Few-layer black phosphorus: emerging 2D semiconductor with high

anisotropic carrier mobility and linear dichroism

 

Jingsi Qiao,1, 2,§ Xianghua Kong,1, 2,§ Zhi-Xin Hu,1,2 Feng Yang,1,2,3 and Wei Ji1,2,*

1Department of Physics, Renmin University of China, Beijing 100872, China

2Beijing Key Laboratory of Optoelectronic Functional Materials & Micro-nano Devices,

Renmin University of China, Beijing 100872, China

3College of Physics and Electronic Engineering, Institute of Solid State Physics,

Sichuan Normal University, Chengdu 610068, China 

Abstract

Two-dimensional crystals are emerging materials for future nanoelectronics. It has

been seeking for candidate channel materials that have sufficient electronic band gap and

high carrier mobility. We present a theoretical investigation of geometric and electronic

structures of few-layer black phosphorus and associated electric and optical properties

were predicted. It has a direct band gap, tunable from 1.51 eV of a monolayer to 0.59 eV

of a 5-layer. Anisotropic mobilities along two directions were found for both electron and

hole. The ``higher mobility” varies from hundreds to thousands of cm2/V•s. The

monolayer hole mobility along a certain direction was predicted 7737 cm2/V•s, over 20

times larger than the corresponding value in the perpendicular direction. Linear

dichroism along two in-plane directions was found for light absorption, which enables

feasibly determining the orientation of BP few-layers. These results make few-layer black

phosphorus a promising candidate for future electronics and optoelectronics.

 

__________ § These authors contributed equally to this work * wji@ruc.edu.cn, http://sim.phys.ruc.edu.cn    

The discovery of graphene opened many new areas of research, among which two-

dimensional (2D) atomic layers, including graphene, transition metal dichalcogenides (TMD)

and others, were intensively investigated as emergent materials for future electronics.1-17 To

realize a high performance device, e.g. field effect transistor (FET), it requires a sufficient

electronic band gap and a reasonably high carrier mobility of the channel material and excellent

electrode-channel contact.5-7,9,11-14 Graphene offers extremely high carrier mobility, due to its

Dirac-like linear dispersion, which thus lead to graphene a promising candidate for, e.g. high

speed FET, however, it is gapless.2-7 Although tremendous research efforts have been made on

how to open a gap in graphene nanostructures, it is still an open issue of relatively large off

current and low on-off ratio.5,6,15 The emergence of monolayer TMDs, e.g. MoS2, as its first

FET recently demonstrated,9 has attracted substantial research interest. Unlike graphene,

monolayer MoS2 is a direct band gap semiconductor with a carrier mobility of approximately

200 cm2/V·s,9 improvable up to 500 cm2/V·s,16 which is fairly good, but orders of magnitude

lower than that of graphene.5,6

An open issue was thus raised that the whole community has been seeking for a 2D

material which is a, preferably direct gap, semiconductor with considerably high carrier

mobility and potentially can form excellent contact with known electrode materials. In this

work, we report a theoretical discovery of high anisotropic carrier mobility and linear dichroism

in a novel category of layered direct band gap semiconductors, namely, few-layer black

phosphorus (BP), an allotrope of phosphorus, as shown in Fig. 1(a). In particular, by density

functional theory calculations we show that few-layer BP, from monolayer up to 5-layer, are

thermally stable, with interlayer interaction energy of -0.44 eV. The bandgap-thickness relation

follows an exponentially decay law that it goes from ~1.5 eV of a monolayer down to ~0.6 eV

of a 5-layer. Carrier mobilities at the room temperature show an explicit anisotropic behavior

that the higher mobility direction with much smaller effective masses was identified. An

exceptionally high hole mobility of monolayer along the ``lower-mobility” direction was found

7737 cm2/V·s, which can be explained by visualizing the wave functions around the band gap.

An explicit linear dichroism was indicated by the computed absorption spectra that the positions

of the first absorption peaks along the two in-plane directions are largely different. These results

show that few-layer BP is a new category of 2D semiconductors that is promising in the

applications of nanoelectronics and optoelectronics.

Geometric and electronic properties of bulk BP

We started from bulk BP to refine the accuracy of our theoretical prediction since there are

no experimental data available for the electronic and geometric properties of few-layer BP.

Figure 1 shows the fully relaxed geometry and associated electronic band structure of bulk BP.

The atomistic structure was fully optimized by the optB88-vdW functional.18,19 Its equilibrium

lattice constants are only about 1% ~ 2% larger than the experiment.20-22 Both the LDA-mBJ

23,24 and HSE06 25,26 methods were employed to predict the electronic band structures, denoted

mBJ (optB88-vdW) and HSE06 (optB88-vdW), as shown in Fig. 1(b). These two combinations

of functionals were found the best-fit among all considered methods on the basis of their results

of the band gap and lattice parameters of bulk BP (see Fig. S1 and Table S1). Both combinations

(methods) predict that bulk BP is a semiconductor with a direct band gap at the Z point of

0.31eV (mBJ) and 0.36 eV (HSE06), highly consistent with the experimental value of

0.31~0.35eV.27-30 The PBE-G06 and optB86b-vdW methods reproduced the experimental

geometry even better in terms of the theory-experiment difference of lattice constants, however,

based on the ``better” geometry, either mBJ or HSE06 failed to reveal a satisfied value to the

bulk band gap. We thus focused our discussion on the geometry of optB88-vdW and the

associated electronic band structures of HSE06 and mBJ.

A valance and a conduction band around the band gap at the Z point along Z-Q and Z-G

directions are nearly linear (Fig. 1b), indicating very small effective masses. We thus fitted

these bands using the nearly-free electron model to derive their effective masses. Carrier

effective masses along the Z-Q direction are rather small and similar to each other, i.e. 0.12 m0

for hole (HSE06 value, the same hereinafter) and 0.11 m0 for electron; while these along Z-G

are slightly larger, i.e. 0.15 m0 and 0.30 m0, respectively. Sufficiently large values were found

along Z-T’-A’ where the carrier effective masses are 1.15 m0 and 0.71 m0, respectively. All

these results are highly consistent with the early experimental results for bulk BP. 31

Corresponding mBJ results are, only within 0.04 m0 (Z-Q and Z-G) and 0.14 m0 (Z-T’-A’),

larger than those of HSE06. Values of direction Z-Q are similar to high-mobility

semiconductors, e.g. ~0.1 m0 of AlGaAs,32 and four times smaller than that of MoS2.33

Geometric and electronic properties of few-layer BP

Few-layer BP is suggested a likely high-mobility direct band gap 2D semiconductor by

the effective masses of bulk BP. We thus inspected the geometric and electronic properties of

BP few-layers. Table I summarizes the changes of geometric properties with respect to the

increased number of layers. Lattice parameter b elongates by 0.11 Å from bulk to monolayer

while lattice constant a only varies within 0.02 Å. There is an abrupt reduction of b from

monolayer to bilayer, which is, we believe, ascribed to the presence of interlayer interactions

in the bilayer. The significant stretching of b in BP few-layers is primarily due to the increase

of bond angle θ1 but not bond lengths.

Electronic structures of BP few-layers were shown in Fig. 2. In a monolayer, the original

Z point in the bulk BZ folds back to the Gamma (G) point, so that the original Z-T’-A’ and Z-

Q directions in the bulk BZ project into the G-X and G-Y directions of a monolayer, along the

a and b directions in real-space, respectively (see Fig. 2a inset). Results carried out with the

mBJ method are quantitatively the same to those of HSE06 and the results with other methods

are also available in Fig. S2. Monolayer BP is still a direct band gap semiconductor, see Fig.

2a, with the gap width of 1.51 eV residing at the G point. Inclusion of spin-orbit coupling does

not change the bandstructure (see Fig. S3). When putting another layer to a monolayer, the gap

reduces to 1.02 eV and two additional bands emerge around the gap at the G point. Together

with the original two, we denote those bands VB1, VB2, CB1, and CB2, as shown in Fig. 2b.

Each of these four states is extended throughout the bilayer, as shown in Fig. 2c. Bands

VB1 and VB2 are different at the interlayer region although they share the same origin of atomic

orbitals. A clear bonding-like feature is available at the interlayer region (marked by red

rectangles) of VB1 which is energetically lower, while that of VB2 shows an anti-bonding-like

feature. Similar bonding- and anti-bonding-like features were also found in CB1 and CB2

where the features were not observable between the layers but across the troughs. These features

indicate that wave function overlap, rather than van der Waals interactions, plays a key role in

the inter-layer interaction, which explains the abrupt reduction of lattice constant b from

monolayer to bilayer. This inter-layer interaction introduces band dispersions of VBs and CBs

in the direction perpendicular to the layer plane, which leads to the band gap dropping by 0.5

eV from monolayer to bilayer.

The thicker the BP few-layer, the stronger the inter-layer interaction, and thus the larger

the dispersion of VBs (CBs) resulting in a smaller band gap. The gap, therefore, continuously

drops when putting more layers to the bilayer and eventually reaches to 0.59 eV in a 5-layer BP,

as illustrated in Fig. 2d. We fitted these values by an exponentially decay relation and found

that the corresponding bulk gap was extrapolated 0.53 eV, being 0.17 eV larger than the

predicted value of 0.36 eV, which was ascribed to the elongated lattice parameter b in BP few-

layers. Results calculated with a series of constrained b values confirms this statement,

implying that the band gap is very sensitive to the lateral strain along the b (y) direction and

could be modulated by varying atomic distances and angles along the direction. During the

preparation of this paper, we come to be aware that a strain modulation of band gaps was

demonstrated by a theory. 34

The small effective masses still remain in the few-layers (see Table II). In the G-Y

direction of the monolayer, equivalent to Z-Q of the bulk, the effective masses are 0.15 m0

(hole) and 0.17 m0 (electron), respectively, only 0.04-0.05 m0 larger than those in bulk BP. It is

remarkable that in the G-X direction, equivalent to Z-T’-A’ of the bulk, the VB appears a nearly

flat band close to the G point with an effective mass of 6.35 m0 (see Fig. 2a), 9 times larger

than its bulk value of 0.71 m0; while that for the CB is 1.12 m0, fairly close to its bulk value of

1.15 m0. The VBM, however, moves by approximately 0.1 Å-1 off the G point with a hole

effective mass of -36.17 m0 if the atomistic structure was optimized by a less accurate structural

relaxation methods, e.g. PBE-G06. Unlike the band gap, only the hole effective mass along G-

X shows a strongly layer-dependent evolution that it decreases from 6.35 m0 of a monolayer to

1.81 m0 of a bilayer and eventually to 0.89 m0 of a 5-layer, close to the bulk value of 0.71 m0.

Carrier mobility

Electric properties of few-layer BP are immensely governed by the carrier mobility, a

property relevant with but not solely determined by carrier effective mass. We thus theoretically

predicted the carrier mobility along both x and y directions of few-layer BP under a phonon

limited scattering model, in which phonon scattering is the primary portion for the scattering of

carriers. 32,35-37 The deformation potential E1 and the elastic modulus C2D in the propagation

direction of the longitudinal acoustic wave are highly relevant with the mobility.38,39 All these

data are available in Table II. A standard 2D model was considered by formula35-37

32D

2D * * 2e 1

B d

e C

k Tm m E (1)

Here, *em is the effective mass along the transport direction and *

dm is the density-of-state mass

determined by yxd mmm and T is the temperature.

Predicted mobilities show that holes generally move faster than electrons in both directions,

suggesting better hole conductivity of few-layer BP. In terms of mobility anisotropy, both

electron and hole mobilities along y are, nearly twice for hole and four times for electron, larger

than those along x, except in the monolayer. The monolayer electron mobility along y is 13

times, rather than four times, larger that along x, i.e. 561 cm2/V·s versus 44 cm2/V·s, consistent

with the trend of other few-layers. Much to our surprise that the hole mobility along y is,

however, 23 times smaller than that along x, i.e. 337 cm2/V·s versus 7737 cm2/V·s, indicating

x is the direction of higher hole conductivity in a monolayer. This value is extraordinary large

for monolayer BP, which is ascribed to the extremely small deformation potential of 0.15 eV,

even if the carrier is very heavy (6.35 m0). This ``monolayer exception” is potentially useful in,

e.g. separation of electrons and holes. We can infer from these values that the carrier mobility

for either hole or electron is at the order of 1000 cm2/V·s for few-nanometer-thick BP thin-

films.

The value of 0.15 eV for1

h

xE (E1 along the x direction for hole) in a monolayer is a very

striking result that is an order of magnitude smaller than typical values of E1, e.g. 5.0 eV for

graphene,39 3.9 eV for MoS2,33 and 3.7 eV for h-BN 35 and 7 eV for AlGaAs quantum wells32.

It abruptly jumps to 1.63 eV in a bilayer and smoothly increases to 2.97 eV of a 5-layer. These

results can be well explained by the wave functions shown in Fig. 2c. The inter- and intra-layer

interactions varnish in monolayer BP, so that the wave function of VB of a monolayer, appeared

similar to that of VB2 of a bilayer, is fairly isolated along x. Small deformations along x scarcely

perturbs its wave function and energy level, thus giving rise to a very small deformation

potential of 0.15 eV. Another jump was found for 1

e

yE from 2.72 eV of a monolayer to 5.02

eV of a bilayer BP. It is substantially enhanced that the perturbation to energy levels of CBs by

structural deformations along y if in-plane bonding- and anti-bonding-like features are available

in BP few-layers. The monolayer exception aside, all values of CB were fairly large, i.e. over 7

eV and 5 eV for directions x and y and almost unchanged with the increase of thickness, which

was ascribed to the strongly overlapped wave functions of CBs, especially along direction x

(Fig. 2c). In terms of VBs, the potentials were, however, smaller, i.e. around 2-3 eV, because of

less overlapped wave functions between different P atoms.

Given the effective masses of 0.1 to 0.2 m0, even higher carrier mobilities were initially

expected. The key issue here is the too small elastic modulus, in other words it is ``too soft”,

along the y direction, which was due to the fact that mostly bond angles scarcely bond lengths

are changed when deforming BP along this direction. Among monolayers, BP’s modulus along

the y direction is 14.5 J/m2 which is 4, 20 and 22 times smaller than that of MoS2 ,33 h-BN, 35

and graphene, 39 respectively. These results indicate that much higher mobilities could be most

likely achieved if few-layer BP can be modified stiffer, i.e. larger elastic modulus, along the y

direction by, e.g. external stress, doping or substrate confinement. The change of the

deformation potential with respect to modified elastic moduli is, however, not clear so far,

which is subject to further investigations. The modulus along direction x is much larger than

that along y, however, the carrier hoping velocity is rather small, resulting in the carrier

mobilities smaller than those along y.

Optical absorption spectra and linear dichroism

In addition, optical absorption spectra were predicted by computing the dielectric function.

Two absorption spectra were available in Figs. 3a and 3b, in which the dielectric polarization

vectors are along the x and y directions, marked xx and yy. It shows explicit linear dichroism

that in a monolayer, for example, the band edge of the first absorption peak sits at 3.14 eV for

incident light polarized along the x direction (red in Fig. 3a) but that along the y direction was

found at 1.55 eV (red in Fig. 3b), in the infrared region, corresponding to the direct band gap

of monolayer BP. The energy of band edges for direction x are slightly reduced when increasing

the thickness, but that for direction y are rapidly goes smaller because it is correlated with the

band gaps. Given this striking property of linear dichroism, we proposed an experimental setup

that may be helpful for determining the orientation of BP few-layers using optical spectroscopy,

as illustrated in Fig. 3c. The incident light shall be linearly polarized along a certain orientation

and near-normally incident to a BP few-layer. The absorption spectrum is expected to vary

when rotating the sample. The a or b direction of the few-layer could be thus identified by

monitoring the absorption signal. After the sample orientation identified, it is much more

feasible to construct electrodes or gates utilizing the highest mobility direction of few-layer BP

in an, e.g. FET, device.

Conclusions

In summary, we found theoretically that few-layer BP is a novel category of 2D

semiconductors offering tunable direct band gap and high anisotropic carrier mobility with

respect to the layer thickness. The band gap decrease from 1.51 eV of a monolayer to 0.59 eV

of a 5-layer, well filling the gap between graphene nanoribbons and TMDCs. Anisotropy is

another unique feature of few-layer BP from other 2D channel materials in terms of electric or

optical properties. Higher conductivity is generally found in the direction oriented

perpendicular to the troughs (the y direction) compared with that along the troughs (the x

direction) and holes usually move faster than electrons. The mobility, of hole for example, along

the y direction increase from 337 cm2/V·s of a monolayer to 2722 cm2/V·s of a 5-layer.

Monolayer BP was found very special that a high hole mobility of 7737 cm2/V·s along direction

x was predicted. Results of mobility were well understood from a wave function point of view,

indicating much stronger interlayer coupling in few-layer BP than that in graphene or TMDC.

Both the band gap and the mobility are highly correlated with the structural properties, which

offers various possibilities to further tune the electric or mechanical properties of few-layer BP.

Furthermore, a linear dichroism behavior was found in BP few-layers, allowing us to feasibly

determine its orientation. All these results make few-layer BP a promising candidate of channel

materials for future 2D electronics and optoelectronics.

Methods

Density functional theory calculation. Density functional theory calculations were performed

using the generalized gradient approximation for the exchange-correlation potential, the

projector augmented wave method,40,41 and a plane wave basis set as implemented in the Vienna

ab-initio simulation package.42 The energy cutoff for plane-wave basis was set to 500 eV for

all calculations. Two k-meshes of 10×8×4 and 10×8×1 were adopted to sample the first

Brillouin zone of the conventional cell of bulk and few-layer black phosphorus, and the mesh

density of k points was kept fixed when calculating bandstructures using primitive cells. In

geometry optimization, van der Waals interactions were considered at the vdW-DF level with

optB88 for exchange functional (optB88-vdW) 18,19. The shape and volume of each supercell

were fully optimized and all atoms in it were allowed to relax until the residual force per atom

was less than 0.001 eV/Å. Electronic band structures were calculated at modified Becke-

Johnson (mBJ) 23,24 and hybrid functional (HSE06)25,26 levels based on the atomistic structures

fully optimized by optB88-vdW. A charge gradient c=1.1574 was used in mBJ calculations,

which was extracted from the bulk value.

Carrier mobility calculation. Carrier mobility in 2D read as 3

2D2D * * 2

e 1

B d

e C

k Tm m E35-37, where

*em is the effective mass along the transport direction and *

dm is the density-of-state mass

determined by yxd mmm . Term E1 represents the deformation potential constant of the

VBM (hole) or CBM (electron) along the transport direction, defined 1 0/ ( / ) iiE V l l .

Here iV is the energy change of the ith band under proper cell compression and dilatation (by

a step 0.5 %), l0 is the lattice constant along the transport direction andΔl is the length of

deformation on l0. Elastic modulus of the longitudinal strain in the propagation direction, both

x and y directions, of the LA wave C2D is derived by 20 0 0( ) / ( / ) / 2 E E S C l l where E

is the total energy, S0 is the lattice volume at the equilibrium for 2D systems. All structural or

energy related properties in mobility calculation were done by optB88-vdW and electronic

related properties were computed with the HSE06 functional. The temperature for mobility

calculation is 300 K.

Absorption spectra calculation. Absorption spectra were calculated from the dielectric

function by formula ( ) z( ) 1 - A e , where 2( )= cn

is the absorption coefficient,

2 21 2 1

2

n is the index of refraction, ε1 and ε2 are the real and imaginary parts of

the dielectric function, ω is the circular frequency of the light, c is the light speed in vacuum

and, Δz represents the cell size of the c direction in simulation. Electronic structures were based

on HSE06 results and the k-mesh was doubled in calculating dielectric functions. Effects of

exciton were not considered so far. Total number of bands was set twice to total energy and

band structure calculations. Since the dielectric function is a tensor, the absorption spectra along

the three, a (xx), b (yy), and c (zz), directions were separately obtained.

 

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Acknowledgements

This work was financially supported by the National Natural Science Foundation of China

(NSFC), Grant Nos. 11004244 and 11274380, the Beijing Natural Science Foundation (BNSF),

Grant No. 2112019, and the Basic Research Funds in Renmin University of China from the

Central Government (Grant Nos. 12XNLJ03, 14XNH060 and 14XNH062). W.J. was supported

by the Program for New Century Excellent Talents in University. Calculations were performed

at the Physics Lab of High-Performance Computing of Renmin University of China and

Shanghai Supercomputer Center.

Author contributions

W.J. conceived this research. J.Q., Z.X. Hu and W.J. performed structural and electronic

structure calculations. X.K. calculated deformation potential, elastic modulus, and carrier

mobility. F.Y. and W.J. predicted the optical absorption spectra. J.Q., X.K. and W.J. wrote the

manuscript and all authors commented on it.

   

Figure 1 | Structural and electronic structures of bulk black phosphorus (BP). a, Crystal

structure of bulk BP marked with coordinate axis, lattice vectors a, b, c and structural parameters

R1, R2, θ1 and θ2. b, Electronic band structures of bulk BP calculated with the HSE06 functional

(red solid line) and the mBJ potential (blue dashed line), together with fitted effective masses

along the Z-T’-A’, Z-Q and Z-G directions. Brillouin zone path and the direct gap at Z point were

shown in the insets.

Table I | Structural information of few-layer BP. Lattice constants a, b and Δc (the interlayer

spacing between two adjacent BP layers) and structural parameters R1, R2, θ1 and θ2 of few-layer

and bulk black phosphorus calculated using optB88-vdW. There are slightly difference in θ1 and θ2

between the outermost and inner layers.

NL a (Å) b (Å) Δc (Å) R1(Å) R2(Å) θ1/θ1’ (°) θ2/θ2’ (°)

1 3.32 4.58 3.20 2.28 2.24 103.51 96

2 3.33 4.52 3.20 2.28 2.24 102.96 96.21/95.92

3 3.33 4.51 3.20 2.28 2.24 102.81/102.74 96.30/95.99

4 3.34 4.5 3.20/3.21 2.28 2.24 102.76/102.67 96.34/96.01

5 3.34 4.49 3.20/3.21 2.28 2.24 102.71/102.63 96.37/96.05

Bulk 3.34 4.47 3.20 2.28 2.25 102.42 96.16

 

 

 

Figure 2 | Electronic structures of few-layer BP. a, b, Band structures of monolayer and

bilayer BP calculated with the HSE06 functional (red solid lines) and the mBJ potential (blue

dashed lines), respectively. Top view of the atomistic structure of the monolayer and the

associated Brillouin zone were shown in a inset. Two valence, VB1 and VB2, and two

conduction, CB1 and CB2, states were marked in b. Side views of the atomistic structure of the

bilayer were shown in b inset. c, Wave functions of the four marked states were visualized

along x and y directions with an isosurface of 0.0025 e/Å3. d, Evolution of the direct band gaps

as a function of the thickness. The method used for structural optimization was presented in

parentheses. Band gap values of the monolayer, the extrapolated and actual bulk BP were listed.

Table II | Predicted carrier mobility. Carrier type ``e” and ``h” stand for ``electron” and

``hole”, respectively. NL represents the number of layers, mx* and my* are the carrier effective

masses along directions x and y, E1x (E1y) and Cx_2D (Cy_2D) are the deformation potential and

2D elastic modulus along the x (y) direction. Mobilities µx_2D and µy_2D were calculated using

formula (1) where the temperature T was set to 300 K.

Carrier

type

NL mx*/m0

G-X

my*/m0

G-Y

E1x

(eV)

E1y

(eV)

Cx_2D

(J/m2)

Cy_2D

(J/m2)

µx_2D

(cm2/V·s)

µy_2D

(cm2/V·s)

e

1 1.12 0.17 7.11 2.72 50.80 14.47 44 561

2 1.13 0.18 7.35 5.02 97.31 28.74 75 299

3 1.15 0.16 7.63 5.85 143.60 42.93 106 389

4 1.16 0.16 7.58 5.92 189.79 57.33 140 504

5 1.18 0.15 7.35 5.78 239.91 73.29 190 737

h

1 6.35 0.15 0.15 2.5 7737 337

2 1.81 0.15 1.63 2.45 825 1301

3 1.12 0.15 2.24 2.49 1324 2383

4 0.97 0.14 2.79 3.16 1449 2363

5 0.89 0.14 2.97 3.4 1839 2722

 

   

 

Figure 3 | Optical absorption spectra. a, b, Optical absorption spectra of BP few-layers for

the dielectric polarization vector along the a (xx) and b (yy) directions, respectively. Band edges

of the first absorption peak were extrapolated. It slightly reduces for those along direction xx,

i.e. from 3.14 eV of monolayer (red) to 2.90 eV of 5-layer (orange), but drops rather faster for

those along direction yy, i.e. from 1.55 eV of monolayer (red) to 0.60 eV of 5-layer (orange). c,

Schematic illustration of a proposed experimental setup which may determine the orientation

of BP few-layers using optical absorption spectroscopy and thus could utilize desired electric

properties of few-layer BP. In the experiment, the light shall be linearly polarized along a certain

orientation and near-normally incident to the sample. The sample is rotatable that the a or b

direction of the BP few-layer could be identified by monitoring the recorded absorption signal,

after which, electrodes could be deposited fabricating an FET device.

Few-layer black phosphorus: emerging 2D semiconductor with high

anisotropic carrier mobility and linear dichroism

 

Jingsi Qiao,1, 2,§ Xianghua Kong,1, 2,§ Zhi-Xin Hu,1,2 Feng Yang,1,2,3 and Wei Ji1,2,*

1Department of Physics, Renmin University of China, Beijing 100872, China

2Beijing Key Laboratory of Optoelectronic Functional Materials & Micro-nano Devices,

Renmin University of China, Beijing 100872, China

3College of Physics and Electronic Engineering, Institute of Solid State Physics,

Sichuan Normal University, Chengdu 610068, China 

*Email: wji@ruc.edu.cn

Supplementary Materials 

5 Sections 

4 Figures 

4 Tables 

12 References 

 

 

 

__________ § These authors equally contributed to this work * wji@ruc.edu.cn, http://sim.phys.ruc.edu.cn

1. Best-fit functionals for few-layer black phosphorus (BP).

Lattice constants and structural parameters of bulk BP optimized by various functionals and

methods were summarized in Table SI, in which PBE, RPBE and HSE06 functionals together

with the consideration of the dispersion force at the DFT-G06 and vdW-DF levels were adopted.

Methods HSE06-G06, PBE-G06 and optB86b-vdW very accurately reproduced the

experimental geometry that the theoretical equilibrium volumes are only 0.1% smaller and 1%

and 0.1% larger than the experiment.1-3 Results of PBE and PBE-G06 are consistent with a

previous work.3 Functional optB88-vdW performs the second best in terms of geometry, which

yields an equilibrium volume only 5% larger than the experiment. The calculated bond lengths

are similar regardless of which method used except HSE06-G06, by which the predicted bond

lengths are sufficiently smaller than the others.

Table SI | Lattice constants and structural parameters of bulk black phosphorus calculated

using different computational methods.

Functional a(Å) b(Å) c(Å) R1(Å) R2(Å) θ1(°) θ2(°)

Expt. 2 3.31 4.38 10.48 2.24 2.22 102.09 96.34

PBE 3.30 4.57 11.33 2.26 2.22 103.59 95.98

PBE 3 3.28 4.54 11.22 - - - -

PBE-G06 3.32 4.43 10.49 2.26 2.23 102.48 96.50

PBE-G06 3 3.30 4.40 10.43 - - - -

PBE-TS 3 3.29 4.39 10.82 - - - -

RPBE-G06 3.33 4.57 10.91 2.27 2.23 103.58 96.42

HSE06-G06 3.30 4.42 10.43 2.23 2.20 102.87 96.96

optPBE-vdW 3.34 4.54 10.95 2.28 2.24 103.02 96.07

optB86b-

vdW 3.33 4.35 10.52 2.27 2.24 101.58 96.09

optB88-vdW 3.34 4.47 10.71 2.28 2.25 102.42 96.16

The Figure S1(a) shows the electronic band structures of bulk BP calculated using the LDA-

mBJ 4,5 or the HSE06 6,7 functionals based on the optB86b-vdW atomistic structure, denoted

mBJ (optB86b-vdW) and HSE06 (optB86b-vdW). Although the geometry was very accurately

reproduced by optB86b-vdW, the associated band gaps are only 0.03 eV (mBJ) and 0.05 eV

(HSE06). Similar very small gaps were obtained in HSE06 or mBJ band structures based on

the PBE-G06 or HSE06-G06 geometries. The PBE-G06 band structure even report a negative

band gap. Therefore, the combinations of mBJ or HSE06 functionals with PBE-G06, HSE06-

G06 and optB86b-vdW are unacceptable for predicting the electronic structures of few-layer

BP. In addition, BP appears a metal with a band cross near the Z point in the PBE-G06

bandstructure; while in that of PBE, it opens a direct gap of 0.15 eV, which was ascribed to the

over estimated volume by PBE, giving rise to a weaker overlapping interaction and thus a larger

gap than that of PBE-G06. Combination of HSE06 or mBJ with the optB88-vdW structures is

thus the best-fit method for such prediction, which offers the second best geometry and the best

electronic band structure.

 

Figure S1 | Electronic band structure of bulk black phosphorus calculated with different

method. (a) HSE06 (red solid lines) and mBJ (blue dashed lines) band structures based on the

atomistic structure optimized by optB86b-vdW. (b) Band structures calculated with HSE06

(PBE-G06) (red), PBE (blue) and PBE-G06 (black).

2. Effect of spin-orbit coupling

Figure S2 | Electronic band structure of bulk black phosphorus with and without spin-

orbit coupling. Band structures were calculated by the mBJ potential with spin-orbit coupling

(green dashed lines) and without spin-orbit coupling (black solid lines).

It was expected that phosphorus, a much heavier element than Carbon, should have stronger

spin-orbit coupling in its 2D forms than that of graphene. We considered full spin-orbit coupling

effect in calculating the electronic band structure of bulk BP, as shown in Fig. S2. It shows that

the spin-orbit coupling has no appreciable effect on the band structure, indicating phosphorus

is not ``heavy” enough.

3. Electronic band structure of few-layer BP.

Figure S3(a) and Table SII show that the predicted band gap is less sensitive to which

functional used to optimize the atomistic structures, while the PBE functional significantly

underestimates the band gap by roughly 0.6 eV.

 

Figure S3 | Electronic band structure of monolayer (a) and bilayer (b). Results were

revealed with the HSE06 functional based on the structure of PBE-G06 (red) and PBE

functional (blue).

 

Table SII | Band gaps of few-layer BP. Exact values, in the unit of eV, of band gaps plotted in

Figs. 2d and S3 were summarized here.

Number of

Layers

HSE06

(optB88-vdW)

mBJ

(optB88-vdW)

HSE06

(PBE-G06)

PBE

(PBE)

1 1.51 1.41 1.53 0.91

2 1.02 0.94 1.01 0.58

3 0.79 0.72 0.73 0.42

4 0.67 0.61 0.61 0.36

5 0.59 0.54 0.52 0.28

∞ 0.53 0.49 0.45 0.24

Bulk 0.36 0.31 0.27 0.15

 

 

 

 

 

 

 

 

 

 

4. Predicted carrier mobility of few-layer BP.

Since the found carrier mobilities of few-layer BP is of high anisotropy, two additional

models were considered to comprehensively capture this anisotropic property. An anisotropic

2D 8-11 and a 1D 12 model read as

21

2*

2_3

AS_D2_y 3

2

yeB

Dy

ETmk

Ce (S1)

2_1D

y_1D 1/2 *3/2 21y

2

( )

y

B

e C

k T m E (S2)

Here, *em is the effective mass along the transport direction and T is the temperature. The

anisotropic 2D model makes the carrier mobilities along the x direction much lower than the

standard 2D model, but those along y are much higher. Mobilities predicted by the 1D model

are relevant to the width of a BP ribbon. The narrower the ribbon, the smaller the 1D elastic

modulus, thus the lower the mobility. We considered a BP ribbon of 10nm in width, which is

very reasonable to be fabricated in experiment. Results show that mobilities are rather large in

the 1D model, compared with those values in 2D models. Reduction of 2D BP sheet into 1D

ribbon could, we believe, significantly enhance the carrier mobilities, but the edge structure and

its role in electron scattering are subject to further investigations.

Table SIII | Carrier mobility predicted by the anisotropic 2D model. Carrier type ``e” and

``h” stand for ``electron” and ``hole”, respectively. NL represents the number of layers, mx* and

my* are the carrier effective masses along directions x and y, E1x (E1y) and Cx_2D (Cy_2D) are the

deformation potential and 2D elastic modulus along the x (y) direction. Mobilities µx_2D_AS and

µy_2D_AS were calculated using formula S1.

Carrier

type

NL mx*/m0

G-X

my*/m0

G-Y

E1x

(eV)

E1y

(eV)

Cx_2D

(J/m2)

Cy_2D

(J/m2)

µx_2D_AS

(cm2/V·s)

µy_2D_AS

(cm2/V·s)

e

1 1.12 0.17 7.11 2.72 50.80 14.47 11 960

2 1.13 0.18 7.35 5.02 97.31 28.74 20 499

3 1.15 0.16 7.63 5.85 143.60 42.93 26 695

4 1.16 0.16 7.58 5.92 189.79 57.33 35 905

5 1.18 0.15 7.35 5.78 239.91 73.29 45 1379

h

1 6.35 0.15 0.15 2.5 793 1461

2 1.81 0.15 1.63 2.45 158 3013

3 1.12 0.15 2.24 2.49 323 4341

4 0.97 0.14 2.79 3.16 367 4147

5 0.89 0.14 2.97 3.4 486 4575

 

Table SIV | Carrier mobility predicted by the 1D model. E1x (E1y) and Cx_1D (Cy_1D) are the

deformation potential and 1D elastic modulus along the x (y) direction. Mobilities µx_1D and

µy_1D were calculated using formula S2.

Carrier

type

NL mx*/m0

G-X

my*/m0

G-Y

E1x

(eV)

E1y

(eV)

Cx_1D

(10-7J/m)

Cy_1D

(10-7J/m)

µx_1D

(cm2/V·s)

µy_1D

(cm2/V·s)

e

1 1.12 0.17 7.11 2.72 5.08 1.45 116  2764

2 1.13 0.18 7.35 5.02 9.73 2.87 205  1480

3 1.15 0.16 7.63 5.85 14.36 4.29 273  1941

4 1.16 0.16 7.58 5.92 18.97 5.73 361  2529

5 1.18 0.15 7.35 5.78 23.99 7.33 473  3730

h

1 6.35 0.15 0.15 2.5 19243  3954

2 1.81 0.15 1.63 2.45 2051  8152

3 1.12 0.15 2.24 2.49 3293  11745

4 0.97 0.14 2.79 3.16 3480  10840

5 0.89 0.14 2.97 3.4 4417  11959

 

 

5. Optical absorbance spectra of BP.  

Figure S4 | Theoretically predicted optical absorption spectra of BP few-layers along the

c (zz) direction. There are no appreciable absorption found in energy below 2.0 eV.

Optical absorption spectra along the c (zz) direction was plotted in Fig. S4. Together with

the results shown in Fig. 3, the absorption exhibits an explicit anisotropic behavior that light

absorption in the infrared range, even up to 2eV, is only observable along a certain direction –

the b (yy) direction. This behavior should be detectable by light spectroscopies and may bring

on novel optoelectronic applications.

 

1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

3.00.08.0

0.08.0

0.06.0

0.0

4.00.0

0.5

Energy of incident light (eV)

Ab

sorb

an

ce (

%)

2.44

2.44

2.53

2.69

2.91

3.201-layer

2-layer

3-layer

4-layer

5-layer

Bulk

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