Few-layer black phosphorus: emerging direct band gap semiconductor

with high carrier mobility

 

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

Abstract

Two-dimensional crystals are emerging materials for future nanoelectronics, the community

of which has been seeking for candidate channel materials that have sufficient electronic band

gap, high carrier mobility, and good channel-lead contact. We present a theoretical investigation

of geometric and electronic structures of few-layer black phosphorus, based on which,

associated electric and optical properties were predicted. Our results show that it has a direct

band gap, which is tunable from 1.51 eV of monolayer to 0.59 eV of 5-layer; while the hole

mobility, for example, increase from 337 cm2/V•s of monolayer to 2722 cm2/V•s of 5-layer. In

addition, directionally selective optical transitions were found that significant light absorption

happens at energies in the infrared range along a certain direction, while there are no appreciable

absorption below 2.0 eV for the other two directions. These results make few-layer black

phosphorus a promising candidate for future electronics and optoelectronics.

 

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

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

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

and others, were intensively investigated as emergent materials for future electronics.1-13 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.5-8,10-13 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.1-7 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.5,6,14

The emergence of monolayer TMDs, e.g. MoS2, as its first FET recently demonstrated,8

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, improvable up to 500

cm2/V·s,8 which is fairly good, but orders of magnitude lower than that of graphene.5,6

Germanane is another candidate for 2D electronics that theory predicts a high carrier mobility

of 18195 cm2/V·s and a finite band gap of 1.56 eV.15 It is, however, covered by H atoms, rather

electronically inert, which doubts whether a good contact can be made between electrode

materials and germanane. Therefore, a very important open problem is to seek 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 discovery of high carrier mobility 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 BPs, 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 monolayer down to ~0.6 eV of 5-layer. Effective masses in the

range from 0.14 m0 to 0.18 m0 for electron and hole were found along the b direction. Carrier

mobilities along the same direction at 300K were theoretically derived that the smallest hole

mobility is 337 cm2/V·s (monolayer) and the smallest electron mobility is 299 cm2/V·s (bilayer)

among all considered few-layers, while they are over or close to a thousand cm2/V·s for the 5-

layer BP. The corresponding values considered in an 1D model are an order of magnitude higher.

A sufficient optical absorbance peak was found at 1.72 eV in the monolayer, or at lower energies

in thicker layers, only along the b direction. All these results strongly suggest that few-layer BP

is a new category of 2D semiconductors that is promising in the applications of nanoelectronics

and optoelectronics.

Results

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

BP. The PBE-G06 and optB88-vdW methods well reproduce the experimental geometry that

the theoretical equilibrium volumes are only 1% and 5% larger than the experiment,16

respectively, as summarized in Table S1. Electronic band structures of bulk BP calculated using

the LDA-mBJ 17,18 or the HSE06 19,20 functionals based on the optB88-vdW atomistic structure,

denoted mBJ (optB88-vdW) and HSE06 (optB88-vdW), were shown in Fig. 1(b), while the

results of other methods are available in Fig.S1. Combinations HSE06 (optB88-vdW) and mBJ

(optB88-vdW) are the best-fit methods among all considered methods according to their results

of the band gap and lattice parameters of bulk BP (see Fig. S1 and Table S1). We thus focused

our discussion on the results of these two methods. Both mBJ and HSE06 methods predict bulk

BP 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 0.33 eV.21,22 BP appears a metal with a band

cross at 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.

The two, a valance and a conduction, bands around the band gap at the Z point are nearly

liner, indicating very small effective masses. We thus fitted these bands using the nearly-free

electron model to derive their effective masses. Both electron and hole effective masses along

the Q-Z direction are rather small and similar to each other, i.e. 0.12 m0 and 0.11 m0 (HSE06

values), respectively; while these along the Z-G direction are larger, i.e. 0.15 m0 and 0.28 m0

for electron and hole, highly consistent with early experimental results for the bulk BP 23.

Effective masses revealed by the mBJ bandstructures are, only within 0.04 m0, larger than those

of HSE06. These values are similar to most high-mobility semiconductors, e.g. AlGaAs,24 and

even four times smaller than that of MoS2.25

Black Phosphorus has a striking property of two dimensionality. Small effective masses in

the bulk BP implies that few-layer BP is a likely high-mobility direct band gap semiconductor.

We thus inspected the geometric and electronic properties of BP few-layers. Table I summarizes

the changes of geometric properties with respect to the increase of the number of layers. Lattice

parameter b elongates by 0.11 Å from the bulk form to monolayer while lattice constant a only

changes within 0.02 Å. There are an abrupt change of lattice constant b from monolayer to

bilayer, which is, we believe, ascribed to the appearance of interlayer interactions in bilayer.

The significant stretching of b in few-layers BPs is largely due to the increase of bond angle θ1

but not the bond lengths.

Figure 2 illustrates the electronic structures of few-layer BPs computed using HSE06 and

mBJ. It shows that results carried out by the mBJ potential are quantitatively the same to those

of HSE06 and the results by other methods are also available in Fig. S2. Figure 2(a) represents

the electronic band structure of monolayer BP, which is still a direct band gap semiconductor

when reducing its thickness to monolayer. The direct band gap at the Gamma point was

predicted 1.53 eV and 1.41 eV by HSE06 and mBJ respectively. Inclusion of spin-orbit coupling

does not change the bandstructure (see Fig. S3). The gap reduces by roughly 0.5 eV when

putting another layer to monolayer, i.e. the bilayer gap of 1.02 eV (HSE06, hereinafter), and

the two bands near the gap split into four bands, denoted bands 1 to 4, as shown in Fig. 2(b).

Visualized wave functions, Fig. 2(b) inset, indicate that each band (among band 1 to 4) can be

found throughout the bilayer. Bands 1 and 2 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 of band 1, while that of band 2, which is energetically higher, shows a non-

bonding-like feature. Similar results were found in bands 3 and 4 where the bonding and non-

bonding features were not observable between layers but troughs along the b direction.

Figure 2(c) shows the evolution of band gaps with respect to the increase of thickness. If

the thickness increase further, the direct band gap rapidly decreases and reaches 0.59 eV in a 5-

layer BP. We fitted these values by an exponentially decay relation and found that the

corresponding bulk gap was extrapolated 0.53 eV, 0.17 eV larger than the predicted bulk value

of 0.36 eV. In few-layers, as shown in Table I, the lattice parameter b substantially elongates

from its bulk value, which is, most likely, the reason why the extrapolated bulk band gap differs

from that of an actual bulk calculation. Results calculated with a series of constrained b values

confirms this statement, which implies that the band gap is very sensitive to the lateral strain

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

direction.

In a monolayer, the original Z point in the bulk BZ folds back to the Gamma point, so that

the small effective masses direction is G-Y in a monolayer, in real-space along the b direction,

equivalently the Z-Q direction in bulk. The effective masses in the G-Y direction of a monolayer

are 0.17 m0 (electron) and 0.15 m0 (hole), respectively, only 0.04-0.05 m0 larger than those in

the bulk, as shown in Table II. It is remarkable that in the G-X direction, the valence band

around the G point appears a nearly flat band. HSE06 band structures calculated with the

atomistic structure optimized by less accurate structural relaxation methods, e.g. DFT-G06,

yield the VBM 0.1 Å-1 off the G point with a hole effective mass of -36.17 m0; while that with

the optB88-vdW structure predicts the VBM at G with the hole effective mass of 6.35 m0.

Unlike the band gap, effective masses along the G-Y direction and the electron effective mass

along G-X are almost unchanged, within 0.06 m0, with respect to the increase of the thickness,

however, the hole effective mass along G-X substantially decreases from 6.35 m0 of monolayer,

to 1.81 m0 of bilayer and eventually to 0.89 m0 of 5-layer.

These results show that the characteristic of symmetric and rather small electron-hole

effective masses remains in the mono- and few-layers, which suggests that few-layer BP is

highly promising for future 2D electronics. Such statement should be, however, supported by

the prediction of the carrier mobility, a property relevant with but not solely determined by

carrier effective mass. Table II shows the carrier mobility of few-layer BP under a phonon

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

carriers. The deformation potential E1y and the elastic modulus Cy in the propagation direction

of the longitudinal acoustic wave, which are highly relevant with the mobility26,27 were also

available in Table II. Since the effective masses are highly anisotropic, both 2D and 1D models

were considered by formulas

3_ 2D

y_2D * * 2e 1

y

B d y

e C

k Tm m E

(1)

21

2*

2_3

AS_D2_y 3

2

yeB

Dy

ETmk

Ce (2)

2_1D

y_1D 1/2 *3/2 21y

2

( )

y

B

e C

k T m E (3)

Formulas (1) 28-30 and (2) 24,31-33 represent the standard and anisotropic 2D models and formula

(3) 34 is the 1D model. 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.

The electron deformation potentials are close to each, from 5-6 eV, except that of

monolayer of 2.72 eV. The hole deformation potential behaviors differently from that of

electron, in which those of the mono- to tri-layer are very similar, roughly 2.5 eV, but those of

the 4- and 5-layer increase up to 3.4 eV, still smaller than those for electron. The elastic modulus

increases, as expected, linearly as a function of the layer thickness while that of the monolayer

is slightly larger, ~0.1 J/m2. Therefore, the mobility is thus unsymmetrical, primarily ascribed

to unsymmetrical values of E1y. With different models, the electron mobility at 300 K ranges

from 299 to 3730 cm2/V·s in mono- to 5-layer BPs and that of hole goes from 337 to over 10000

cm2/V·s, indicating both rather high electron and hole mobilities of few-layer BPs. These results

of mobility calculation are compelling evidence for suggesting that few-layer BP is a direct

band gap semiconductor with very high mobility.

Discussion

Given the effective masses of 0.1 to 0.2 m0, the carrier mobility was initially expected to

be even higher than the values we obtained. Here we discuss the deformation potential and

elastic modulus. According to formulas (1) to (3), the mobility is proportional to the elastic

modulus and reversely proportional to the square of the deformation potential. The key issue

that few-layer BP’s mobility was not as high as expected is the too small elastic modulus, in

other words it is ``too soft”, along the b direction, which is due to the fact that mostly bond

angles scarcely bond lengths are changed when deforming BP along this direction. In terms of

monolayers, BP’s modulus along the b direction is 14.5 J/m2 which is 4, 20 and 22 times smaller

than that of MoS2 ,25 h-BN, 28 and graphene.27 These results indicate that much higher mobilities

are most likely achieved if the BP can be modified stiffer, i.e. larger elastic modulus, along the

b direction by, e.g. external stress, doping or substrate confinement. The deformation potentials

of the monolayer BP of 2.5 eV (hole) and 2.7 eV (electron) are rather small compared with

those of single layer graphene of 5.0 eV,27 MoS2 of 3.9 eV,25 and h-BN of 3.7 eV 28 and AlGaAs

quantum wells of 7 eV.24 It is still quite reasonable for the few-layers that the potentials are

roughly 3 eV (hole) and 5-6 eV (electron). The change of the deformation potential with

modified elastic moduli is, however, not clear so far, which is subject to further investigations.

In addition, optical absorption properties was evaluated by the computed imaginary part

of the dielectric function. Figure 3 plots the absorption spectra of few layer BPs along the b (y)

direction. It shows that the absorption happens in the infrared range for all considered few-

layers, which reflects their direct band gap. The absorbance is up 4 % for the first peak in the

monolayer. It is remarkable that the absorbance is highly directional that there are no

appreciable absorption in directions a and c in the energy range below 2.0 eV for all considered

few-layer BPs, as shown in Fig. S4.

In summary, we reported a theoretical investigation of geometric and electronic structures

of few-layer BPs, based on which, the associated electric and optical properties were predicted.

Our results suggest that few-layer BP is a novel category of direct band gap 2D semiconductor

of very high carrier mobility. Either the band gap or the carrier mobility is tunable with

respective to the thickness. The band gap decrease from 1.51 eV of monolayer to 0.59 eV of 5-

layer, while the mobility of hole, for example, along the b direction increase from 337 cm2/V•s

of monolayer to 2722 cm2/V•s of 5-layer. It was also found that both the band gap and the

mobility are highly correlated with the structural properties, i.e. bond angle and elastic modulus,

along the b direction, which offers various possibility to further tune the electric or mechanical

properties of few-layer BPs. In addition, few-layer BP is an elementary semiconductor, which

implies a likely good contact with metal or other electrode materials. Furthermore, directionally

selective optical transitions were found in few-layer BPs that along the b direction, significant

light absorption happens at energies lower than 1.65 eV, in the infrared range, while there are

no appreciable absorption below 2.0 eV for the other two directions. All these striking results

make the few-layer BP a promising candidate 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,35,36 and a plane wave basis set as implemented in the Vienna

ab-initio simulation package.37 The energy cutoff for plane-wave basis was set to 500 eV for

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

zone of the conventional cell of bulk and few layers 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).38,39 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)

17,18 and hybrid functional (HSE06) 19,20 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

_ 2Dy_2D * * 2

e 1

y

B d y

e C

k Tm m E28-30，

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

dm is the density-of-state

mass determined by yxd mmm . Term E1y represents the deformation potential constant of

the VBM (hole) or CBM (electron) defined by 1 0/ ( / ) iiE V l l . Here i

V is the energy

change of the ith band under proper cell compression and dilatation (by a step 0.5 %) and l0 is

the lattice constant along the transport direction. Elastic modulus of the longitudinal strain in

the propagation direction, the b direction, of the LA wave Cy_2D 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. If we consider the fact that the effective mass of few layer BP is

highly anisotropic, a 2D model could be revised as 2

12*

2_3

AS_D2_y 3

2

yeB

Dy

ETmk

Ce 24,31-33 where the

md is replaced with the effective mass along the transport direction, or the 2D model could be

simplified into a 1D model that reads 2

_1Dy_1D 1/2 *3/2 2

1

2

( )

y

B y

e C

k T m E34. 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 300K and the set width in the 1D model is 10 nm.

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 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, b, and c, 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. Crystal structure of bulk black phosphorus (BP) (a) and associated electronic band

structures (b) calculated using the HSE06 functional (red solid line) and the mBJ potential (blue

dashed line). Lattice vectors a, b, c and structural parameters R1, R2, θ1 and θ2 were marked in

(a).

Table I. Lattice constants a, b and Δc 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 band structure of monolayer (a) and bilayer (b) calculated with the HSE06

functional (red solid lines) and the mBJ potential (blue dashed lines). Top views of few-layer

atomistic structure and the associated Brillouin zone, as well as the a and b lattice parameters

(directions), were shown in (a) inset. Wave functions of the valence and conduction states near

the gap and around the Gamma point were visualized in (b) inset with an isosurface of 0.003

e/Å3. Evolution of the direct band gaps as a function of the thickness was shown in (c), in which

the method for structural optimization was indicated in the parentheses.

Table II. Theoretically predicted carrier mobility. NL represents the number of layers, mx* and

my* are the carrier effective masses along directions a and b, E1y is the deformation potential

and Cy_2D (Cy_1D) are the 2D (1D) elastic modulus along the b direction. According to formulas

(1) to (3) and the values listed here, carrier mobilities of the standard (µy_2D) and anisotropic

(µy_2D_AS) 2D and 1D (µy_1D) models were predicted at 300 K.

Carrier

type

NL mx*/m0

G-X

my*/m0

G-Y

E1y

(eV)

Cy_2D

(J/m2)

Cy_1D

(10-7J/m)

µy_2D

(cm2/V·s)

µy_2D_AS

(cm2/V·s)

µy_1D

(cm2/V·s)

e

1 1.12 0.17 2.72 14.47 1.45 561 960 2764

2 1.13 0.18 5.02 28.74 2.87 299 499 1480

3 1.15 0.16 5.85 42.93 4.29 389 695 1941

4 1.16 0.16 5.92 57.33 5.73 504 905 2529

5 1.18 0.15 5.78 73.29 7.33 737 1379 3730

h

1 6.35 0.15 2.5 337 1461 3954

2 1.81 0.15 2.45 1301 3013 8152

3 1.12 0.15 2.49 2383 4341 11745

4 0.97 0.14 3.16 2363 4147 10840

5 0.89 0.14 3.4 2722 4575 11959

 

 

 

 

 

  

Figure 3. Theoretically predicted optical absorption spectra of BP few-layers along the b

direction. The energy of the first absorption peak redshifts with respect to the increase of layer

thickness, i.e. from 1.72 eV of monolayer (red) to 0.76 eV of 5-layer (pink). The energy of the

peak edges is roughly 0.2 eV smaller than that of the peak, which is consistent with the direct

band gaps of these BP few-layers. 

Few-layer black phosphorus: emerging direct band gap semiconductor

with high carrier mobility

 

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 

4 Sections 

4 Figures 

2 Tables 

5 References 

 

 

 

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

1. The choice of the best-fit calculation method for few-layer BP.

Lattice constants and structural parameters of bulk black phosphorus were shown 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 The results of optB88-vdW are 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 which method adopted except that of HSE06-G06 where 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(°)

Exp.1 3.31 4.38 10.50 2.28 2.22 101.91 96.37

PBE 3.30 4.57 11.33 2.26 2.22 103.59 95.98

PBE-G06 3.32 4.43 10.49 2.26 2.23 102.48 96.50

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

Figure S1(a) shows the electronic bandstructures of bulk BP calculated using the

LDA-mBJ 2,3 or the HSE06 4,5 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 bandstructures

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

negative 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. The 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) Bulk black phosphorus of optB86b-vdW calculated with the HSE06 functional

(red solid lines) and the mBJ potential (blue dashed lines). (b) Bulk black phosphorus of

different method, e.g. HSE06 functional with the structure of optB88-vdW (red solid lines),

mBJ potential with the structure of optB88-vdW (blue dashed lines), HSE06 functional with

the structure of PBE-G06 (violet dashed lines), PBE (bule dashed lines), PBE-G06 (black

dashed lines).

2. Effect of spin-orbit coupling

Figure S2. Electronic band structure of bulk black phosphorus calculated by the mBJ

potential with spin-orbit coupling (green dashed lines) and without spin-orbit coupling (black

solid lines). Inclusion of spin-orbit coupling does not change the bandstructure.

It was expected that phosphorus, a much heavy element, should have stronger spin-orbit

coupling in 2D forms than that of grapheme. 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 effect on the band structure, indicating phosphorus is not ``heavy”

enough.

3. Electronic band structure of few-layer black phosphorus revealed with other methods.

  

Figure S3. Electronic band structure of monolayer (a) and bilayer (b) calculated with the

HSE06 functional with the structure of optB88-vdW (red solid lines), mBJ potential with the

structure of optB88-vdW (blue dashed lines), HSE06 functional with the structure of

DFT-G06 (green dashed lines) and PBE functional (black dashed lines). 

 

Table SII. Band gap of few-layer black phosphorus.

Electronic band gap of few-layer black phosphorus (eV)

Layer HSE06

(optB88-vdW)

mBJ

(optB88-vdW)

HSE06

(DFT-G06) 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 

 

Exact values plotted in Fig. 2(c) were summarized in Table SII and other associated

electronic band structures of monolayer and bilayer were plotted in Fig. S3.

     

4. Optical absorbance spectra of BP.

 

 

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

and c directions. There are no appreciable absorption found in energy below 2.0 eV, different

from that along the b direction.

Optical absorption spectra along the a and c directions were 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 direction. This behavior should be detectable by light spectroscopies and

may bring on novel optoelectronic applications.

 

References  

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684‐685, doi:10.1107/S0365110X65004140 (1965). 

2  Becke, A. D. &  Johnson, E. R. A  simple effective potential  for exchange.  J Chem Phys 124, 

221101, doi:10.1063/1.2213970 (2006). 

3  Tran, F. & Blaha, P. Accurate Band Gaps of Semiconductors and  Insulators with a Semilocal 

Exchange‐Correlation Potential. Physical Review Letters 102, 226401 (2009). 

4  Heyd,  J., Scuseria, G. E. & Ernzerhof, M. Erratum: “Hybrid  functionals based on a  screened 

Coulomb potential” [J. Chem. Phys. 118, 8207 (2003)]. The Journal of Chemical Physics 124, 

219906, doi:10.1063/1.2204597 (2006). 

5  J. Heyd, G. E. S., and M. Ernzerhof. J. Chem. Phys. 124 (2006).