# Universal co-existence of photovoltaics and ferroelectricity from two-
dimensional 3R bilayer BX (X=P, As, Sb)

Qiang Wang,a,b,c Yan Liang,d Hui Yao,a Jianwei Li,a Tianwei Liu,a and Bin
Wang,∗a a Shenzhen Key Laboratory of Advanced Thin Films and Applications,
College of Physics and Optoelectronic Engineering, Shenzhen University,
Shenzhen, 518060, People’s Republic of China. E-mail<EMAIL_ADDRESS>b
Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education
and Guangdong Province, College of Optoelectronic Engineering, Shenzhen
University, Shenzhen 518060, China.  c School of Materials Science and
Engineering, Hanshan Normal University, Chaozhou 521041, People’s Republic of
China.  d School of Physics, State Key Laboratory of Crystal Materials,
Shandong University, 250100 Jinan, People’s Republic of China.

###### Abstract

The intertwined ferroelectricity and photovoltaics in two-dimensional (2D)
materials will enable the favorable improvement and control of photovoltaic
preformances. In this paper, we take 2D 3R bilayer BX (X=P, As, Sb) as model
systems to study the photovoltaic characteristics of intrinsic 2D out-of-plane
(OOP) ferroelectric material, and try to explore a strategy to regulate the
photoelectric properties by changing the strength of ferroelectric
polarization. Due to the spatial inversion symmetry broken caused by special
3R stacking, spontaneous OOP ferroelectric polarization will appear in the 3R
bilayer BX, which can be swichable through a specific interlayer sliding. The
OOP ferroelectricity leads to charge transfer between layers, realizes
efficient spatial separation between holes and electrons, and forms the
characteristics of type-II band alignment. Moreover, due to perfect lattice
match on account of two identical layers, the 3R bilayer BX is more stable and
easy to realize in experiments than most of traditional 2D heterostructures
made up of different materials. The 3R bilayer BX shows moderate band gap,
ultra-high carrier mobility and efficient optical absorption, and its nano-
devices present large photocurrent, high photon responsivity and excellent
external quantum efficiency. More importantly, all these photoelectric
parameters depend on the intrinsic OOP ferroelectric strength. By changing the
interlayer distance of bilayer BX, the ferroelectric polarization can be
regulated effectively to achieve the optimal photoelectric performance.
Finally, we emphasize the importance and universality of spatial inversion
symmetry broken in layered materials beyond the 3R bilayer BX to realize the
co-regulation of ferroelectric and photovoltaics.

###### pacs:

63.22.-m,65.80.CK, 72.80.Vp

##  I Introduction

The reversible spontaneous polarization of ferroelectric materials enable them
desirable for high-density data storage and fast device operation without
quantum tunneling or high power consumption/dissipation.Haeni _et al._ (2004);
Valasek (1921) Traditional ferroelectric materials were commonly based on some
complex oxides including ABO3 perovskitesCohen (1992), lithium niobate
(LiNbO3) ferroelectric crystalZheng _et al._ (2010); Baeumer _et al._ (2015),
BaBi2Ta2O9Peng _et al._ (2014), and plenty of their investigations and
commercial applications have been proposed in the last few decadesBona and
Smith (1975); Moya _et al._ (2014). With continue miniaturization of
electronic devices and rapid development of 2D materials, more efforts have
been made toward exploring ferroelectricity in 2D thin film materials.Wang _et
al._ (2018). More strikingly, the out-of-plane (OOP) ferroelectricity in these
2D thin films is more favorable than the in-plane one because the vertical
polarizability is more desirable for most practical technologiesJunquera and
Ghosez (2003); Ghosez and Rabe (2000). The most direct idea is to process the
bulk ferroelectric materials to thin film by experimental methods. However,
due to the depolarizing electrostatic field and electron reconstruction, the
OOP ferroelectricity will vanish in experiments when the thickness of the film
is below several to tens of nanometersJunquera and Ghosez (2003); Fong _et
al._ (2004). Therefore, it is reasonable to explore 2D ferroelectric materials
with intrinsic OOP polarization. So far, a series of such materials have been
proposed including In2Se3Ding _et al._ (2017), MoTe2Yuan _et al._ (2019),
WTe2Fei _et al._ (2018), Sc2CO2Chandrasekaran _et al._ (2017), and even the
ferroelectric property has been found in 2D elemental materials, such as
bilayer phosphorene, arsenene, and antimonene.Liang _et al._ (2021) To data,
exploring 2D thin films with excellent OOP ferroelectricity is still in its
infancy, and hence of great significance to clarify the microscopic origin
inside and further assess its fascinating facilitation in the modern micro-
electronics industry.

The utilization of solar energy by photovoltaic conversion is an effective and
promising method to meet the urgent requirements of energy crisis. In view of
different generation process of electron-hole pairs, the photovoltaic systems
can be generally classified into two groups. One is the traditional bulk
inorganic semiconductors such as Si, CdTe, and GaAs,Ginley _et al._ (2008)
where the photo-excited electrons and holes are generated without intermediate
steps but also the missing of long-range Coulomb interactions. The other
refers to the excitonic solar cells (XSCs), in which the electron-hole pairs
are generated upon illumination.Mayer _et al._ (2007); Nelson (2011) As
indicated in previous studies, the different ionization potentials and
electron affinity at the donor-acceptor interfaces of XSCs would induce a high
yield dissociation of excitons, and hence a substantial power conversion
efficiency (PCE).Hong _et al._ (2014) However, the PCE of most recent XSCs are
still below 12% although considerable efforts have been dedicated to improve
their relevant performances.Tsai _et al._ (2014); Britnell _et al._ (2013)
Besides, more critical factors are non-negligible for high efficient XSCs,
including high visible light optical absorptionLong _et al._ (2013), carrier
mobility,Long (2013); Wang _et al._ (2019) small exciton binding
energy,Scharber _et al._ (2006) and moderate direct band gap (1.2-1.6 eV) of
the donor.Liang _et al._ (2018); Wang _et al._ (2018)

Very recently, the staggered band offset of type-II heterostructures has been
confirmed to be favorable for the effective separation of the excitons,
implying potential applications in the XSCs.Massicotte _et al._ (2016); Wang
_et al._ (2019) However, the type-II heterostructures can not be always
guaranteed due to the different chemical potentials between different
components (there exist three different types of semiconductor
heterostructures).Wang _et al._ (2020) In addition, most of the proposed type-
II heterostructures are composed of different monolayer materials which are
probably instability in experimental manipulation due to lattice mismatch and
large potential barrier between two components. Fortunately, this problem can
just be overcome in recently proposed 2D bilayer ferroelectric films because
they are stacked by the same layered materials.Zhang _et al._ (2020); Wang _et
al._ (2015) Due to special stacking mode, the spatial reflection symmetry is
broken, and thus a spontaneous vertical polarization between two layers is
induced.Liu _et al._ (2020); Liang _et al._ (2021) This intrinsic
electrostatic field is an exciting realm in designing 2D photovoltaic devices
because the electrons and holes can be separated effectively in space.
Moreover, recent experimental technique of growing high-quality atomic
crystals layer by layer also guarantees the feasibility of stacking individual
thin film into different layered materials with van der Waals (vdWH)
interaction.Yasuda _et al._ (2020); Zhang _et al._ (2020); Wang _et al._
(2015) In light of the charge transfer between the interfacial layers, these
newly combined 2D layered materials can enrich the exceptional properties of
the isolated components and break the limitations of their applications in
many fields, and one of the potentially inspiring combinations is
ferroelectric and photovoltaic. Therefore, it is very necessary to explore the
2D ferroelectric/photoelectric hybrid materials, where the photoelectric
efficiency can be naturally improved and adjusted by the intrinsic
ferroelectricity. Moreover, it is significant to propose a fundamental
mechanism to design stable and high-efficient 2D ferroelectric/photoelectric
hybrid materials and understand their inherent physical mechanisms for
diversity and efficiency.

2D hexagonal group-VA boride BX (X=P, As, Sb), derivatives of h-BN, is one of
the most promising families of 2D materials, which has been successfully
fabricated experimentally and widely studied in recent years.Xie _et al._
(2016) Plenty of prominent physical properties have been discovered in this
family, including ultra-high carrier mobilities, moderate direct band gaps and
novel electronic, thermal and optical properties _et.al._. _ect._ Xie _et al._
(2016); Wang _et al._ (2019) In this work, we take the special 3R stacking
configurations of bilayer BX (X=P, As, and Sb) as examples to explore the
coexist of OOP ferroelectricity and photovoltaics by using the first
principles calculation method. Due to the charge redistribution induced by the
unique lattice stacking, sizeable and switchable OOP ferroelectric
polarizations could be detected in these crystals accompany with intrinsic
type-II band alignments between two semiconducting layers. Moderate quasi-
direct band gaps, ultra-high carrier mobilities and efficient optical
absorptions indicate appealing applications of these materials in the stable
and novel 2D XSCs. The photovoltaic performance can be adjusted effectively by
changing the strength of OOP ferroelectric polarization. The decrease of OOP
dipole moment can enhance PCE efficiently, while its increase can induce an
obvious red-shift of optical absorption in the visible light region. Moreover,
extremely large photon responsivities and external quantum efficiencies are
obtained in the 3R BP based few-layer optoelectronic devices, which can be
further enhanced by adjusting the OOP dipole moment. Our numerical results
show that 2D OOP ferroelectricity can promote the improvement of photoelectric
performance of 3R bilayer BX materials omnibearingly. More importantly, the
fundamental mechanisms of all above conclusions can also be extended into
other layered photovoltaic ferroelectrics, and hence all these fascinating
findings predicted in 3R bilayer BX systems may shed a new guidance in future
experimental design and practical application of 2D photoelectric conversion
devices and XSCs.

Figure 1:  (color online) (a) Top and side views of schematic structures of
bilayer 3R BX (X=P, As, Sb), where X and B atoms are labeled by orange and
green balls. The black arrow denotes the direction of charge transfer and
ferroelectric polarization. The primitive cell is marked by the rhombus. (b)
Charge density deformation of plane projection and three-dimensional
integration for the 3R bilayer BP, BAs and BSb. In each panel, the red
horizontal dotted lines denoted the Fermi level, the green vertical lines
indicate edges of the top and bottom layers, and the yellow and cyan
isosurfaces represent the charge accumulation and depletion at the interface
with an isovalue of 0.001 eÅ-3 (BP and BAs) and 0.045 eÅ-3 (BSb). (c) The
reaction energy barriers and (d) the magnitudes of dipole moment along the
ferroelectric switching pathway. The three inserts in (c) are side view atomic
structures at initial, middle and final states in the ferroelectric switching
pathway.

The rest parts of this paper are organized as follows. In Section 2, the
computational methods used in this work are briefly introduced. In Section 3,
the numerical results accompany with physical mechanisms are presented,
including experimental feasibility, electric polarization, ferroelectric
switching and tunable electronic/photovoltaic performances of the 3R bilayer
BX structures. The tunable photocurrent and photovoltaic revelent quality
factors in the 3R BP based nano-devices are also performed. In section 4, a
brief summary of this work is presented.

##  II Model and Numerical Method

In this study, the structural and electronic properties of the 2D BX (X=P, As,
Sb) lattices are implemented by the Vienna _ab initio_ simulation package
(VASP) based on the density functional theory (DFT).Kresse and Hafner (1993)
The generalized gradient approximation (GGA) in the form of Perdew-Burke-
Ernzerhof (PBE)Perdew _et al._ (1996) and Heyd-Scuseria-Ernzerhof (HSE06)Heyd
_et al._ (2003) are used to deal with the exchange-correlation functional. The
projected augmented wave (PAW) approachKresse and Joubert (1999) is employed
to manipulate the ion-electron interaction. A cut-off energy of 500 eV is set
for the plane wave function, and a vacuum larger than 20 Å is used to
eliminate the spurious interaction perpendicular to the 2D plane. $15\times
15\times 1$ k-meshed grid is set in the first Brillouin zone to describe the
periodicity of the 2D plane. The convergence criteria of structural relaxation
is defined less than $0.01eV/$Å for force and $10^{-5}eV$ for energy. The
phonon spectrum is implemented by the Phonopy code,Togo and Tanaka (2015) and
the _ab initio_ molecular dynamics simulation is performed within a $3\times
3\times 1$ supercell at a time step of 1 fs under finite temperature. The
energy barriers during ferroelectric switching are calculated by using the
method of nudged elastic band (NEB),Henkelman _et al._ (2000) and the
ferroelectric polarization is evaluated by the Berry phase approach.Li and Wu
(2017); Lin _et al._ (2019)

The photovoltaic transport in the 2D BP-based nano-device is performed by the
first principles quantum transport package NanoDcal,Taylor _et al._ (2001);
Waldron _et al._ (2006); Wang _et al._ (2009) which is based on the
combination of non-equilibrium Green’s function (NEGF) and DFT.Taylor _et al._
(2001) In this calculation, the norm-conserving nonlocal pseudo-potential is
used to define the atomic coreHamann _et al._ (1979), and the wave function is
expanded by the atomic orbital basis set with double-zeta polarization (DZP).
The exchange-correlation potential is handled at the PBE level.Perdew _et al._
(1996) $24\times 1$ k-meshed gride is set in the self-consistent calculation,
and $256\times 1$ k-meshed gride is adopted in the photocurrent calculation.
Finally, the convergence criteria of self-consistence is set less than
$10^{-5}$ eV.

##  III Numerical Results and Discussions

###  III.1 Electric polarization and ferroelectric switching in 3R bilayer BX
(X=P, As, Sb)

Figure 2: (color online) (a-c) Phonon band dispersion curves of 2D bilayer 3R
BP, BAs, and BSb. (d-f) The variation of total energy of each BX bilayer as a
function of time when T=500 K. Insets: top and side views of a snapshot for
each 3$\times$3 BP supercell at 10 ps.

Monolayer BX is a planer hexagonal lattice material similar to graphene, where
B atoms and X atoms evenly distribute in the vertices of the hexagon
adjacently. There are two typical stacking pattens for bilayer BX, namely 1T
stacking and 3R stacking. For the 1T bilayer BX, the top and the bottom layers
overlap completely without any dislocation, and thus no vertical electric
polarization occurs owning to the mirror symmetry between two layers (see
Fig.S1 (a)). While for the 3R stacking, the X atoms in the bottom layer are
right below the hexagon centers of the top layer, and the B atoms in the
bottom layer are coincidentally below the X atoms in the top layer as shown in
Fig. 1(a). Due to this staggered stacking configuration, the horizontal mirror
symmetry between the top layer and the bottom layer is broken, and a
spontaneous electric polarization perpendicular to the plane is induced. With
increase of interlayer interaction from BP to BSb, the planer bilayer
structures gradually evolve into wrinkled ones especially for BSb as shown in
Fig. 1(a), which further increases the spatial reflection asymmetry and
induces larger polarization. In order to obtain the polarization direction,
the renormalized real space charge density difference $\Delta\rho(\textbf{r})$
was calculated.
$\Delta\rho(\textbf{r})=[\rho_{3R}-\rho_{top}-\rho_{bottom}]/n$, where
$\rho_{3R}$, $\rho_{top}$ and $\rho_{bottom}$ are charge density of 3R bilayer
BX, individual top and bottom monolayer, respectively, and $n=2$ indicates
bilayers. Both the 1D plane integrated $\Delta\rho(z)$ and 3D isosurface plot
of $\Delta\rho(\textbf{r})$ are displayed in Fig. 1(b). Obviously, charge
depletes in the top layer, while accumulates in the bottom layer for each 3R
BX. Bader charge analysis suggests that there are 0.014, 0.023 and 0.257
electrons transferred from the top layer to the bottom layer for 3R BP, BAs
and BSb, respectively. The increased charge accumulation at the interface from
BP to BSb indicates more interlayer interaction of the latter than the former.
This asymmetry induced interlayer electric polarization has been reported in
some other 2D vdWHs and confirmed a 2D out-plane ferroelectric phase.Yasuda
_et al._ (2020); Liang _et al._ (2021); Liu _et al._ (2020) In order to
quantitatively characterize the polarization intensity, the magnitude of
interfacial polarization density $P$ was calculated based on the Berry phase
method, which is roughly equal to the polarized surface charge density
multiplied by the interlayer distance.Qiao _et al._ (2017) The calculated
values are $P_{BP}=1.6\times 10^{-12}C/m$, $P_{BAs}=2.2\times 10^{-12}C/m$ and
$P_{BSb}=13.9\times 10^{-12}C/m$, which are comparable or larger than those of
bilayer BN ($2.0\times 10^{-12}C/m$)Zhao _et al._ (2020) and 1T MoS2 ($0.22\mu
C/m^{2}$)Shirodkar and Waghmare (2014), indicating strong out-plane
ferroelectricity of 3R bilayer BX. The increased $P$ from BP to BSb is
attributed to the discontinuity of electrostatic potentials ($\Delta V$)
between the top and bottom layers, which is equal to 0.15 V, 0.18 V and 0.47
V, respectively, for 3R BP, BAs and BSb as shown in Fig.S2. The much larger
difference of $\Delta V$ between BSb and BAs than that between BAs and BP is
attributed to the wrinkled structure of BSb.

In order to confirm the ferroelectric phase of this out-plane polarization,
the modest energy barrier $\Phi$ is detected by evaluating the total energies
at different transition states for each 3R BX. A series of transition states
can be obtained by sliding the top layer along the armchair direction until
the bilayer BX returns to its initial 3R configuration. It is easy to find
that when the sliding distance is equal to two times of B-X bond length, the
final stacking is the same as the top-bottom flipped initial 3R stacking.
Using the nudged elastic band (NEB) method, the evolution of total energies
along the reaction path is shown in Fig. 1(c). Due to the interlayer vdW
interaction, the reaction energies increase firstly and then decrease
symmetrically for all the bilayer BX owning to the same initial and final 3R
stacking configurations. In other words, the modest energy barriers of
$\Phi_{BP}$=19.5 meV/f.u., $\Phi_{BAs}$=28.4 meV/f.u., and
$\Phi_{BSb}$=128.5/f.u. meV should be surmounted during the interlayer
translation from the initial 3R stacking to the final 3R stacking. The
gradually enlarged energy barriers from BP to BSb can be ascribed to the
enhanced interlayer vdW interaction. These values of reaction energy are
comparable or lower than that of bilayer 3R MoS2 (17.3 meV/f.u.),Park _et al._
(2019) $\alpha-$In2Se2 (50 meV/f.u.),Jiang _et al._ (2019) functionalized
bismuth layer (120-548 meV/f.u.),Kou _et al._ (2018) and $\gamma-$GeSe (887
meV/f.u.),Liu _et al._ (2019)), implying entirely feasible experimental
feasibility.

Figure 3: (color online) Band structures (left panels), PDOS (middle panels),
and partial charge densities at the CBM and VBM (right panels) of 3R bilayer
BP (a), 1T bilayer BP (b), 3R bilayer BAs (c), and 3R bilayer BSb (d),
respectively. In each panel, the Fermi level of each system is set to zero and
marked by the green vertical dotted lines. The band structures at PBE (HSE06)
levels are labeled by black (red) lines and the value of the isosurface in
each middle panel is set at 0.0007 eÅ-3.

Furthermore, the ferroelectric switching of the 3R BX structures along the
reaction path is shown in Fig. 1(d). Owing to the top-bottom reversion between
the initial and the final stackings, the magnitudes of interfacial dipole
moments are centrosymmetric along the total reaction path indicating a
reversion of out-plane ferroelectricity. Based on the obtained $P$, the
interlayer voltage $U$ can be estimated coarsely by using the formula
$U=P/\varepsilon_{0}$, where $\varepsilon_{0}$ is vacuum permittivity. For 3R
stacking BX, $U$ is equal to $\pm 0.12V$, $\pm 0.15V$ and $\pm 0.57V$,
respectively, for BP, BAs and BSb. By dragging either monolayer along the
armchair direction, these interlayer voltages can be reversed periodically.
Such intriguing vertical ferroelectric switching properties can be used as a
nanogenerator, where the electron flow and energy harvesting will be
manipulated with the mechanism akin to the previously designed triboelectric
nanogenerator.Fan _et al._ (2012); Zhang _et al._ (2016, 2015)

###  III.2 Crystal stabilities and electronic properties of 3R BX (X=P, As,
Sb)

Before further evaluating the electric properties of ferroelectric 3R bilayer
BX, their structural stability should be examined firstly. Firstly, the real
space projection of electron localized functions (ELFs) is calculated to
characterize the bonding types of 3R BX. The magnitude of ELF from 0 to 1
describes the spatial localization extent of the reference electrons, where
ELF = 1 or 0 represents complete localization or full decentralization of the
electrons. For each 3R bilayer BX, substantial concentration of electrons is
located between B and X atoms as shown in Fig.S3, indicating strong covalent
bonding feature and stable configuration. Next, the interfacial binding energy
is calculated to examine the energy stability of the 3R bilayer BX by
$E_{b}=(E_{3R}-E_{top}-E_{bottom})/S$, where $E_{3R}$, $E_{top}$ and
$E_{bottom}$ are energies of the unit cell of the 3R bilayer BX, the top and
the bottom monolayer, respectively, and $S$ is the area of unit cell. The
interfacial binding energies are found to be $E_{b,BP}=-0.27J/m^{2}$,
$E_{b,BAs}=-0.35J/m^{2}$ and $E_{b,BSb}=-0.12J/m^{2}$, which are comparable to
those of some recently reported vdWHs including WTe2/HfS2 ($-0.204J/m^{2}$)Lei
_et al._ (2019), phosphorene/SnSe2 ($-0.019J/m^{2}$),Xia _et al._ (2018),
KAgSe/SiC2 ($-0.327J/m^{2}$),Wang _et al._ (2020) _et.al._. Moreover, the
stability of 3R bilayer BX is further confirmed by phonon spectrum and _ab
initio_ molecular dynamics simulations. As shown in Fig. 2 (a-c), no imaginary
phonon mode is observed in the entire Brillouin zone, confirming the dynamical
stability of 3R BX structures. Finally, the temporal evolution of the total
energies of 3$\times$3 3R BX supercells are evaluated to examine their thermal
stabilities. The temperature is set equal to 500K and the evolutionary time is
up to 10 ps with a time step of 1 fs. As shown in Fig. 2 (d-f), the total
energy of each 3R BX oscillates around a fixed value with only small
fluctuations, indicating the thermal stability of each 3R BX during the
temporal evolution. Meanwhile, the structural integrity of dynamic
configurations maintains well without broken bond or geometry reconstruction,
revealing excellent thermal stability of each 3R BX structure even at high
temperature. In summary, the structural stability of the three 3R bilayer BX
is confirmed from several aspects, which provides a possible guidance for
their experimental synthesis.

Figure 4: (color online) (a) Schematic diagrams of the energy levels at the
$K$ point of 2D bilayer 3R BP, where the Fermi level is marked by green
horizontal line. (b) Optical absorption coefficients versus energy of the
incident light for different 2D bilayer 3R BX (X=P, As, Sb) systems (left
panel) and different configurations of BP (right panel). Here, the
polarization vector is set parallel to the plane and along the zigzag edges.
The region of visible light in each panel are denoted by the colored
translucent shadow areas between two vertical dash lines.

In the following, we examine the electronic behaviors of the 3R bilayer BX.
Fig. 3 (a) shows the band structure, projected density of states (PDOS) and
partial charge densities of 3R bilayer BP with the exchange-correlation
interaction considered at both PBE and HSE06 levels. Obviously, the band
structures based on both exchange-correlation interactions are qualitatively
consistent. The 3R bilayer BP shows indirect band gap semiconducting behavior
with band gap equal to 0.73 eV at PBE level and 1.15 eV at HSE06 level. The
moderate band gap at HSE06 level is very close to the optimum range of
1.2$\sim$1.6 eV for excitonic solar cells (XSCs).Liang _et al._ (2018) The
conduction band minimums (CBM) is located at K point, while the valence band
maximum (VBM) is slightly deviated away from K point accompany with tiny
upward from B point to C point. Considering that the bands of bilayer should
be roughly a superposition of those from two individual monolayers (see
Fig.S1(b)) due to very weak vdWH interaction, we expect that this indirect
band gap can be ascribed to the interlayer built-in electric field caused by
the spontaneous ferroelectric polarization. To confirm this conjecture, the
electronic behaviors of the 1T bilayer BP is also plotted in Fig. 3 (b).
Obviously, direct band gap behavior is observed for either PBE or HSE
calculation, where both CBM and VBM are located at K point. Its band gap is
equal to 0.42 eV at PBE level and 0.78 eV at HSE06 level, which is smaller
than that of corresponding 3R bilayer BP, indicating the stronger interlayer
vdWH interaction of 1T BP. As shown by PDOS in the middle panels of Fig. 3 (a)
and (b), the CBMs are dominated by the B-$p_{z}$ orbitals, while the VBMs are
mainly contributed by the P-$p_{z}$ for both 3R and 1T bilayer BP indicating
their similarity. However, further calculation of partial charge densities as
shown in the right panels of Fig. 3 (a) and (b) points out their difference.
For the 1T bilayer BP, both the CBM and the VBM are equally contributed by the
top layer and the bottom layer, and therefore no interlayer charge
polarization. While, for the 3R bilayer BP, the CBM is mostly dominated by the
bottom layer, and the VBM is mainly contributed by the top layer, indicating
an obvious spatial separation between the CBM and the VBM. As a result, a
type-II band alignment is formed for the 3R bilayer BP owing to the real space
charge separation between CBM and VBM. Similar behavior of type-II band
alignment and real space electron-hole separation are also found for 3R
bilayer BAs and BSb as shown in Fig. 3 (c) and (d), respectively, apart from
the enlarged delocalizations of VBM and the decreased band gaps due to the
stronger electric polarization. In Fig. 4 (a), we provide a more essential and
intuitive physical image to present the type-II band alignment of 3R BP, where
the CBM and VBM of top and bottom layer are dominated by B-$p_{z}$ and
P-$p_{z}$, respectively. Obviously, driven by the interlayer built-in electric
field, the energy levels of B-$p_{z}$ and P-$p_{z}$ from the top layer are
higher than those of the bottom layer.

For this type-II 3R bilayer BP ferroelectric system, a series of intriguing
photovoltaic excellences will be formed naturally. (1) Different from 1T
bilayer BP, the built-in electric field in 3R bilayer BP will facilitates the
separation of the free electrons and holes into different layers, accompanied
with reduction of the photo-excited electron-hole recombination opportunity.
(2) As for the heterojunctions with distinct 2D materials, lattice mismatch
would induce large potential barriers which limits their stability and
experimental synthesis. In addition, the possible type-I or type-III band
alignments between layers may restrict their application in photovoltaic field
someway.Wang _et al._ (2019, 2020) Instead, the 3R bilayer BP possesses
identical component materials, intrinsic type-II band alignment and nearly
complete real space electron-hole separation. (3) The unique vertical
ferroelectric switching properties enable the realization of electron-hole
separation and reversion in real space by relative translation between layers.
These properties of 3R bilayer BP are appealing for experimental design,
application and manipulation in the field of optoelectronic devices and solar
energy conversions.Cheng _et al._ (2018) More strikingly, these ideal
performances of ferroelectric photovoltaic materials can also be extended to
other 2D layered crystals beyond the 3R bilayer BX, hence providing a new
strategy for advancing the fundamental research, experimental design and
unique industrial applications of future photovoltaic conversion devices.

Figure 5: Band structures of the 3R bilayer BP under various interlayer
distances $d$ at HSE06 level. In each panel, the bands are projected to the
top and the bottom layers with different magnitudes indicated by different
colors. The green horizontal lines denote the Fermi level, and the red arrows
indicate the indirect-to-direct band gap variation. (b) Electron transfer
($\Delta$q) from the top layer to the bottom layer and binding energy (Eb) as
functions of $d$. (c) Top panel: isosurface of charge density difference under
three different $d$. The yellow and green regions correspond to charge
accumulation and depletion, respectively. Bottom panel: dipole moment $P$
changes over $d$. (d) Band gap $\Delta E$, conduction band offset $\Delta
E_{c}$ and band gap of the donor segment $E_{g}$ versus interlayer distance
$d$. In (b) and (d), The white vertical line represents the equilibrium
interfacial distance of the system..

###  III.3 Carrier mobilities and optical absorptions of 3R BX (X=P, As, Sb)

In light of potential application as 2D photoluminescence device, the carrier
mobility and optical absorption coefficient are two fundamental factors to be
explored. The carrier mobilities of the 3R bilayer BX are calculated based on
the deformation potential (DP) theory, which can be expressed as
follows,Bardeen and Shockley (1950)

$\mu_{2D}=\frac{e\hbar^{3}C_{2D}}{k_{B}T|m_{e/h}^{*}|(E_{1})^{2}}.$ (1)

where $C_{2D}$ is the elastic modulus, $k_{B}$ is the Boltzmann constant, $T$
is the temperature, $m_{e/h}^{*}$ is the effective mass of electrons or holes,
and $E_{1}$ is the deformation potential. This model of deformation potential
theory has been used extensively to examine the $\mu_{2D}$ of 2D atomic
structures. As shown in Table 1, ultra-high carrier mobilities have been found
for the proposed 3R bilayer BX structures. For example, the maximum $\mu_{2D}$
of the 3R bilayer BP can reach $5.72\times 10^{5}cm^{2}\cdot V^{-1}\cdot
s^{-1}$ for electrons along armchair direction. These ultra-high carrier
mobilities of 3R bilayer BX can further highlight their potential application
prospects in electronics and photoelectronics.

Table 1:  carrier mobilities ($\mu_{2D}$) of bilayer 3R BX crystals for electrons and holes along armchair and zigzag directions. crystal structures | zigzag |  | armchair
---|---|---|---
e | h |  | e | h
$BP(10^{4}cm^{2}V^{-1}s^{-1})$ | 5.72 | 1.26 |  | 3.73 | 1.46
$BAs(10^{4}cm^{2}V^{-1}s^{-1})$ | 2.49 | 1.56 |  | 1.07 | 1.85
$BSb(10^{4}cm^{2}V^{-1}s^{-1})$ | 1.44 | 1.41 |  | 0.62 | 1.61
Figure 6: (color online) (a) Optical absorption coefficients $a(\omega)$
versus energy of incident light for 3R bilayer BP under different interlayer
distances $d$. The polarization vector is set parallel to the plane and along
the zigzag edges. The two vertical dash lines indicate the visible light
region. (b) Phase diagram of PEC as functions of donor bandgap $E_{g}$ and
conduction band offset $\Delta E_{c}$ under different interlayer distance $d$.

The optical absorption coefficient $a(\omega)$ is calculated based on the
following formula,Bassani _et al._ (1976)

$a(\omega)=\frac{\sqrt{2\omega}}{c}\sqrt{\sqrt{\epsilon_{1}^{2}(\omega)+\epsilon_{2}^{2}(\omega)}-\epsilon_{1}(\omega)},$
(2)

where $\omega$ is the frequency of incident light, $c$ is the light velocity,
$\epsilon_{1}$ and $\epsilon_{1}$ are the real and imaginary parts of
frequency-dependent dielectric function, respectively. $\epsilon_{1}$ can be
calculated by Kramers-Kronig transformation, and $\epsilon_{2}$ is the
summation over the empty states.Bassani _et al._ (1976) In this calculation,
the directions of polarization vectors of the incident light are set parallel
to the plane. As shown in the left panel of Fig. 4 (b), $a(\omega)$ maintains
zero for each 3R BX until $\omega$ of the incident light is larger than the
cut-off frequency, and then increases versus the energy of incident light.
Besides, the curves of $a(\omega)$ obtained at HSE06 level are qualitatively
consistent with the curves obtained at PBE level with proper blue shifts,
which is accordant with the information given by the band structures as shown
in Fig. 3. Intriguingly, all $a(\omega)$ curves possess wide peaks over the
range of visible light ($\sim 1.6-3.2eV$). The maximum values can reach $\sim
4.95-7.75\times 10^{5}cm^{-1}$ for 3R BP, BAs, and BSb, respectively, which
are one order of magnitude larger than those of ideal 2D layered KAgSe and
TiNX (X=F, Cl, Br).Wang _et al._ (2018); Liang _et al._ (2018) To further
understand the mechanism of the large absorption coefficients of these 3R
bilayer BX, the absorption spectrums of the monolayer and 1T bilayer BP are
also evaluated to make a comparison as shown in the right panel of Fig. 4 (b).
Obviously, $a(\omega)$ of the 3R bilayer BP is the largest over the visible
light range, indicating most excellent optical absorption performances.

###  III.4 Tunable effects of interfacial distance on the 3R bilayer BP

In the following, we regulate the electric and photoelectric properties of the
3R bilayer BX by changing their interfacial distance, which can be implemented
through applying a vertical strain by vacuum thermal annealing or imposing
pressure with a scanning tunneling microscopy tip in experiments.Tongay _et
al._ (2014); Yankowitz _et al._ (2016) As shown in Fig. 5 (a), when the
interfacial distance $d$ of the 3R bilayer BP increases from 2.9 Å to 4.1 Å,
the band gap increases from 0.91 eV to 1.29 eV (HSE06), accompany with a
conversion from indirect band gap to direct band gap. This variation of band
gap is desired to enhance the photovoltaic performance of the material. To
understand the variation of band gap, charge transfer $\Delta q$ and binding
energy $E_{b}$ under different $d$ are evaluated and presented in Fig. 5 (b).
$E_{b}$ remains negative within the entire range of $d$, indicating the
energetic stability of the 3R bilayer BP under all vertical strain, and the
minimum value appears at the equilibrium distance $d=3.5$ Å. Meanwhile,
$\Delta q$ decreases monotonically versus $d$, revealing declined charge
transfer from the top layer to the bottom layer. More visually, real space
charge transfer under three typical interfacial distances are shown in the top
panel of Fig. 5 (c), where charge accumulation at the interface attenuates
with increase of interlayer distance, indicating decaying interfacial
interaction. It is the decaying interfacial interaction that induces
monotonically decreased dipole moment $P$ as shown in the bottom panel of Fig.
5 (c) and therefore the variation from indirect band gap to direct band gap.

Next, we explore the tunable photovoltaic performances of the 3R bilayer BP
under various vertical strain. Fig. 6 (a) presents the optical absorption
coefficients $a(\omega)$ under different interlayer distance at both PBE and
HSE06 levels. The curves obtained at HSE06 level are roughly accordant with
the blue-shifted curves obtained at PBE level with qualitative agreement. More
intriguingly, both magnitudes and positions of the peaks of $a(\omega)$ are
changed obviously under different vertical strains. When $d$ is enlarged to
4.1 Å, larger magnitude of peak can be observed over the visible light range,
revealing the stronger capture performance of visible light. In contrast, when
$d$ is compressed to 2.9 Å, a significant red-shift is generated due to the
band recombination under vertical strain. PCE is another important factor to
describe the efficiency of XSC, which can be calculated by Scharber _et al._
(2006),

$PCE=\frac{0.65(E_{g}-\Delta
E_{c}-0.3)\int_{E_{g}}^{\infty}\frac{P(\hbar\omega)}{\hbar\omega}d(\hbar\omega)}{\int_{0}^{\infty}P(\hbar\omega)d(\hbar\omega)},$
(3)

where 0.65 is the assumed value of the band-fill factor; $E_{g}$ is the band
gap of the donor segment; $\Delta E_{c}$ is the magnitude of conduction band
offset; $(E_{g}-\Delta E_{c}-0.3)$ is the estimated value of the maximum open
circuit voltage; and $P(\hbar\omega)$ is the $AM1.5$ solar energy flux with
the photon energy of $\hbar\omega$. As shown in Fig. 5 (d), the band gap
increases gradually with stretch of interlayer distance $d$, while $E_{g}$ and
$\Delta E_{c}$ decrease monotonically due to the reduced vdWH interaction. As
a result, PCE increases with interlayer distance. Fig. 6 (b) shows the phase
diagram of PCE versus $E_{g}$ and $\Delta E_{c}$ of the 3R bilayer BP under
different interlayer distance. At the equilibrium interlayer distance (3.5 Å),
$E_{g}$ and $\Delta E_{c}$ are 1.74 eV and 0.60 eV, respectively, and PCE can
reach 11.7%. This large PCE is coincidentally equal to the best certified
efficiency of the organic solar cell (11.7%)Zhao _et al._ (2016), and is
larger than that of the typical MoS2/p-Si heterojunction solar cell
(5.23%)Tsai _et al._ (2014). When $d$ is widened from 3.5 Å to 4.5 Å, PCE
grows from 11.7% to 22.6%. Such high PCE has outpaced those of most available
materials so farZhou _et al._ (2020), such as recently reported TiNX1/TiNX2
XSCs (X${}_{1}\neq X_{2}$=F, Cl, Br ) ($\sim 18\%-22\%$).Liang _et al._ (2018)
These excellent tunable performances of solar absorption and PCE indicate that
our new designed 3R bilayer BP structure can behave as ideal XSCs.

Figure 7: (color online) (a) Schematic structures of 2D two-probe devices
based on the monolayer BP (top panel) and the bilayer 3R BP (bottom layer).
The drain and source are marked by the left red box and right green region,
respectively. The yellow zone in the center scattering region stands for
lighting area. (b) Photocurrent of the two-probe systems based on the
monolayer (1L) and bilayer BP under different vertical strains. The incident
photon energy of each system is set equal to the corresponding band gap.

###  III.5 Photocurrent in different types of 2D BP based nanodevices

Although favorable photovoltaic properties of 2D 3R bilayer BX has been
discovered, however, it is still necessary to explore the practical
photovoltaic performances in 3R bilayer BP based electronic devices
considering of the quantum scattering in nanoscale devices due to aperiodic
characteristics. Fig. 7 (a) shows two schematic structures of pure monolayer
and 3R bilayer BP based 2D two probe devices, where both leads are constructed
by periodic extension of the scattering region. Compared with the traditional
metal-simiconductor tunneling junctions, there are more advantages of these
pure BP devices including simple structure, lattice matching, smooth and
continuous interface and easily to be prepared. As reported in previous works,
such simplified device model has been confirmed effective to evaluate the
photo-electronic transport in practical devices,Wang _et al._ (2019, 2018);
Liang _et al._ (2018) and has been employed extensively in experimental
battery designWang _et al._ (2020); Xia _et al._ (2018); Wang _et al._ (2019,
2018); Liang _et al._ (2018). For each nanodevice shown in Fig. 7 (a), the
whole scattering region is irradiated by a perpendicular linearly polarized
light, and a small bias voltage is applied between the source and drain to
drive the flowing of photocurrent. When the photon energy is greater than the
band gap, electrons are excited from the valence band to the conduction band
accompany with the generation of electron-hole pairs. Under the driven force
of potential difference between source and drain, the electron-hole pairs will
be further separated to electrons and holes transferring along opposite
directions, and finally a photocurrent occurs. In our calculation, the power
density of incident light is set to 103 $\mu W/mm^{2}$, and the bias voltage
is equal to 0.2 V, which is far less than the band gap of BP to ensure only
the photocurrent is generated. At the first order of Born approximation, the
photocurrent flowing into the left probe can be expressed as follows,Zhang _et
al._ (2014); Xie _et al._ (2015); Henrickson (2002)

$\displaystyle
I_{L}^{ph}=\frac{ie}{h}\int\textbf{Tr}\left[\Gamma_{L}\\{G^{<(ph)}+f_{L}(E)(G^{>(ph)}-G^{<(ph)})\\}\right]dE,$
(4)

where $\Gamma_{L}$ is the line-width function, $f_{L}$ is the Fermi
distribution function, and $G^{</>(ph)}$ is the lesser/greater Green’s
function of the two probe system including electron-photon
interaction.Henrickson (2002) More detailed theoretical formalism of
photocurrent can be found in Ref.[74].

Fig. 7 (b) shows the photocurrent density $J_{ph}=I_{L}^{ph}/S$ of monolayer
BP and 3R bilayer BP nanodevices with three different interlayer distances $d$
when the incident photon energy $E_{ph}$ is equal to the band gap $\Delta E$
of corresponding BP. $J_{ph}$ of different BP devices can reach 13.8-213.5
$\mu A/mm^{2}$, which are large enough for experimental detection. For the 3R
bilayer BP device at equilibrium interlayer distance (3.5 Å), $J_{ph}$ is
roughly 3.5 times larger than that of the monolayer BP device due to the
stronger absorption ability and type-II band alignment. For the bilayer 3R BP
devices, $J_{ph}$ is largely improved from 13.8 $\mu A/mm^{2}$ to 213.5 $\mu
A/mm^{2}$ with increasing $d$ from 2.9 Å to 4.1 Å. This phenomenon can be
attributed to increasing PCE and conversion from indirect to direct band gap
as functions of $d$, considering of their similar optical absorptions and
type-II band alignments. Compared to the indirect band gap systems, the
excitation of electrons in the direct band gap devices no longer needs the
assistance of phonons, hence suggesting a much larger photocurrent. On the
contrary, the obvious indirect band gap and ultra-low PCE of the bilayer
device with $d$=2.9 Å reduce $J_{ph}$ even smaller than that of the monolayer
device. Note that all results in Fig. 7 (b) are calculated at PBE level
considering of the huge computational cost of photocurrent in HSE06
calculation. We believe that the photocurrents obtained at HSE06 level should
be accordant qualitatively with those obtained at PBE level with quantitative
difference.

Figure 8: (color online) Photocurrent density Jph versus incident light energy
Eph and polarization angle $\theta$, for (a) monolayer BP and bilayer 3R BP
with interlayer distance $d$ is equal to (b) 4.1 Å, (c) 3.5 Å, and (d) 2.9 Å
at PBE level. The power density Pin of the incident light is 103 $\mu
Wmm^{-2}$.

In the following, we discuss the variation of $J_{ph}$ _versus_ energy
$E_{ph}$ and polarization angle $\theta$ of linearly polarized incident light.
As shown in Fig. 8, broad peaks of $J_{ph}$ are detected in the visible light
region with the maximum value equal to of 72 $\mu A/mm^{2}$ for monolayer BP
device and 108-144 $\mu A/mm^{2}$ for three 3R bilayer devices. These energies
of maximum $J_{ph}$ roughly correspond to the peaks of optical absorption
coefficient a($\omega$) as shown in Fig. 6 (a), and are also close to the
energy difference between the two maximum values of the density of states near
the Fermi energy (see Fig.S4). The detected maximum values of $J_{ph}$ are far
larger than those of the MoS2 and KAgSe devices under the same external
conditionsStewart and Léonard (2004); Wang _et al._ (2018), indicating an
appealing photoelectric conversion performances of these 2D BP devices in the
visible light range. For three 3R bilayer BP devices, an obvious red-shift of
$J_{ph}$ peak is observed when interlayer distance is stressed from 4.1 Å to
2.9 Å. In addition, $J_{ph}$ of each BP device is roughly symmetrical relative
to $\theta=90^{\circ}$ with the maximum values locating at
$\theta=90^{\circ}$. This is reasonable because $J_{ph}$ can be expressed as a
linear superposition of three terms, which are proportional to
$\sin^{2}\theta$, $\cos^{2}\theta$, and $\sin 2\theta$, respectively, and the
final magnitude of $J_{ph}$ is determined by the competition of these three
trigonometric terms.Xie _et al._ (2015) In view of the enlarged band gaps and
right-shift a($\omega$) peak under HSE06 calculations, we predict that the
peaks of $J_{ph}$ obtained at HSE06 level should be pushed toward the
ultraviolet direction as what happened in the previous reports.Wang _et al._
(2018)

Finally, the photon responsivity $R_{ph}$ and external quantum efficiency
$\tau_{eqe}$ are evaluated based on $J_{ph}$, which are also essential
parameters in describing the photovoltaic performance in experiments. $R_{ph}$
is defined by the ratio of $J_{ph}$ to the incident photon flux
$F_{ph}$,Koppens and Mueller (2014); Wang _et al._ (2015)

$R_{ph}=\frac{J_{ph}}{eF_{ph}},$ (5)

where $e$ is the electron charge, and the photon flux $F_{ph}=P_{in}/(e\cdot
E_{ph})$ describes the number of incident photons pear unit area and time.
$\tau_{eqe}$ describes the ratio of the excited photoelectrons to the incident
photons, which can be calculated byWang _et al._ (2015, 2019)

$\tau_{eqe}=R_{ph}E_{ph}/e.$ (6)

In our calculation, the maximum values of aforementioned parameters are
located at $E_{ph}$ = 2.62 eV, 2.61 eV, 2.31 eV, and 2.11 eV for monolayer and
3R bilayer BP with d=4.1Å, 3.5Å, and 2.9Å, respectively. In Table 2, the
$R_{ph}$ and $\tau_{eqe}$ are listed including our designed BP devices and
previously reported experimental and theoretical parameters. We can find that
these parameters in our designed BP devices are comparable and even larger
than most of the previous reported ones. Especially, both $R_{ph}$ and $\tau$
are strongly enhanced in the 3R bilayer BP with large interlayer distance. All
these results suggest that the 3R bilayer BP structures are appealing 2D solar
cell materials, and its tunable photoelectric performances provide a new
strategy for future experimental design of 2D photoelectric conversion
devices.

Table 2:  The photon responsivity (Rph) and the external quantum efficiency ($\tau_{eqe}$) of 2D BP based device. For comparison, the parameters of some other pervious experimental and theoretical reported 2D systems are also listed. - | Thickness | $R_{ph}(AW^{-1})$ | $\tau_{eqe}(\%)$ | Ref.
---|---|---|---|---
BP | ML | 0.069 | 11.2 | this work
3R BP${}_{d=4.1\mathring{A}}$ | ML-ML | 0.127 | 20.7 | this work
3R BP${}_{d=3.5\mathring{A}}$ | ML-ML | 0.091 | 13.1 | this work
3R BP${}_{d=2.9\mathring{A}}$ | ML-ML | 0.113 | 14.9 | this work
BP-BAs | ML-ML | 0.03 | 7.4 | [27]
BAs-BSb | ML-ML | 0.058 | 14.7 | [27]
KAgSe | ML | 0.056 | 17.92 | [30]
MoS2 | ML | 0.016 | - | [79]
MoS2-MoS2 | ML-ML | 0.030 | - | [80]
MoS2-WSe2 | 11 nm | 0.010 | 1.5 | [81]
InSe-InTe | ML-ML | 0.030 | 7.1 | [82]
BP-MoS2 | 11 nm-ML | 0.418 | 0.3 | [83]

##  IV Summary

In summary, we carried out a first principles calculation to investigate the
photovoltaic properties of 3R bilayer BX(X=P, As, Sb) and explore their
potential applications as 2D XSCs. Due to the intrinsic OOP ferroelectricity
induced by inversion symmetry broken, the CBM and VBM are dominated by
different layers and a type-II band alignment is formed spontaneously. Lots of
intriguing performances are demonstrated for these systems, including moderate
band gaps, ultrahigh carrier mobilities, efficient optical absorption in
visible light region and large PCE. These essential parameters of
photoelectric materials can be further enhanced and a desirable red-shift of
optical absorption peak is presented by increasing the interlayer distance.
Excellent photoelectric conversion behaviors are also detected in the 3R
bilayer BP based nanodevices. Remarkable photocurrents, photon responsivities
and external quantum efficiencies are also obtained. These ideal properties
reveal an appealing application of 3R bilayer BX in the photoelectric
conversion and XSCs. Moveover, the efficient and adjustable photovoltaic
performance driven by intrinsic OOP ferroelectricity in 3R bilayer BX provides
a new strategy for designing various nanoscale ferroelectric-photovoltaic
devices.

## Conflicts of interest

The authors declare no competing financial interest.

## Acknowledgements

This work was financially supported by grants from the National Natural
Science Foundation of China (Grant No. 11774238), Key Program Project of
National Natural Science Foundation of China (Grant No. 12034014), and
Shenzhen Natural Science Foundations (Grant No. JCYJ20190808150409413).

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