driving long-ranged interactions and a further low thermal conductivity. Note that different vdW functionals might lead to different results due to the specific implementation. In this line, it might be expected that the thermal conductivity of phosphorene could be enlarged by breaking the intrinsic resonant bonding, such as by doping or by applying external tensile strain along the armchair direction. For other systems, in principles, if the covalent bond is formed between atoms, the vdW interactions can be neglected due to the much weaker strength than the covalent bond interactions. Some test works have been conducted and it is found that the vdW interactions have no significant effect on the thermal conductivity of graphene, silicene, and monolayer GaN [⁶⁰]. However, if there are non-covalent-bonding atoms close to each other in some systems, it would be safe to test the vdW to see whether there exist vdW interactions and what is the effect. In most situations, there should be always some difference considering the vdW or not, and the difference can be accepted considering even the results from PBE and LDA are not always exactly the same [⁵³].

Another interesting topic is the size dependent (convergence/divergence with Q-grid) behavior in 2D materials, which has raised a lot of debates for a long time [⁶¹]. For example, it was claimed by Gu and Yang[⁶²]. and Xie et al. [⁶³] that the thermal conductivity of silicene diverged with the sample size. Similar conclusions were also drawn for graphene [⁶⁴]. On the other hand, it was reported that the thermal conductivities of graphene and silicene were size independent [⁶²,⁶⁵,⁶⁶]. As for the case of phosphorene, it was claimed by Zhu et al. [³⁰] that there existed the a coexistence of size-dependent and independent thermal conductivity along zigzag and armchair directions, respectively (Fig. 5(b)), which was distinctly different from the isotropically divergent thermal conductivities in two-dimensional (2D) graphene and silicone [⁶¹–⁶³]. The size-dependent thermal conductivity was concluded due to the quickly blowing up of the lifetime of phonon modes when approaching the G-point in the Brillouin zone (BZ) [⁶⁴]. The contribution of acoustic (FA, TA, LA) and optical phonon branches to the total thermal conductivity of phosphorene were further studied to understand the dependence on the size of the q mesh of thermal conductivity along the zigzag direction [³⁰]. It is found that the divergent thermal conductivity of phosphorene along the zigzag direction is mainly caused by the TA phonon branch, while the thermal conductivity contributed from FA, LA, and all other (optical) phonon branches converge with the increasing size of the q mesh. However, as shown in Fig. 5(a), the thermal conductivity of phosphorene along both the zigzag and armchair directions is size (Q-grid) independent at a large cutoff distance (7.88 Å). The discrepancy was analyzed lying in the far from large enough cutoff radius (4.4 Å) used in the third-order IFCs calculations by Zhu et al. [³⁰,³⁸]. The quickly blowing up of the phonon lifetime of low frequency phonon modes can be suppressed when the long-ranged interactions (>~6 Å) are effectively involved. The so-called coexistence of size-dependent and independent behavior of thermal conductivity no longer exists when the cutoff distance is large enough to exceed the range of physically relevant interactions of anharmonicity [³⁸].

Resonant bonding in phosphorene

So far, the thermal transport properties of phosphorene were reviewed, which show a huge deviation in the reported thermal conductivity. The mechanism underlying the deviation was discussed by reviewing the effect of different functionals and cutoff distance on the thermal transport properties of phosphorene. It was found that the vdW interactions played a key role in the formation of resonant bonding, which led to long-ranged interactions. It is important to take into account the vdW interactions, including the long-ranged interactions caused by the resonant bonding with a large cutoff distance, to get an accurate and converged thermal conductivity of phosphorene [³⁸]. A weak sp-hybridization is generally a premonitor for the emergence of resonant bonding [⁶⁷,⁶⁸]. Due to the weak sp-hybridization, s-orbital is fully filled and the bonding is only formed by the p-orbitals. Generally, the resonant bonding is formed in group IVVI compounds [⁶⁹]. Given the rocksalt-like structure (each atom has six nearest neighbors) and the three valence p electrons per atom in average, the bonding configuration based on the electron occupation is not unique. Consequently, the real bonding state is a hybridization among all the possible bonding configurations for the three electrons forming the six bonds. Phosphorene possesses the hinge-like structure, which is puckering. The hinge-like structure of phosphorene can be actually regarded as a deformed rocksalt structure in two-dimensional [³⁸], where there are 5 bonds to be formed for each P atom. Due to the weak sp hybridization in phosphorene (Fig. 6(a)), only three p-orbitals (electrons) are available for the bonding, and the covalent bonding in phosphorene is unsaturated. Thus, the real bonding state in phosphorene is a hybridization or resonance among different electronic configurations of p-electrons occupations.

Considering the collinear resonant bonding with the hinge-like structure, long-ranged interactions would exist in phosphorene due to the resonant bonding. The microscopic picture can be intuitively understood [⁵⁴]. The resonant bonding is a superposition of different bonding configurations. Due to the more bondings than those allowed by the 8-N rule, the single, half-filled p-band forms two bonds to the left and right simultaneously. Thus one p-electron is shared by the two bonds, which leads to long-ranged interactions. For example, if one atom is displaced along the+x direction, it perturbs the p_x orbital of the adjacent atom. Thus, the bonding electrons on the -x side of the adjacent atom can easily move to the+x side since both sides are in the same p_x orbital due to the resonant bonding. This perturbation can persist over long ranges owing to the large electronic polarizability and the collinear bonding characteristics [⁷⁰]. Figure 6(b) shows the phonon dispersion of phosphorene where the TOz phonon branch is softened with frequency decreasing [³⁸]. The softness of the TOz phonon branch is due to the long-ranged interactions in phosphorene, which is caused by the resonant p-bonding as discussed above and can be understood based on the 1D lattice chain model [⁵⁴].

The long-ranged interactions caused by the resonant bonding have a remarkable effect on phonon anharmonicity and further on thermal conductivity, especially on the convergence process with cutoff distance. As discussed in Section 4, the convergence test of thermal conductivity with respect to the cutoff distance (Fig. 4(a)) reported by Qin et al. [³⁸] shows a distinct exponential-like decreasing trend and toward the converged thermal conductivity of phosphorene. A large decrease of thermal conductivity at the cutoff distance of ~6 Å is observed, which is due to the long-ranged interactions caused by the resonant bonding in phosphorene [³⁸]. Another point of the effect of long-ranged interactions on the thermal transport is the problem of convergence/divergence with the sample size (Q-grid). Generally, a cutoff distance not large enough would give a relatively larger thermal conductivity than the real value or even yield a diverged thermal conductivity, in particular for low-dimensional systems [³⁰,³⁸,⁶¹]. As for the case of phosphorene, it was claimed by Zhu et al. [³⁰] that there existed the coexistence of size-dependent and independent thermal conductivity along the zigzag and the armchair directions, respectively (Fig. 5(b)). The claimed size-dependent behavior lay in the small cutoff radius (4.4 Å) used in the calculations of the third-order IFCs, which was reproduced by another work lately with the same cutoff distance [³⁸]. The size-dependent thermal conductivity in the case of using a small cutoff distance primarily lay in the quickly blowing up of the lifetime of acoustic phonon modes when approaching the G point [⁶⁴]. However, the lifetime of low frequency phonon modes can be suppressed when long-ranged interactions (~6 Å) caused by the resonant bonding are effectively involved.

The so-called coexistence of size-dependent and independent behavior of thermal conductivity in phosphorene no longer exists when the cutoff distance is large enough to exceed the range of physically relevant interactions of anharmonicity [³⁸]. As shown in Fig. 5(a), the thermal conductivity of phosphorene along both the zigzag and the armchair directions show size (Q-grid) independent behaviors with the cutoff distance of 7.88 Å.

Summary and perspectives

In summary, in this mini-review the unique thermal transport properties of phosphorene caused by the hinge-like structure were examined. There existed a huge deviation in the reported thermal conductivity of phosphorene. Besides, the mechanism underlying the deviation was discussed by reviewing the effect of different functionals and cutoff distance on the thermal transport properties of phosphorene. It was found that the vdW interactions played a key role in the formation of resonant bonding, which led to long-ranged interactions. It was important to take into account of the vdW interactions and include the long-ranged interactions due to resonant bonding with a large cutoff distance to get an accurate and converged thermal conductivity of phosphorene. Then, a fundamental insight into the thermal transport was provided based on the review of resonant bonding in phosphorene. Moreover, the progresses of the thermal transport in phosphorene were summarized, which would be of great significance to the design and development of efficient phosphorene based nano-electronics.

Despite the significant accomplishment that has been gained in studying the thermal transport properties of phosphorene in the past few years, some important physical fundamentals and issues still remain to be clarified. For instance, the actual thermal conductivity of phosphorene should be further confirmed by testing more computational parameters and especially various exchange-correlation functionals (such as PBEsol, LDA, rPBE, PW91, optB86b). Besides, the thickness dependent thermal conductivity of few layer phosphorene is on emerging demand to get the whole trend of the thermal conductivity from monolayer to finite thickness phosphorene films, and further to bulk form. The effect of different substrates on the phonon transport in 2D phosphorene based structures also demands further systematic studies. Moreover, the 2D phosphorene based heterostructures could possess even more fantastic phonon transport behavior, which could be the new building blocks for advanced functional devices in the next generations.

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