Few-layer black phosphorus: emerging 2D semiconductor with ...
Transcript of Few-layer black phosphorus: emerging 2D semiconductor with ...
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 * [email protected], 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.
References 1 Katsnel′son, M. I. Carbon in Two Dimensions. Cambridge University Press,New York (2012).
2 Novoselov, K. S. et al. Two‐dimensional gas of massless Dirac fermions in graphene. Nature
438, 197‐200 (2005).
3 Geim, A. K. & Novoselov, K. S. The rise of graphene. Nat. Mater 6, 183‐191 (2007).
4 Berger, C. et al. Ultrathin Epitaxial Graphite: 2D Electron Gas Properties and a Route toward
Graphene‐based Nanoelectronics. J. Phys. Chem. B 108, 19912‐19916 (2004).
5 Liao, L. et al. High‐speed graphene transistors with a self‐aligned nanowire gate. Nature 467,
305‐308 (2010).
6 Schwierz, F. Graphene transistors. Nat. Nanotechnol. 5, 487‐496 (2010).
7 Hwang, E. & Das Sarma, S. Acoustic phonon scattering limited carrier mobility in two‐
dimensional extrinsic graphene. Phys. Rev. B 77, 115449 (2008).
8 Wang, Z. M. MoS2 . Springer International Publishing Switzerland (2014).
9 Radisavljevic, B., Radenovic, A., Brivio, J., Giacometti, V. & Kis, A. Single‐layer MoS2
transistors. Nat. Nanotechnol. 6, 147‐150 (2011).
10 Wang, Q. H., Kalantar‐Zadeh, K., Kis, A., Coleman, J. N. & Strano, M. S. Electronics and
optoelectronics of two‐dimensional transition metal dichalcogenides. Nat. Nanotechnol. 7,
699‐712 (2012).
11 Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419‐425 (2013).
12 Wang, H. et al. Integrated circuits based on bilayer MoS2 transistors. Nano Lett. 12, 4674‐
4680 (2012).
13 Yoon, Y., Ganapathi, K. & Salahuddin, S. How good can monolayer MoS2 transistors be? Nano
Lett. 11, 3768‐3773 (2011).
14 Popov, I., Seifert, G. & Tománek, D. Designing Electrical Contacts to MoS2 Monolayers: A
Computational Study. Phys. Rev. Lett. 108, 156802 (2012).
15 Xia, F., Farmer, D. B., Lin, Y. M. & Avouris, P. Graphene field‐effect transistors with high on/off
current ratio and large transport band gap at room temperature. Nano Lett. 10, 715‐718
(2010).
16 Fivaz, R. & Mooser, E. Mobility of Charge Carriers in Semiconducting Layer Structures. Phys.
Rev. 163, 743‐755 (1967).
17 Bianco, E. et al. Stability and Exfoliation of Germanane: A Germanium Graphane Analogue.
ACS Nano 7, 4414‐4421 (2013).
18 Dion, M., Rydberg, H., Schröder, E., Langreth, D. C. & Lundqvist, B. I. Van der Waals Density
Functional for General Geometries. Phys. Rev. Lett. 92, 246401 (2004).
19 Klimeš, J., Bowler, D. R. & Michaelides, A. Van der Waals density functionals applied to solids.
Phys. Rev. B 83, 195131 (2011).
20 Hultgren, R., Gingrich, N. S. & Warren, B. E. The Atomic Distribution in Red and Black
Phosphorus and the Crystal Structure of Black Phosphorus. J. Chem. Phys. 3, 351 (1935).
21 Brown, A. & Rundqvist,S. Refinement of the crystal structure of black phosphorus. Acta Cryst.
19, 684‐685 (1965).
22 Cartz, L., Srinivasa, S. R., Riedner, R. J., Jorgensen, J. D. & Worlton, T. G. Effect of pressure on
bonding in black phosphorus. J. Chem. Phys. 71, 1718 (1979).
23 Becke, A. D. & Johnson, E. R. A simple effective potential for exchange. J. Chem. Phys. 124,
221101 (2006).
24 Tran, F. & Blaha, P. Accurate Band Gaps of Semiconductors and Insulators with a Semilocal
Exchange‐Correlation Potential. Phys. Rev. Lett. 102, 226401 (2009).
25 Heyd, J., Scuseria, G. E. & Ernzerhof, M. Hybrid functionals based on a screened Coulomb
potential. J. Chem. Phys. 118, 8207 (2003).
26 Heyd, J., Scuseria, G. E. & Ernzerhof, M. Erratum: “Hybrid functionals based on a screened
Coulomb potential” [J. Chem. Phys. 118, 8207 (2003)]. J. Chem. Phys. 124, 219906 (2006).
27 Keyes, R. The Electrical Properties of Black Phosphorus. Phys. Rev. 92, 580‐584 (1953).
28 Warschauer, D. Electrical and Optical Properties of Crystalline Black Phosphorus. J. Appl. Phys.
34, 1853 (1963).
29 Maruyama, Y., Suzuki, S., Kobayashi, K. & Tanuma, S. Synthesis and some properties of black
phosphorus single crystals. Physica B+C 105, 99‐102 (1981).
30 Akahama, Y., Endo, S. & Narita, S.‐i. Electrical Properties of Black Phosphorus Single Crystals.
J. Phys. Soc. Jpn. 52, 2148‐2155 (1983).
31 Morita, A., Asahina, H., Kaneta, C. & Sasaki, T. Anisotropic Mobility and Electron‐Phonon
Interaction in Black Phosphorus. Proceedings of the 17th International Conference on the
Physics of Semiconductors, 1320‐1324 (1985).
32 Walukiewicz, W., Ruda, H., Lagowski, J. & Gatos, H. Electron mobility in modulation‐doped
heterostructures. Phys. Rev. B 30, 4571‐4582 (1984).
33 Kaasbjerg, K., Thygesen, K. S. & Jacobsen, K. W. Phonon‐limited mobility in n‐type single‐layer
MoS2 from first principles. Phys. Rev. B 85, 115317 (2012).
34 Rodin , A. S., Carvalho, A. & Castro Neto , A. H. Strain‐induced gap modication in black
phosphorus. arXiv:1401.1801 (2014).
35 Bruzzone, S. & Fiori, G. Ab‐initio simulations of deformation potentials and electron mobility
in chemically modified graphene and two‐dimensional hexagonal boron‐nitride. Appl. Phys.
Lett. 99, 222108 (2011).
36 Takagi, S.‐i., Toriumi, A., Iwase, M. & Tango, H. On the universality of inversion layer mobility
in Si MOSFET's: Part I‐effects of substrate impurity concentration. IEEE Trans. Electr. Dev. 41,
2357‐2362 (1994).
37 Fiori, G. & Iannaccone, G. Multiscale Modeling for Graphene‐Based Nanoscale Transistors.
Proceedings of the IEEE 101, 1653‐1669 (2013).
38 Bardeen, J. & Shockley, W. Deformation potentials and mobilities in non‐polar crystals. Phys.
Rev. 80, 72‐80 (1950).
39 Xi, J., Long, M., Tang, L., Wang, D. & Shuai, Z. First‐principles prediction of charge mobility in
carbon and organic nanomaterials. Nanoscale 4, 4348‐4369 (2012).
40 Blöchl, P. E. Projector augmented‐wave method. Phys. Rev. B 50, 17953‐17979 (1994).
41 Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented‐wave
method. Phys. Rev. B 59, 1758‐1775 (1999).
42 Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total‐energy calculations
using a plane‐wave basis set. Phys. Rev. B 54, 11169‐11186 (1996).
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: [email protected]
Supplementary Materials
5 Sections
4 Figures
4 Tables
12 References
__________ § These authors equally contributed to this work * [email protected], 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
References
1 Hultgren, R., Gingrich, N. S. & Warren, B. E. The Atomic Distribution in Red and Black
Phosphorus and the Crystal Structure of Black Phosphorus. J. Chem. Phys. 3, 351 (1935).
2 Rundqvist, A. B. a. S. Refinement of the crystal structure of black phosphorus. Acta Cryst. 19,
684‐685 (1965).
3 Appalakondaiah, S., Vaitheeswaran, G., Lebègue, S., Christensen, N. E. & Svane, A. Effect of van
der Waals interactions on the structural and elastic properties of black phosphorus. Phys. Rev.
B 86, 035105 (2012).
4 Becke, A. D. & Johnson, E. R. A simple effective potential for exchange. J. Chem. Phys. 124,
221101 (2006).
5 Tran, F. & Blaha, P. Accurate Band Gaps of Semiconductors and Insulators with a Semilocal
Exchange‐Correlation Potential. Phys. Rev. Lett. 102, 226401 (2009).
6 Heyd, J., Scuseria, G. E. & Ernzerhof, M. Hybrid functionals based on a screened Coulomb
potential. J. Chem. Phys. 118, 8207 (2003).
7 Heyd, J., Scuseria, G. E. & Ernzerhof, M. Erratum: “Hybrid functionals based on a screened
Coulomb potential” [J. Chem. Phys. 118, 8207 (2003)]. J. Chem. Phys. 124, 219906 (2006).
8 Northrup, J. Two‐dimensional deformation potential model of mobility in small molecule
organic semiconductors. Appl. Phys. Lett. 99, 062111 (2011).
9 Takagi, S.‐i., Hoyt, J. L., Welser, J. J. & Gibbons, J. F. Comparative study of phonon‐limited
mobility of two‐dimensional electrons in strained and unstrained Si metal‐oxide‐
semiconductor field‐effect transistors. J. Appl. Phys. 80, 1567‐1577 (1996).
10 Walukiewicz, W., Ruda, H., Lagowski, J. & Gatos, H. Electron mobility in modulation‐doped
heterostructures. Phys. Rev. B 30, 4571‐4582 (1984).
11 Price, P. Two‐dimensional electron transport in semiconductor layers. I. Phonon scattering. Ann.
Phys. 133, 217‐239 (1981).
12 Beleznay, F., Bogar, F. & Ladik, J. Charge carrier mobility in quasi‐one‐dimensional systems:
Application to a guanine stack. J. Chem. Phys. 119, 5690‐5695 (2003).