Strongly anisotropic in-plane thermal transport in single-layer black phosphorene

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## Strongly anisotropic in-plane thermal transport in single-layer black phosphorene

Ankit Jain & Alan J. H. McGaughey

Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213.

wo-dimensional (2D) materials (e.g., graphene, MoS2, silicene) are a focus of intense research because of
their rich physics and potential for integration into next-generation electronic and energy conversion
devices 1–4 . As opposed to their bulk counterparts, the optical, electronic, mechanical, and thermal properties
# T
of 2D materials can be easily tailored through the application of external strain, by introducing defects, or by
stacking multiple layers of the same or different 2D materials. For example, the thermal conductivity of freelysuspended single-layer graphene is reduced from 3000–5000 W/m-K to 600 W/m-K by depositing it on amorphous SiO25.

Using first principles calculations, we predict the thermal conductivity of the two-dimensional materials
black phosphorene and blue phosphorene. Black phosphorene has an unprecedented thermal conductivity
anisotropy ratio of three, with predicted values of 110 W/m-K and 36 W/m-K along its armchair and zigzag
directions at a temperature of 300 K. For blue phosphorene, which is isotropic with a zigzag structure,
the predicted value is 78 W/m-K. The two allotropes show strikingly different thermal conductivity
accumulation, with phonons of mean free paths between 10 nm and 1 mm dominating in black
phosphorene, while a much narrower band of mean free paths (50–200 nm) dominate in blue phosphorene.
Black phosphorene shows intriguing potential for strain-tuning of its thermal conductivity tensor.

ous SiO25.

Recently, a new and promising 2D semiconductor, black phosphorene, was fabricated by exfoliating a few
layers from bulk black phosphorus 6,7 . Similar to graphene, black phosphorene has a honeycomb-like structure,
but it is non-planar [Fig. 1(a)]. Single-layer black phosphorene is a direct-gap semiconductor with a predicted
band gap of 2 eV 8 . The band gap decreases with an increasing number of layers and is 0.3 eV for the bulk phase.
Performance characteristics similar to or better than other 2D materials have been achieved for a black phosphorene-based transistor 7 . What distinguishes black phosphorene from other 2D materials is its anisotropic
structure, which leads to direction-dependent optical and electronic properties that vary by as much as 50% 8,9 .

Our objective is to study the lattice thermal conductivity of single-layer black phosphorene, for which there is
no existing experimental or theoretical data available. Ong et al. investigated the effect of strain on the ballistic
thermal conductance of black phosphorene using non-equilibrium Green’s functions based on harmonic lattice
dynamics calculations 10 . Our aim is to predict the thermal conductivity of black phosphorene by including
anharmonic phonon-phonon scattering. Thermal transport characterization is important for application in most
devices, where large electrical currents can lead to Joule heating, non-radiative recombination, and potentially
high operating temperatures. We find that the in-plane thermal transport in black phosphorene is strongly
anisotropic, with thermal conductivity varying by a factor of three over the two orthogonal directions. While
strong anisotropy in thermal conductivity is observed for van der Waals layered materials when comparing the
in-plane and cross-plane directions (e.g., two orders of magnitude in graphite 11 ), no other covalently-bonded 2D
or 3D materials show the in-plane anisotropy we predict for black phosphorene.

We also investigate the thermal transport in blue phosphorene, another single-layered allotrope of phosphorus

[Fig. 1(b)], which was recently predicted to be nearly as stable as black phosphorene 12 . First principles calculations
predict a band gap in blue phosphorene in excess of 2 eV 12, but unlike black phosphorene, blue phosphorene is
isotropic.

Methods
Thermal transport in semiconductors like black phosphorene and blue phosphorene is dominated by atomic vibrations whose energy is
quantized as phonons. The phonon contribution to thermal conductivity in the l direction can be calculated using a solution to the

Figure 1 | Crystal structure of (a) black phosphorene and (b) blue
phosphorene. Atoms in different planes are denoted by different colors
and the unit cell and the in-plane lattice vectors are shown in the top views.
Black phosphorene has a four-atom unit cell with an armchair structure
along a1 and a zigzag structure along a2. Blue phosphorene has a two-atom
unit cell with the zigzag structure along a1 and a2. The side views show the
armchair and zigzag structures. The lattice constants are | a1 | 5 4.43 A ˚ and

| a2 | 5 3.28 A ˚ for black phosphorene, and | a1 | 5 | a2 | 5 3.15 A ˚ for blue
phosphorene.

Boltzmann transport equation (BTE) that uses the Fourier law, giving 13,14

kl~

cph,ivg 2,l,i tl,i: ð1Þ

The summation in Eq. (1) is over all the phonon modes in the first Brillouin zone. The

The summation in Eq. (1) is over all the phonon modes in the first Brillouin zone. The� k
, enumerates phonon wave vector, k, and polarization, n. On the

Figure 2 | Phonon dispersion in the high symmetry directions for (a)
black phosphorene and (b) blue phosphorene. The slope of the
longitudinal acoustic phonon branches at the Gamma point (shown as
dashed straight lines), which represents the sound velocity, is anisotropic
for black phosphorene and isotropic for blue phosphorene.

allotropes (14.0 THz in black and 16.3 THz in blue), but the
phonon band gap in blue phosphorene (4.9 THz) is almost double
that in black phosphorene (2.5 THz). The longitudinal acoustic
phonon group velocity close to the C point (i.e., the sound speed),
is 7,733 m/s in the C 2 Y (armchair) direction and 4,168 m/s in the
C 2 X (zigzag) direction for black phosphorene, an indication of
anisotropic phonon transport. For blue phosphorene, the sound
velocity is 8,287 m/s in both the C 2 M and C 2 K directions.

Thermal Conductivity. The thermal conductivities of black phosphorene and blue phosphorene for temperatures between 200 and
500 K are plotted in Fig. 3(a). For black phosphorene, thermal
transport is anisotropic and we plot the thermal conductivity in
both the armchair and zigzag directions [see Fig. 1(a)]. Predicting
the thermal conductivity of a 2D material requires specification of
the layer thickness. We choose the bulk layer separation, which is
5.25 A ˚ for black phosphorus 19 and 5.63 A ˚ for blue phosphorus 12 . As
the thermal conductivity scales linearly with the layer thickness, the
values reported here can be easily modified for other choices.

Thermal conductivity decreases with increasing temperature, as
expected for a phonon-dominated crystalline material. At a temperature of 300 K, the predicted thermal conductivities are 110 W/m-K
(zigzag) and 36 W/m-K (armchair) for black phosphorene and
78 W/m-K for blue phosphorene. For black phosphorene, the thermal conductivity in the zigzag direction is three times higher than
that in the armchair direction. This anisotropy could be useful in the
design of heat channeling in micro- and nano-devices. We attribute
this anisotropy in thermal conductivity to the anisotropic phonon
dispersion, which leads to direction-dependent group velocities

[Fig. 2(a)]. Based simply on the zone-center longitudinal acoustic
group velocities Eqn 1 predicts a thermal conductivity anisotropy of

k
mode index, i:
n

right-hand side of Eq. (1), cph,i is the volumetric specific heat, vg,l,i is the lcomponent of the phonon group velocity vector vg,i, and tl,i is the phonon lifetime.
The phonon mean free path is jvg,ijtl,i. The specific heat is calculated using BoseEinstein statistics. The� group velocity vector is related to the mode frequency,
vi:v k n , as vg,i~Lvk n Lk. The phonon frequencies are obtained by diagonalizing

Einstein statistics. The� group velocity vector is related to the mode frequency,
vi:v k n , as vg,i~Lvk n Lk. The phonon frequencies are obtained by diagonalizing

the dynamical matrix and the phonon lifetimes are obtained using an iterative
(full) solution of the linearized BTE for phonons 15 . The phonon-phonon scattering rates are obtained by considering three-phonon interactions in the Fermi
golden rule. At a temperature of 300 K, the electronic contribution to the thermal
conductivity of black phosphorene is predicted to be less than 3 W/m-K for a
typical carrier concentration of 10 12 cm 22 16 .

In our thermal conductivity calculations, the only required inputs are harmonic
and anharmonic force constants, which are obtained from density functional theory
(DFT) and density functional perturbation theory (DFPT) calculations. We use a
scalar relativistic pseudopotential generated using the projector augmented-wave
method as implemented in the plane wave-based quantum-chemistry package
Quantum Espresso 17 . The plane wave energy cutoff is 50 Ry. To remove inter-layer
interactions due to the periodicity of the computational cell, we use a vacuum of 30 A ˚
for black phosphorene and 17 A ˚ for blue phosphorene. For black phosphorene, the
harmonic force constants are calculated on phonon and electronic wave-vector grids
of 14 3 12 3 1. The cubic force constants are obtained by finite differencing of
Hellmann-Feynman forces on a 144-atom supercell with a Gamma-point electronic
wave-vector grid. For blue phosphorene, the electronic and phonon wave-vector grids
for the harmonic force constants are 10 3 10 3 1 and the Hellmann-Feynman forces
are obtained using a 128-atom supercell with a Gamma-point electronic wave-vector
grid. For the thermal conductivity calculation, the phonon wave vector grid is 50 3 50
3 1 for both allotropes. Translational invariance (i.e., the acoustic sum rule) for the
cubic force constants is enforced using the Lagrangian approach presented by Li
et al. 18 . We note that the thermal conductivities are converged within 20% (10%) for
black (blue) phosphorene for the above choice of parameters. Further details
regarding these choices are provided in the supplementary information (SI).

Results
Phonon Dispersion. The phonon dispersion in the high-symmetry
directions of the first Brillouin zone for black phosphorene and blue
phosphorene are plotted in Figs. 2(a) and 2(b) and closely match
those reported by Zhu et al. 12 . Black(blue) phosphorene has a
four(two)-atom unit cell, resulting in twelve(six) dispersion
branches The maximum phonon frequency is similar in both

k n , as vg,i~Lvk n

Figure 3 | (a) Temperature-dependent thermal conductivity of black
phosphorene and blue phosphorene. The thermal conductivities are
obtained using an iterative solution of the linearized BTE. The predictions
(symbols) are connected using lines to guide the eye. (b) Thermal
conductivity accumulation functions for black phosphorene and blue
phosphorene at a temperature of 300 K.

3.5, which is comparable to that from the full calculation. For blue
phosphorene, the thermal conductivity is isotropic and is up to 1.8
times lower than the zigzag direction thermal conductivity of black
phosphorene.

Our thermal conductivities are obtained using an iterative solution
of the linearized BTE. For single-layer graphene, Lindsay et al.
showed that the commonly used relaxation time approximation
(RTA) of the BTE under-predicts the thermal conductivity by more
than a factor of five at a temperature of 300 K 22 . We find that the RTA
under-predicts the thermal conductivity by up to a factor of 1.3 for
black phosphorene and 2.0 for blue phosphorene at a temperature of
300 K (see SI).
The thermal conductivity of black phosphorene has recently also
been predicted by other researchers. Zhu et al. used the RTA to

predict values of 84 W/m-K (24 W/m-K) in the zigzag (armchair)
direction 20 . These values closely match our RTA predictions of
81 W/m-K (30 W/m-K) (see SI). Qin et al. predict black phosphorene thermal conductivities a factor of three lower than our values 21 .
This difference may be due to (i) their use of the RTA, (ii) their choice
of the cubic force constant cutoff, and/or (iii) the implementation of a
translational invariance constraint on the third-order force constants. Our predictions of the effects of these factors on thermal
conductivity are presented in the SI.

In Fig. 3(b), we plot the thermal conductivity accumulation functions for black phosphorene and blue phosphorene at a temperature
of 300 K. The thermal conductivity accumulation function describes
the contribution of different mean free path phonons towards the
total thermal conductivity of a material 23 . Phonons with mean free
paths spanning over two orders of magnitude (10 nm to 1 mm)
contribute towards the thermal conductivity in black phosphorene.
For blue phosphorene, however, the accumulation function closely
resembles a step function, with the major contribution coming from
phonons with mean free paths between 50 and 200 nm. This steep
thermal conductivity accumulation in blue phosphorene is similar to
that in silicene (which also has buckled hexagonal structure), where
phonons with mean free path between 5 and 20 nm contribute more
than 80% to the thermal conductivity 24 . The thermal conductivity
accumulation functions indicate that, unlike in graphene, where
thermal conductivity is predicted to increase with sample sizes even
greater than 10 mm, changing the sample size beyond 10 mm will
have a minimal effect on the thermal conductivity of either phosphorene allotrope at a temperature 300 K.

As mentioned above, the structure of blue phosphorene is similar
to that of silicene. Blue phosphorene, however, has a thermal conductivity of 78 W/m-K at a temperature of 300 K, which is more
than eight times higher than that predicted for silicene (9.4 W/mK) 24 . This higher thermal conductivity of blue phosphorene is the
result of a larger sound velocity and the larger frequency gap in its
phonon dispersion [Fig. 2(b)], which reduces the number of threephonon scattering processes that can satisfy the energy conservation
selection rule 25–28 . The zigzag direction thermal conductivities of
black and blue phosphorene are comparable in magnitude to that
predicted for a 10 mm MoS2 sample prediction at a temperature of
300 K (108 W/m-K) 29 .
At a temperature of 300 K, Lindsay et al. 30 predicted the thermal
conductivity of graphene to be 3,600 W/m-K, which is more than 30
times higher than the thermal conductivity of either black phosphorene or blue phosphorene. We believe that the lower thermal conductivity of the phosphorene allotropes is due to their: (i) smaller sound
velocities (4,000–8,000 m/s compared to 21,300 m/s in graphene 31 ),
(ii) lower Debye temperatures [500 K (see SI) compared to 2,300 K
for graphene 32 ], resulting in higher phonon-phonon scattering rates
as more phonon modes are active at a given temperature, and (iii)
non-planar structure, which breaks the out-of-plane symmetry. This
out-of-plane symmetry exists in graphene and restricts the participation of odd numbers of ZA phonons in phonon-phonon scattering
events 22 . As a result, the predicted contribution of ZA phonons to
thermal conductivity in the phosphorene allotropes varies from 12–

Table 1 | Contribution of different phonon modes branches [longitudinal acoustic (LA), transverse acoustic (TA), out-of-plane acoustic (ZA),
and all optical] towards the total thermal conductivity in black phosphorene, blue phosphorene, MoS2 (10 mm sample 29 ), and graphene
(10 mm sample 30 ) at a temperature of 300 K

Material Total thermal conductivity (W/m-K) LA (%) TA (%) ZA (%) Optical (%)

Black phosphorene (zigzag) 110 32 22 31 15
Black phosphorene (armchair) 36 28 33 12 27
Blue phosphorene 78 26 27 44 3
MoS229 108 28 24 39 9

Graphene 30 3600 8 15 76 1

Figure 4 | (a) Stress and (b) sound velocity in black phosphorene and blue
phosphorene under bi-axial strain. The predicted values (symbols) are
connected using lines to guide the eye.

44% as compared to 76% in graphene at a temperature of 300 K 30, as
presented in Table 1. We note that the thermal conductivity contributions of the different acoustic phonon branches in black phosphorene and blue phosphorene are similar to those predicted for
MoS229, which also has a non-planar structure. More information

is provided in the SI.

The thermal conductivity of freely-suspended single-layer graphene at a temperature of 300 K is reduced from 3000–5000 W/
m-K to 600 W/m-K when it is deposited on amorphous SiO25.

This more than a factor of five reduction in the thermal conductivity is due to the increased scattering of ZA phonons in supported graphene. The results presented in this work are for
suspended phosphorene. As can be seen in Table 1, the contribution of ZA phonons is 31% (12%) in the zigzag (armchair)
direction in black phosphorene and 44% in blue phosphorene.

We expect a comparable decrease in the thermal conductivity of
supported phosphorene samples.

Strain Tuning of Thermal Conductivity. We now consider the
possibility of strain-tuning the thermal conductivity of black
phosphorene and blue phosphorene. In Fig. 4(a), we plot the
stresses in both allotropes when they are subjected to a bi-axial
tensile strain. For black phosphorene, the stress is anisotropic and
is three times lower in the armchair direction compared to the zigzag
direction. For blue phosphorene, which has a zigzag structure, the
stress is isotropic and is 1.5 times larger than the stress in the zigzag
direction of black phosphorene.

As can be seen from Eqn. 1, thermal conductivity scales as the
square of the phonon group velocities. We plot the strain-dependence of the sound velocities in Fig. 4(b) for black phosphorene and
blue phosphorene under the bi-axial strain conditions. The sound
velocities are maximum at zero strain for both materials and decrease
with increasing strain. Using this reduction as a guide, we estimate
that the thermal conductivities of both allotropes may decrease by a
factor of 1.7 at a strain of 8%. Similar calculations for uni-axial strains
suggest the possibility for strain-tailoring the thermal conductivity
anisotropy in black phosphorene. For example, the estimated anisotropy in the thermal conductivity of black phosphorene is a factor of
4.6 (2.2) for uni-axial strain of 6% along the armchair (zigzag) direction (see SI).

Anisotropy. To the best of our knowledge, no other 2D material
displays anisotropic in-plane thermal transport. Furthermore, the
thermal conductivity anisotropy in black phosphorene is striking
when compared to that found in layered, wurtzite, and
orthorhombic three-dimensional crystal structures, as compiled
in Table 2. We quantify anisotropy, r, by the ratio of the
maximum and minimum direction-dependent thermal conductivities. The maximum r is in the layered structures graphite
and hexagonal boron nitride. These layers are weakly bonded
through van der Waals interactions compared to the strong
covalent bonds within the layers, which results in poor
interlayer heat transport. Of the remaining materials in Table 2,
which are all covalently bonded, the maximum anisotropy is for
our prediction for black phosphorene, which is twice as large as
the next highest value.

Summary
We predicted the thermal conductivity of black phosphorene and
blue phosphorene using first-principles-driven lattice dynamics calculations and a full (iterative) solution of the BTE. We found a factor
of three anisotropy in the thermal conductivity of black phosphorene, which could potentially be tuned up to 4.6 using strain. At a
temperature of 300 K, the predicted thermal conductivities of both
phosphorene allotropes are larger than that of silicene, similar to that
of MoS2, and are up to two order of magnitude smaller than that of
graphene.

Table 2 | Anisotropy in thermal conductivity for selected materials with layered, wurtzite, and orthorhombic crystal structure at a temperature of 300 K. ‘exp’ and ‘pred’ in the first column denote experimental measurements and simulation predictions

Material kmax (W/m-K) kmin (W/m-K) r

Graphite (exp 33 ) (1000–2000) 6,300
h-BN (exp 33 ) (200–300) 2,100
Black Phosphorene 110 36 3.1
SnSe (exp 34 ) 0.70 0.46 1.52
GaN (pred 14 ) 401 385 1.04
AlN (pred 14 ) 322 303 1.06
NdFeO3 (pred 35 ) 3.01 2.68 1.12
NdAlO3 (pred 35 ) 6.61 5.72 1.16

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Author contributions

A.J. performed the calculations. A.J. and A.J.H.M. analyzed the data. A.J. wrote the
manuscript. A.J.H.M. reviewed and edited the manuscript.

Additional information
[Supplementary information accompanies this paper at http://www.nature.com/](http://www.nature.com/scientificreports)
[scientificreports](http://www.nature.com/scientificreports)

Competing financial interests: The authors declare no competing financial interests.

How to cite this article: Jain, A. & McGaughey, A.J.H. Strongly anisotropic in-plane
thermal transport in single-layer black phosphorene. Sci. Rep. 5, 8501; DOI:10.1038/
srep08501 (2015).

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