Applied Surface Science 356 (2015) 626–630
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Applied Surface Science
journal homepage: www.elsevier.com/locate/apsusc
Strain-induced band structure and mobility modulation in graphitic
blue phosphorus
L.Z. Liu a,b , X.L. Wu a,b,∗ , X.X. Liu a,b , Paul K. Chu c
a
b
c
Key Laboratory of Modern Acoustics, MOE, Institute of Acoustics, National Laboratory of Solid State Microstructures, Nanjing 210093, PR China
Department of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, PR China
Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, PR China
a r t i c l e
i n f o
Article history:
Received 23 June 2015
Received in revised form 14 August 2015
Accepted 15 August 2015
Available online 19 August 2015
Keywords:
Blue phosphorus
Electronic structure
First-principles calculation
Strain
a b s t r a c t
The effects of external strain on the electronic structure and carrier mobility of graphitic blue phosphorus
are investigated theoretically. Symmetry breaking induced by the in-plane strain not only modulates the
band structure, but also changes the carrier population at the valence band maximum and conduction
band minimum so that the transport current density can be regulated. Compressed deformation in the
direction normal to the plane transforms the blue phosphorus into an in-plane structure and the superfluous electrons reduce the band gap giving rise to a semiconductor–metal transition. Our theoretical
assessment reveals that strain engineering is a useful method to design electronic devices.
© 2015 Elsevier B.V. All rights reserved.
1. Introduction
Elemental phosphorus is stable as white, red, violet, blue,
and black allotropes and its color is defined by the fundamental
band gap [1,2]. The most stable one is black phosphorus which
has recently attracted much attention because of its potential in
high-performance electronic devices. Since black phosphorus has
high anisotropic hole-dominated mobility [3–6], directional factors
must be considered in the design of devices making it challenging
to integrate into current manufacturing technology. In addition, the
band gap of black phosphorus is 0.59–1.51 eV [4] and most visible
light cannot be properly utilized by optoelectronic devices. Another
stable allotrope with a wider band gap is blue phosphorus in which
the in-plane hexagonal structure and bulk layer stacking are very
similar to those in graphite [7] which has a wide fundamental band
gap of more than 2 eV. Due to the weak interlayer interaction, blue
phosphorus can be exfoliated easily to form quasi-two-dimensional
(2D) structures and has promising applications in optoelectronics.
If the carrier mobility is high, blue phosphorus with a few layers
is a contender in the emerging field of post-graphene 2D electronics. In a 2D structure, the carrier mobility is related to the band
structure and hence, regulation of the band structure at the valence
∗ Corresponding author at: Department of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, PR China.
E-mail address: [email protected] (X.L. Wu).
http://dx.doi.org/10.1016/j.apsusc.2015.08.125
0169-4332/© 2015 Elsevier B.V. All rights reserved.
band maximum (VBM) and conduction band minimum (CBM) (valley position) is very important. External strain, which can be easily
implemented by introducing a specific substrate during the fabrication of blue phosphorus devices, is an effective way to modulate
the electronic structure and carrier mobility. In fact, if the effects
of external strain on carrier mobility can be exploited, the transfer current can be controlled by strain engineering. However, our
understanding of strain-dependent electronic structures and carrier mobility in blue phosphorus devices is still quite limited.
In this work, the relationship between external strain and carrier
mobility in graphitic blue phosphorus is investigated theoretically. The results disclose that symmetry breaking induced by the
in-plane strain not only regulates the band structure and carrier
mobility, but also changes the carrier population at the valley. The
carrier mobility and concentration alter the transport current density. Compressed deformation in the direction normal to the plane
decreases the band gap and induces a semiconductor–metal transition.
2. Theoretical methods
The theoretical assessment is based on the density functional theory in the Perdew–Burke–Ernzerhof generalized gradient
approximation using the Vienna ab initio simulation package
(VASP) code with projector augmented wave pseudopotentials
[8–10]. The plane-wave energy cutoff of 500 eV is used to expand
the Kohn–Sham wave functions and relaxation is carried out until
L.Z. Liu et al. / Applied Surface Science 356 (2015) 626–630
Fig. 1. (a) Top view and (b) side view of the (2 × 2) blue phosphorus monolayer; calculated electronic density of (1 × 1) blue phosphorus monolayer at VBM (c) and CBM
(d) in the absence of strain; calculated electronic density of (1 × 1) blue phosphorus
monolayer at VBM (e) and CBM (f) with 3% uniaxial strain along the x direction.
all forces on the free ions converge to 0.01 eV/Å. The vacuum
space is at least 15 Å, which is large enough to avoid the interaction between periodical images. The Monkhorst-Pack k-points grid
(10 × 10 × 1) has been tested to converge well. Moreover, since the
standard density functional theory may fail to describe the band
structure, state-of-the art hybrid functional calculation based on
the Hey–Scuseria–Ernzerhof (HSE06) functional is adopted in this
work [11].
3. Results and discussion
The optimized hexagonal (2 × 2) supercell of an isolated blue
phosphorus monolayer with the 3.243 Å lattice constant is spanned
by lattice vectors in the x and y directions (Fig. 1a). By considering the long-range van der Waals correction in the DFT-D2
methods [12], the influence of the interlayer interaction on the inlayer structure can be neglected because of the small change from
x = y = 3.240 Å in the bulk to 3.3243 Å in the isolated monolayer. As
shown in Fig. 1b, the side view of blue phosphorus displays an ideal
in-layer connection by dislocation, and the relaxed thickness of
the monolayer is 1.218 Å (marked by h0 ). The electron density at
VBM and CBM is evenly distributed among adjacent phosphorus
atoms (P1 P2 bond) in the absence of external strain (Fig. 1c and
d). When uniaxial strain is applied along the x direction (Fig. 1e
and f), asymmetric deformation induced by the external strain
breaks the intrinsic symmetry, which makes the electronic density
at VBM deviate from the P1 P2 bond to P1 P3 and P1 P4 bond
vicinity. However, asymmetric deformation induced by uniaxial
strain cannot alter the electronic density distribution at CBM. The
results demonstrate that the electronic density distribution at VBM
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is associated with structural deformation which can be exploited
to regulate the band structure and carrier mobility.
For verification, the band gap of the (1 × 1) blue phosphorous
monolayer with different in-plane strain and band structure along
the high symmetry points are shown in Fig. 2a and the inset of
Fig. 2b, respectively. In the absence of external strain (xy = 0%), the
calculated band structure shows that a blue phosphorus monolayer
is semiconducting with an indirect band gap of 2.72 eV, which is
close to that derived by the GW approach (2.62 eV). The results
generated by the HSE06 method are adopted in this work, because
they are more exact (the DFT band gap is underestimated to be
1.65 eV). As shown in Fig. 1c and d, the electron density associated
with VBM and CBM is mostly dispersed at the P P bond (not confined tightly to phosphorous atoms). Owing to the asymmetrical
distribution, the VBM and CBM (valley position) deviate from the
high symmetry points to be located between K–G (G–B) (marked
by C1 and C2) and G–F (G–B) (marked by V1 and V2). Such a band
structure indicates that the conduction/valence bands have two
minima/maxima at equal energy but different momentum (marked
by the red line). This provides the possibility to vary the electron quantity in each valley. Therefore, electronic devices can be
designed by using the valley degree of freedom. Tensile deformation induced by biaxial strain (xy = 3%) can reduce the CBM values
(C1 and C2) (marked by the black line) and shift the VBM position
(V1 and V2) slightly. Because the bond length is stretched to 2.274 Å
(xy = 3%) from 2.229 Å (xy = 0%), symmetrical deformation reduces
the overlapping of the electron wave function. Symmetry breaking induced by the uniaxial strain (x = 3%) not only decreases C1
(keeping C2 constant), but also enlarges the discrepancy between
V1 and V2 (VBM) to 0.11 eV (xy = 3%) from 0.04 eV (xy = 0%). The
energy splitting between CBM (C1 and C2) and VBM (V1 and V2)
adjusts the electron population at each valley thus playing a crucial
role in the carrier injection and quantum transport.
To compare the structural stability, the formation energy is
expressed as Ef = Er − E0 , where Er and E0 represent the deformed
energy and free energy. The maximum deformation energies are
calculated to be 0.1453 eV (x = 9%) and 0.1553 eV (xy = 9%), and so
the discussed strain range can be achieved without a large energy
penalty. As shown in Fig. 2b, the band gap which depends on the
applied in-plane strain is decreased to 1.4 eV (xy = 9%) from 2.7 eV
(x = 0%). The possibility to change the band gap by ∼50% in the
strained geometries further suggests that strain-engineered blue
phosphorus has potential applications in electronic devices. The
obvious change in the band gap can be explained as follows: deformation induced by strain breaks the original balance and symmetry,
and redistributes the electron wave function to form a metastable
structure. Meanwhile, the atomic population is increased from 1.00
(xy = 0%) to 1.03 (xy = 9%) and s and p electrons are changed from
1.82 e and 3.18 e (xy = 0%) to 1.84 e and 3.15 e (xy = 9%).
As shown in Fig. 2c, compressed deformation [z = (h − h0 )/h0 ,
where h0 is the thickness of the free layer] in the direction normal to the plane results in an in-plane expansion of the unit cell.
The bonding structure of blue phosphorus is retained up to z = 40%
and the structure of the strained materials approaches that of a
puckered graphene layer. However, the CBM becomes as low as
VBM marking the transition from indirect band gap semiconductor
to metallic and it is related to the symmetry transformation from
a beta structure to planar one. Fig. 2d shows the variation in the
band gap until a point where the CBM descends below the VBM. In
a narrow range of z between 30% and 35%, the deformed blue phosphorus displays a small density of state at the Fermi level and can
be considered as a semi-metal. If there is more compression, the
original valence band and conduction band finally cross (z > 40%)
deeply and the materials become metallic. The results indicate that
the band structure and physical characteristics can be regulated by
strain engineering.
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L.Z. Liu et al. / Applied Surface Science 356 (2015) 626–630
Fig. 2. (a) Calculated band structure of the (1 × 1) blue phosphorus monolayer under different external strain. (b) Band gap of the (1 × 1) blue phosphorus monolayer as
functions of strain along the x and xy directions. The inset shows the high symmetry points of the Brillouin zone. (c) Calculated band structure of the (1 × 1) blue phosphorus
monolayer under different external strain in the direction normal to the plane. (d) Band gap as a function of strain in the direction normal to the plane. (For interpretation
of the references to color in text, the reader is referred to the web version of this article.)
The carrier mobility depends on the band structure which can
∗
carbe calculated by = e3 C/kB Tm∗e md E 2 [13,14], where
me ∗is the
rier effective mass in the strain direction and md =
mx m∗y is the
average effective mass. E = V/(l/l0 ) is the deformation potential constant of CBM for electron and VBM for holes along the
transport direction obtained by varying the lattice constant along
the transport direction and checking the change of the band energy
under compression and strain. The elastic module is represented
as C = (E − E0 )/S0 , where E − E0 is the change in the total energy
obtained by varying the lattice constant by a small amount (0.5%), S0
Fig. 3. (a) Mobility and (b) effective mass of electrons and holes as functions of uniaxial strain along the x direction; (c) mobility and (d) effective mass of electrons and holes
as functions of biaxial strain along the xy direction.
L.Z. Liu et al. / Applied Surface Science 356 (2015) 626–630
629
Fig. 4. (a) Schematic illustration of the electronic device model. (b) Schematic illustration of the band structure and carrier density in the absence of strain. (c) Schematic
illustration of the band structure and carrier density change induced by external strain. (For interpretation of the references to color in text, the reader is referred to the web
version of this article.)
is the lattice area in the xy plane, and T = 300 ◦ C is the room temperature. It is noted that accurate calculation of electrical conductance is
very complicated and many extrinsic factors may decrease the calculated mobility. However, this formula based on electron–phonon
coupling is sophisticated enough to take into account the conductance of the deformed blue phosphorus system.
The carrier mobility of the blue phosphorus monolayer under
uniaxial and biaxial strains is calculated and shown in Fig. 3a and
c. Firstly, the hole mobility is much larger than that of electron,
indicating that blue phosphorus is hole-dominated. Secondly, in
intrinsic blue phosphorus, the hole mobility is only ∼30 cm2 /V s
which is much smaller than that of black phosphorus. This is
because of the in-direct band structure and CBM and VBM deviating from the high symmetry points (Fig. 2). Thirdly, deformation
induced by uniaxial strain can enhance the hole mobility to
130 cm2 /V s at x = 2%, which is close to those of other 2D semiconductors such as MoS2 and BN [15,16]. In the beginning, the
deformation potential constant E and elastic module C are sensitive
to symmetry breaking and consequently, the mobility increases.
When asymmetric deformation is large enough, the response of E
and C to external strain becomes weak and therefore, after reaching
a maximum, the mobility begins to diminish slowly. Deformation induced by uniaxial strain can lead to more serious symmetry
breaking than that by biaxial strain and the mobility values in Fig. 3a
are larger than those in Fig. 3c. In addition, the effective masses of
electrons and holes as functions of uniaxial strain (x direction) and
biaxial strain (xy direction) are calculated and shown in Fig. 3b and
d. The electron effective mass is larger than that of holes, implying
that holes are more mobile than electrons. The mobility changes can
be ascribed to the band structure change as a result of symmetry
breaking by strain engineering.
To investigate the application potential of blue phosphorus, the
electronic device model is illustrated in Fig. 4. The transport current density in a semiconductor can be calculated by j = nF, where
and n are the carrier mobility and concentration at the valley
and F is the applied electric field. As shown in Fig. 4a, the blue
phosphorus (marked by red region) is glued onto a piezoelectric
stack actuator to obtain tunable strain [17]. When a bias is applied
to the piezo stack, the blue phosphorus is expanded and compressed correspondingly. Therefore, the band structure and carrier
mobility are regulated. In the absence of external strain, band structure analysis (Fig. 2) discloses that electrons can evenly occupy two
conduction valleys at the equivalent C1 (V1) and C2 (V2) points and
the carrier concentration at each valley is also equal as shown in
Fig. 4b. Application of symmetry-breaking strain splits the energy
at C1 (V1) and C2 (V2) transferring charges from one valley to the
other. The carrier concentration at each valley becomes unequal
(Fig. 4c). In addition to the carrier mobility, the carrier concentration is another physical factor that can be also employed to control
the transport current density.
4. Conclusions
The band structure and carrier mobility of a blue phosphorus
monolayer can be controlled by external strain. With increasing
strain, the hole mobility increases to 130 cm2 /V s (x = 2%) and then
diminishes gradually to 30 cm2 /V s (x = 8%). Meanwhile, the two
equivalent valleys begin to split because of symmetry breaking.
The carriers are transferred from one valley to another resulting
in concentration changes. The carrier mobility and concentration
synergistically modulate the transport current density. In addition,
compressed deformation in the direction normal to the plane converts it into an in-plane structure and the superfluous electrons
effectively decrease the band gap inducing a semiconductor–metal
transition. The results indicate that blue phosphorus has large
potential in next-generation electronic devices.
Acknowledgements
This work was supported by National Basic Research Programs
of China under Grants Nos. 2013CB932901 and 2014CB339800
and National Natural Science Foundation (No. 11374141 and
630
L.Z. Liu et al. / Applied Surface Science 356 (2015) 626–630
11404162). Partial support was also from Natural Science Foundations of Jiangsu Province (No. BK20130549) and City University
of Hong Kong Strategic Research Grant (SRG) No. 7004188. We
also acknowledge computational resources of High Performance
Computing Center of Nanjing University.
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