# Low-symmetry two-dimensional BNP2 and C2SiS structures
with high and anisotropic carrier mobilities

Shixin Song  School of Physics, Southeast University, Nanjing, 211189, PRC
Jie Guan<EMAIL_ADDRESS>School of Physics, Southeast University, Nanjing,
211189, PRC  David Tománek<EMAIL_ADDRESS>Physics and Astronomy Department,
Michigan State University, East Lansing, Michigan 48824, USA

###### Abstract

We study the stability and electronic structure of previously unexplored two-
dimensional (2D) ternary compounds BNP2 and C2SiS. Using ab initio density
functional theory, we have identified four stable allotropes of each ternary
compound and confirmed their stability by calculated phonon spectra and
molecular dynamics simulations. Whereas all BNP2 allotropes are
semiconducting, we find C2SiS, depending on the allotrope, to be
semiconducting or semimetallic. The fundamental band gaps of the
semiconducting allotropes we study range from $1.4$ eV to $2.2$ eV at the
HSE06 level and display carrier mobilities as high as $1.5{\times}10^{5}$
cm2V-1s-1. Such high mobilities are quite uncommon in semiconductors with so
wide band gaps. Structural ridges in the geometry of all allotropes cause a
high anisotropy in their mechanical and transport properties, promising a wide
range of applications in electronics and optoelectronics.

## Introduction

Two-dimensional (2D) materials have intrigued researchers around the world
since the successful mechanical exfoliation of graphene [1]. Even though
graphene remains unsurpassed in terms of high charge carrier mobility, the
vanishing band gap in the pristine material precludes its use from
semiconducting circuitry [2, 3]. Other 2D materials including transition metal
dichalchogenides (TMDs) such as MoS2 have sizable band gaps, but are not as
useful due to their low carrier mobility [4, 5]. Phosphorene, a monolayer of
black phosphorus, combines high and anisotropic carrier mobility with a
sizeable and tunable band gap [6, 7, 8], but is unstable under ambient
conditions [9]. In spite of significant efforts to improve the performance of
2D materials in semiconducting devices, the progress has been moderate. There
is a need to find new 2D semiconductors with substantial band gaps and high
carrier mobilities.

Phosphorus carbide (PC), a recently proposed 2D material, has been predicted
to be stable [10] and to display promising electronic behavior including high
carrier mobility [11]. Of the stable allotropes, the semiconducting
$\alpha_{1}$-PC phase, also called black phosphorus carbide ($b$-PC), has been
successfully synthesized. It shows a high field-effect mobility $\mu=1995$
cm2V-1s-1 of holes at room temperature [12], good infrared response [13] and
tunable anisotropic plasmonic performance [14]. The narrow band gap, however,
limits the performance of PC-based field-effect transistors (FETs) due to a
relatively low ON/OFF ratio [12]. Experimental data for 2D GeP [15], another
low-symmetry IV-V compound with strongly anisotropic conductance, indicate a
high field-effect ON/OFF ratio ${\approx}1\times 10^{4}$, but a low carrier
mobility ${\mu}=0.35$ cm2V-1s-1. Theoretical explorations have been extended
to other 2D IV-V compounds, which are not isoelectronic to PC. These include
germanium triphosphide [16] GeP3, tin triphosphide [17] SnP3, and phosphorus
hexacarbide [18] PC6. However, all these systems have narrow band gaps similar
to $\alpha_{1}$-PC and share a hexagonal honeycomb lattice structure and thus
a weak transport anisotropy. 2D structures with a substantial band gap, high
carrier mobility and strong in-layer anisotropy are still missing.

In search of such 2D materials inspired by anisotropic 2D PC structures, we
have applied an effective design strategy known as “isoelectronic
substitution”. This process involves substituting certain elements in the
structure by their neighbors in the periodic table, yet keeping the total
valence electron count unchanged. This approach, which has been successfully
applied in both 3D and 2D systems, allows to change physical and chemical
properties of the system without drastically changing the structure. In this
way, the diamond structure of bulk silicon with a diatomic unit cell can be
changed to the isoelectronic Si3AlP, when one Si atom is substituted by Al and
the other by P in every other unit cell, thus significantly increasing light
absorption in the visible region [19]. In semimetallic 2D graphene with a
diatomic basis, substituting one C atom by B and the other by N forms the
$h$-BN structure, a wide-gap insulator. In a similar way, substituting every
other atom in phosphorene by Si and S atoms results in the 2D SiS structure
[20, 21]. The same isoelectronic substitution in 2D group V systems leads to
2D group IV-VI compounds including GeS, GeSe, and SnS with a lower symmetry
and a wide range of physical properties [22, 23, 24, 25].

Figure 1: Ball-and-stick models of relaxed monolayer structures of (a)
$\alpha_{1}$, (b) $\beta_{1}$, (c) $\alpha_{2}$ and (d) $\beta_{2}$ allotropes
of BNP2, and (e) $\alpha_{1}$, (f) $\beta_{1}$, (g) $\alpha_{2}$ and (h)
$\beta_{2}$ allotropes of C2SiS in top and side view. The lattice vectors are
indicated by red arrows.

In this study, we propose isoelectronic substitution in 2D structures of
phosphorus carbide that leads to previously unknown ternary compounds BNP2 and
C2SiS. Our ab initio density functional calculations identify four stable
allotropes of each compound that share the 2D geometry with PC. Calculated
phonon spectra and ab initio molecular dynamics (MD) simulations confirm the
stability of each of these allotropes. Due to structural ridges in the
geometry, all allotropes considered exhibit a significant anisotropy in the
elastic response. Electronic structure calculations indicate that $\alpha_{1}$
and $\beta_{1}$ phases of C2SiS, as well as all four BNP2 allotropes, are
semiconductors with a wide range of band gap values. Carrier mobility
calculations show that most of the semiconducting allotropes exhibit high and
strongly anisotropic carrier mobilities. The remaining $\alpha_{2}$-C2SiS and
$\beta_{2}$-C2SiS allotropes were found to be semimetallic, displaying an
elliptically distorted Dirac cone in their band structure caused by their
structural anisotropy.

Table 1: Calculated equilibrium properties of BNP2 and C2SiS 2D allotropes.

| BNP2 | C2SiS
---|---|---
| $\alpha_{1}$ | $\beta_{1}$ | $\alpha_{2}$ | $\beta_{2}$ | $\alpha_{1}$ | $\beta_{1}$ | $\alpha_{2}$ | $\beta_{2}$
$|\vec{a}_{1}|$ (Å)a | 8.50 | 9.39 | 9.44 | 10.44 | 8.07 | 4.75 | 9.83 | 10.61
$|\vec{a}_{2}|$ (Å)a | 5.97 | 5.97 | 5.14 | 5.14 | 5.87 | 5.86 | 5.03 | 5.03
$d_{P-P}$ (Å)b | 2.25 | 2.25 | 2.27h | 2.27h |  |  |  |
|  |  | 2.24i | 2.23i |  |  |  |
$d_{P-N}$ (Å)b | 1.78 | 1.78 | 1.79 | 1.79 |  |  |  |
$d_{P-B}$ (Å)b | 1.94 | 1.94 | 1.95 | 1.95 |  |  |  |
$d_{B-N}$ (Å)b | 1.40 | 1.40 | 1.46 | 1.46 |  |  |  |
$d_{Si-S}$ (Å)b |  |  |  |  | 2.49 | 2.46 | 2.41h | 2.40h
|  |  |  |  |  |  | 2.35i | 2.35i
$d_{C-S}$ (Å)b |  |  |  |  | 1.77 | 1.77 | 1.77 | 1.77
$d_{C-Si}$ (Å)b |  |  |  |  | 1.94 | 1.94 | 1.95 | 1.95
$d_{C-C}$ (Å)b |  |  |  |  | 1.36 | 1.36 | 1.42 | 1.42
$E_{coh}$ (eV/atom)c | 4.74 | 4.74 | 4.98 | 4.98 | 5.25 | 5.26 | 5.54 | 5.54
$E_{form}$ (eV/atom)d | -0.01 | -0.01 | -0.25 | -0.24 |  0.68 | 0.68 | 0.39 | 0.39
$c_{11}$ (N/m)e | 7.17 | 40.59 | 10.24 | 57.53 | 6.42 | 33.79 | 14.89 | 53.35
$c_{22}$ (N/m)e | 156.02 | 144.13 | 189.59 | 173.68 | 145.94 | 124.02 | 199.72 | 185.35
$E_{g-PBE}$ (eV)f | 0.52 | 0.80 | 1.39 | 1.26 | 0.71 | 0.70 | 0.00 | 0.00
$E_{g-HSE06}$ (eV)g | 1.38 | 1.69 | 2.15 | 2.06 | 1.57 | 1.40 | 0.00 | 0.00

  * 1

a $|\vec{a_{1}}|$ and $|\vec{a_{2}}|$ are the in-plane lattice constants
defined in Fig. 1.
b $d$ is the equilibrium bond length between the respective species.
c $E_{coh}$ is the cohesive energy per average atom with respect to isolated
atoms.
d $E_{form}$ is the formation energy per average atom with respect to
elemental structures.
e $c_{11}$ ($c_{22}$) are the 2D elastic constants along the $x$ ($y$)
direction.
f $E_{g}$-PBE is the band gap value at the DFT-PBE level.
g $E_{g}$-HSE06 is the band gap value at the DFT-HSE06 level.
h Length of bonds within the structural plane.
i Length of bonds out of the structural plane.

Figure 2: Effect of uniaxial in-layer strain on the relative binding energy
${\Delta}E$ in monolayers of (a) BNP2 and (b) C2SiS. Results for different
allotropes are distinguished by color and symbols. Results for strain along
the $x$-direction are shown by solid lines and for strain along the
$y$-direction by dashed lines.  Figure 3:  Electronic structure of a BNP2
monolayer. Shown are the calculated band structure $E($k) and the density of
states (DOS) including its projection on individual atoms (left panels), the
partial charge density distributions at the valence band maximum (VBM) and the
conduction band minimum (CBM) (right panels) of the (a) $\alpha_{1}$, (b)
$\alpha_{2}$, (c) $\beta_{1}$, and (d) $\beta_{2}$ allotropes. All results are
based on the DFT-PBE functional with the exception of DFT-HSE06 data, which
are shown by the red dashed lines in the band structure plots. The blue arrows
indicate the fundamental band gap. Contributions of different atoms are
distinguished by color in the PDOS plots. Isosurface plots of the partial
charge density are presented at the isosurface value of $8.0{\times}10^{-3}$
e/Bohr3 for all allotropes.

## Computational Techniques

We have used ab initio density functional theory (DFT), as implemented in the
VASP code [26, 27, 28], throughout the study. We applied periodic boundary
conditions, with 2D structures separated by a vacuum region in excess of $20$
Å. The reciprocal space was sampled by a fine grid [29] of
$8{\times}6{\times}1$ $k$-points in the Brillouin zone of 8-atom unit cells or
its equivalent in supercells. We used projector-augmented-wave (PAW)
pseudopotentials [30] and the Perdew-Burke-Ernzerhof (PBE) [31] exchange-
correlation functionals. Selective band structure calculations were performed
using the hybrid HSE06 functional [32, 33] with the default mixing parameter
${\alpha}=0.25$. We utilized the DFT-D2 [34] method to represent van der Waals
(vdW) corrections to the total energy. We used a cutoff energy of 500 eV for
the plane-wave basis set and considered the electronic structure to be
converged once the total energy difference between subsequent electronic
structure iterations would not exceed $10^{-5}$ eV. All geometries were
optimized using the conjugate-gradient method [35], until none of the residual
Hellmann-Feynman forces exceeded $10^{-2}$ eV/Å. The phonon calculations were
carried out using the density functional perturbation theory [36, 37, 38], as
implemented in the PHONOPY code [39]. The system dynamics was studied using
canonical ab initio MD simulations with $3$ fs time steps. Our results are for
supercells containing more than 100 atoms, which were kept at temperatures of
$300$ K, $500$ K or $1000$ K for periods exceeding $15$ ps.

## Results and Discussion

### Structure of BNP2 and C2SiS 2D allotropes

As introduced above, the 2D compounds BNP2 and C2SiS are isoelectronic to
previously reported PC monolayer structures [10]. We have identified four 2D
allotropes, called $\alpha_{1}$, $\beta_{1}$, $\alpha_{2}$ and $\beta_{2}$,
for each of the systems. The most stable structures of each allotrope are
shown in Fig. 1. All atoms are 3-fold coordinated, causing a coexistence of
$sp^{2}$ and $sp^{3}$ bonding and leading to structural ridges in the geometry
of all allotropes. When viewed from the side, the $\alpha$ allotropes have an
armchair profile, whereas the $\beta$ allotropes have a zigzag profile. With
the exception of $\beta_{1}$-C2SiS with 8 atoms in the unit cell, all
allotropes have rectangular unit cells containing 16 atoms. Additional
metastable structures with different atomic arrangements in the unit cell are
discussed in the Appendix.

As seen in Fig. 1, the optimized $\alpha_{1}$ and $\beta_{1}$ allotropes of
BNP2 structures consist of isolated P-P and B-N dimers forming a 2D hexagonal
structure. Since the orientation of the B-N dimers alternates in the plane of
the system, there is no net dipole moment in the system. P atoms prefer the
$sp^{3}$ configuration with a lone electron pair, whereas B and N atoms prefer
the $sp^{2}$ configuration in all BNP2 allotropes. The $\alpha_{1}$ and
$\beta_{1}$ allotropes represent two equivalent ways to achieve optimum
configuration with the same topology, in analogy to black and blue
phosphorene. [40]

The $\alpha_{2}$ and $\beta_{2}$ allotropes of BNP2 contain alternating chains
of P and BN along the $y$-direction. The mismatch between the equilibrium P-P
and B-N bond lengths leads to the formation of pentagon-heptagon pairs instead
of hexagons.

The structure of all 2D allotropes of C2SiS closely resembles that of BNP2 due
to the coexistence of $sp^{2}$-bonded C atoms and $sp^{3}$-bonded Si and S
atoms. Similar to BNP2, the $\alpha_{1}$ and $\beta_{1}$ allotropes of C2SiS
contain isolated Si-S and C-C dimers. Also in this case, there are two
distinguishable, topologically equivalent geometries. Even though the polar
Si-S bonds are out of plane in this compound, their alternating orientation
eliminates a net dipole moment. The $\alpha_{2}$ and $\beta_{2}$ allotropes of
C2SiS, analogous to those of BNP2, contain alternating C and SiS chains along
the $y$-direction. Bond length mismatch leads to a preferential formation of
pentagon-heptagon pairs in the layer.

A summary of the structural characteristics and the cohesive energy of all
these allotropes is presented in Table 1. In the BNP2 system, the B-N dimers
in $\alpha_{1}$ and $\beta_{1}$ allotropes are connected by typical double
bonds with $d_{BN}{\approx}1.40$ Å, close to the $1.403$ Å value in amino
borane [41]. The bond length of the B-N chains in $\alpha_{2}$ and $\beta_{2}$
allotropes is slightly longer, at about $1.46$ Å, close to the $1.45$ Å value
in 2D $h$-BN [42]. These results are consistent with the $sp^{2}$
configuration of B and N atoms in 2D BNP2.

Figure 4: Contour plots of the energy dispersion $E$(k) of the top valence
band (left panels) and the lowest conduction band (right panels) in (a)
$\alpha_{2}$-BNP2 and (b) $\beta_{2}$-BNP2. The Fermi level is set as the
reference energy of zero.  Figure 5: Electronic structure of a C2SiS
monolayer. The calculated band structure $E($k) and the density of states
(DOS) including its projection on individual atoms are shown in the left
panels. The partial charge density distributions at the valence band maximum
(VBM) and the conduction band minimum (CBM) for the semiconducting allotropes
and for the frontier states in the semimetallic allotropes are shown in the
right panels. Results are presented for the (a) $\alpha_{1}$, (b)
$\alpha_{2}$, (c) $\beta_{1}$, and (d) $\beta_{2}$ allotropes. All results are
based on the DFT-PBE functional with the exception of DFT-HSE06 data, which
are shown by the red dashed lines in the band structure plots. The blue arrows
indicate the fundamental band gap in the semiconducting allotropes.
Contributions of different atoms are distinguished by color in the PDOS plots.
Isosurface plots of the partial charge density are presented at the isosurface
value of $8.0{\times}10^{-3}$ e/Bohr3 for all allotropes.  Figure 6: Band
dispersion in the semimetallic allotropes $\alpha_{2}$-C2SiS (left) and
$\beta_{2}$-C2SiS (right) near the Fermi level. $E($k) results are presented
as 3D plots in (a),(b) and 2D contour plots in (c),(d). The energy with
respect to $E_{F}=0$ is represented by color.

We find similar structural characteristics in the C2SiS system. As expected
for $sp^{2}$-bonded C atoms, the C-C bond lengths are comparable to the double
bonds in ethylene and the conjugated bonds in benzene or graphene. The length
of all the other single bonds appears rather insensitive to the structure.

It is also noteworthy that P-N bonds are slightly shorter than P-B bonds in
BNP2 due to the smaller atomic radius of the more electronegative element. The
same is true, for the same reason, with the C-S bonds that are slightly
shorter than C-Si bonds in C2SiS allotropes.

For the sake of easy comparison, we define the cohesive energy $E_{coh}$ of an
“average” atom by dividing the total atomization energy by the total number of
atoms. Our results for $E_{coh}$ are listed in Table 1. We found that for both
BNP2 and C2SiS, $\alpha$ and $\beta$ allotropes with the same index are almost
equally stable. Our results also indicate that for both BNP2 and C2SiS,
$\alpha_{2}$ and $\beta_{2}$ allotropes are energetically more stable than
$\alpha_{1}$ and $\beta_{1}$ allotropes by ${\Delta}E_{coh}{\approx}0.2-0.3$
eV/atom. We find the C2SiS structures to be generally more stable than BNP2
structures. The listed values of $E_{coh}{\approx}5$ eV/atom here are
comparable to cohesive energies found in PC monolayers [10].

We have further confirmed the stability of each allotrope by calculating the
phonon spectra and by performing ab initio MD simulations. Details of all
these calculations are presented in the Appendix.

We also calculated the formation energy per ‘average atom’ $E_{form}$ to
investigate the relative stability of the ternary allotropes comparing with
respect to their elemental components. We defined $E_{form}$ as

$\displaystyle E_{form}(X_{l}Y_{m}Z_{n})$ $\displaystyle=$ $\displaystyle
E(X_{l}Y_{m}Z_{n})$ (1) $\displaystyle-$
$\displaystyle\frac{l{\cdot}E(X)+m{\cdot}E(Y)+n{\cdot}E(Z)}{l+m+n},$

where $E(X_{l}Y_{m}Z_{n})$, $E(X)$, $E(Y)$, $E(Z)$ are the respective total
energies per average atom of the XlY${}_{m}Z_{n}$ compound and the elemental
structures of X, Y and Z. In the particular case of 2D BNP2 and C2SiS
structures, the elemental structures we consider are bulk boron, gas of N2
molecules, phosphorene, graphene, bulk silicon and bulk sulfur. The results of
$E_{form}$ for all the allotropes are listed in Table 1. $E_{form}>0$
indicates instability and possibility of exothermic decomposition, whereas
$E_{form}<0$ indicates stability.

Our results for $E_{form}$ in Table 1 indicate that all BNP2 allotropes are
stable and will not decompose below ${\approx}1000$ K, at which point bonds
may be broken. The values of $E_{form}$ for C2SiS allotropes considered here
are positive, but rather small. The values $E_{form}=0.68$ eV/atom for the
$\alpha_{1}$ and $\beta_{1}$ phases and $E_{form}=0.39$ eV/atom for the
$\alpha_{2}$ and $\beta_{2}$ phases are an order of magnitude smaller than
cohesive energies and comparable to $E_{form}=0.54$ eV/atom found in 2D-PC
[10]. Structural changes would require overcoming a significant activation
barrier associated with breaking bonds and protect C2SiS allotropes from
structural changes at room temperature. As mentioned above, interatomic bonds
may break and structural changes may become possible at extremely high
temperatures of ${\approx}1000$ K. We find these estimates confirmed by our MD
simulation results presented in the Appendix.

### Anisotropic in-plane stiffness of 2D BNP2 and C2SiS allotropes

To investigate the elastic response of the 2D ternary structures, we subjected
all allotropes in this study to in-layer uniaxial strain along the $x$\- and
$y$-direction and display differences ${\Delta}E_{coh}$ of the average binding
energy with respect to the most stable allotrope in Fig. 2. The presence of
structural ridges renders all allotropes much softer along the $x$-direction
normal to the ridge direction. The $\alpha$ allotropes of both BNP2 and C2SiS
are particularly soft along the $x$-direction, with ${\Delta}E{\lesssim}$18
meV/atom when subjected to ${\leq}10$% compressive or tensile strain. The 2D
elastic moduli [43] $c_{ii}$ along the $x$\- and $y$-direction, obtained using
these calculations, are summarized in Table 1. For the semiconducting
allotropes, these values play an important role in carrier mobilities.

### Electronic structure of BNP2 2D allotropes

Results of our DFT calculations for the electronic structure of BNP2
monolayers are presented in Fig. 3. Our DFT-PBE results show all the four
allotropes to be semiconductors. The fundamental band gap is direct in
$\alpha_{1}$-BNP2 and indirect in all the other BNP2 allotropes. Band gap
values $E_{g}$ based on DFT-PBE Kohn-Sham energies, ranging from $0.52$ eV to
$1.39$ eV, are listed in Table 1. We should note here that the interpretation
of Kohn-Sham eigenvalues as self-energies is strictly incorrect and that DFT-
based band gaps are typically underestimated. We find this to be the case when
comparing band gap values based on DFT-PBE and the hybrid DFT-HSE06 functional
in Table 1. As seen when comparing band structure results based on DFT-PBE,
given by the solid lines in Fig. 3, and DFT-HSE06, shown by the red dashed
lines, the main difference is the opening of the fundamental band gap in DFT-
HSE06, whereas the band dispersion is unaffected. We also present the total
electronic density of states (DOS) and its projection onto the different
species next to the band structure plots.

Contour plots of the projected DOS (PDOS) of all BNP2 allotropes at the
valence band maximum (VBM) and conduction band minimum (CBM) are shown in Fig.
3, with the contributions to VBM and CBM distinguished by color. We find the
character of the VBM to be dominated by the lone-pair states of the $sp^{3}$
hybridized P atoms in all BNP2 allotropes. The character of the CBM is very
different. The dominant contribution in $\alpha_{1}$ and $\beta_{1}$
allotropes of BNP2 comes from out-of-plane $p$ orbitals of $sp^{2}$ hybridized
B and N atoms. In the $\alpha_{2}$ and $\beta_{2}$ allotropes, $p$ orbitals of
all the atoms contribute to the CBM character.

We also observe an unusually flat band in the band structure of $\alpha_{2}$
and $\beta_{2}$ allotropes of BNP2 at the top of the valence region, along the
${\Gamma}-X$ line. To further explore the origin of this flat band, we show 2D
contour plots of the dispersion within the top valence and bottom conduction
bands in Fig. 4.

In both allotropes, the CBM energy minimum occurs at the $\Gamma$ point and
the band dispersion around is almost isotropic and free-electron like. In
contrast, the VBM is displaced from the $\Gamma$ point in the direction
towards $X$ for both allotropes. Whereas the band dispersion is almost flat
along the $x$-direction, it is significantly higher along $y$-direction near
the VBM, indicating a significant anisotropy in the carrier effective mass. As
seen in Fig. 3(b) and (d), this is caused at the VBM by a larger separation
and lower hybridization between the P lone pair states along the $x$-direction
than along the $y$-direction.

### Electronic structure of C2SiS 2D allotropes

Results of our DFT calculations for the electronic structure of C2SiS
monolayers are presented in Fig. 5. Our findings indicate that only the
$\alpha_{1}$ and the $\beta_{1}$ allotropes are semiconducting, whereas the
$\alpha_{2}$ and $\beta_{2}$ allotropes are semimetallic. The band gap is
indirect in $\alpha_{1}$-C2SiS and direct in $\beta_{1}$-C2SiS. Numerical band
gap values based on DFT-PBE and DFT-HSE06 are listed in Table 1 and fall in
the range of values found in BNP2.

According to Fig. 5, the character of VBM and CBM states in the semiconducting
allotropes $\alpha_{1}$ and $\beta_{1}$ is very similar. In analogy to results
for BNP2, the VBM of C2SiS is dominated by lone-pair orbitals of Si atoms and
the CBM contains mostly out-of-local-plane $p$ orbitals of C atoms.

The $\alpha_{2}$ and $\beta_{2}$ allotropes of C2SiS are found to be
semimetallic by both DFT-PBE and DFT-HSE06. Dirac “cones” are seen in the band
structure in Fig. 5(b) and (d), with the Dirac point at $E_{F}$ located
between $\Gamma$ and $Y$ in the Brillouin zone. The frontier states at $E_{F}$
are dominated by out-of-plane $p$ orbitals on C atoms.

The Dirac “cones” of $\alpha_{2}$\- and $\beta_{2}$-C2SiS are visualized in
more detail in Fig. 6. The band dispersion in both allotropes is linear around
$E_{F}$, but anisotropic. As seen in Fig. 6(c) and (d), the cross-section of
the “cone” at $E<E_{F}$ or $E>E_{F}$ is not a circle, but rather an ellipse
elongated in the $k_{x}$-direction. The higher steepness of the band
dispersion along the $y$-direction near the Dirac point is a consequence of a
stronger interaction between C$2p$ orbitals within zigzag chains of carbons,
aligned with the $y$-direction. These C chains in $\alpha_{2}$-C2SiS and
$\beta_{2}$-C2SiS are clearly visible in Figs. 1(g) and (h). Otherwise, these
“cones” are very similar in Dirac point location, band dispersion and its
anisotropy in both allotropes.

Table 2: Calculated carrier mobilities and related quantities in
semiconducting BNP2 and C2SiS allotropes.

|  |  $m_{x}^{*}$a |  $m_{y}^{*}$a |  $E_{1x}$b |  $E_{1y}$b |  $\mu_{x}$c |  $\mu_{y}$c
---|---|---|---|---|---|---|---
|  | ($m_{0}$) | (eV) | (103 cm2V-1s-1)
$\alpha_{1}$-BNP2 | $e$ | 0.51 | 0.34 | 1.36 | 0.40 | 0.40 | 151.21
$h$ | 0.75 | 0.39 | 1.83 | 1.60 | 0.11 | 6.16
$\beta_{1}$-BNP2 | $e$ | 0.67 | 0.34 | 0.69 | 1.37 | 5.89 | 10.51
$h$ | 0.67 | 0.57 | 0.37 | 1.26 | 15.82 | 5.62
$\alpha_{2}$-BNP2 | $e$ | 0.88 | 0.78 | 1.01 | 0.88 | 0.30 | 8.29
$h$ | 3.13 | 0.50 | 0.10 | 5.07 | 5.67 | 0.26
$\beta_{2}$-BNP2 | $e$ | 0.53 | 0.75 | 2.41 | 0.68 | 0.65 | 17.17
$h$ | 1.63 | 0.52 | 0.90 | 5.96 | 1.03 | 0.22
$\alpha_{1}$-C2SiS | $e$ | 0.53 | 0.28 | 0.39 | 1.37 | 4.38 | 15.66
$h$ | 1.72 | 0.76 | 2.54 | 0.87 | 0.01 | 4.85
$\beta_{1}$-C2SiS | $e$ | 0.14 | 1.57 | 2.51 | 6.31 | 1.69 | 0.09
$h$ | 0.78 | 2.24 | 1.87 | 1.42 | 0.20 | 0.45

  * 1

Electrons are denoted by $e$ and holes by $h$.
a $m_{x}^{*}$ ($m_{y}^{*}$) are the carrier effective masses along the $x$
($y$) direction.
b $E_{1x}$ ($E_{1y}$) are the deformation potentials along the $x$ ($y$)
direction.
c $\mu_{x}$ ($\mu_{y}$) are the carrier mobilities along the $x$ ($y$)
direction at $300$ K.

### Carrier mobilities in semiconducting BNP2 and C2SiS 2D allotropes

Anisotropy in semiconducting BNP2 and C2SiS 2D allotropes is not only limited
to the geometry, the elastic response and the electronic structure, but is
also present in the carrier mobility. In absence of defects and external
scattering centers, the mobility of carriers in 2D semiconductors is limited
by acoustic phonons. We calculated the carrier mobilities of BNP2 and C2SiS 2D
systems along the $x$\- and $y$-direction using the deformation potential
theory expression [44, 45, 11]

${\mu}_{i}=\frac{e{\hbar}^{3}c_{ii}}{k_{B}Tm_{i}^{*}m_{d}{E_{1i}}^{2}}\;.$ (2)

Here, $i$ represents the Cartesian direction, with $i=1$ standing for $x$ and
$i=2$ for $y$, and $e$ is the carrier charge. $c_{ii}$ is the 2D elastic
modulus [43] along the direction $i$, obtained from the strain energy curve in
Fig. 2. $T$ is the temperature, $m_{i}^{*}$ is the effective mass along the
$i$-direction, and $m_{d}$ is the average effective mass given by
$m_{d}=({m_{x}^{*}}{m_{y}^{*}})^{1/2}$. The deformation potential $E_{1i}$
along the $i$-direction is determined at the valence band maximum (VBM) for
holes and the conduction band minimum (CBM) for electrons. It is defined by
$E_{1i}={\Delta}V/({\Delta}a_{i}/a_{i})$, where ${\Delta}V$ is the energy
shift of the band edge with respect to the vacuum level under a small change
${\Delta}a_{i}$ of the lattice constant $a_{i}$. Our results for room-
temperature carrier mobilities in all semiconducting 2D allotropes are
summarized in Table 2.

We found that most allotropes in our study exhibit very high and remarkably
anisotropic carrier mobilities. The highest carrier mobility value we found is
${\mu_{y}}=1.51{\times}10^{5}$ cm2V-1s-1 for electrons in $\alpha_{1}$-BNP2
along the $y$-direction, over two orders of magnitude higher than the value
${\mu_{x}}=0.4{\times}10^{3}$ cm2V-1s-1 along the $x$-direction. An important
contributing factor to the mobility is the 2D elastic modulus [43]. The
superior electron mobility along its $y$-direction is a result of increased
stiffness along this direction. According to Table 1, $c_{22}$ is indeed much
larger than $c_{11}$ for most allotropes we study.

Unlike for electrons, hole mobility is highest along the $x$-direction in
$\beta_{1}$, $\alpha_{2}$ and $\beta_{2}$ allotropes of BNP2. The change of
preferential transport direction is caused by substantial deformation
potential anisotropy $E_{1x}<E_{1y}$ at the VBM according to Table 2. Carrier
mobilities in semiconducting 2D allotropes of C2SiS are comparable to those in
BNP2.

Carrier mobilities ${\mu}>10^{5}$ cm2V-1s-1 in $\alpha_{1}$-BNP2 with
$E_{g}>0.5$ eV and ${\mu}{\gtrsim}10^{4}$ cm2V-1s-1 in $\alpha_{2}$ and
$\beta_{2}$ allotropes of BNP2 with $E_{g}>1$ eV, evaluated at the PBE level
of DFT are unusually high in view of the wide band gaps. This is particularly
evident when compared to the values found in other similar 2D materials, as
discussed in the Appendix. Even though these mobilities will be reduced due to
inevitable defects and the interaction with the substrate under realistic
conditions, we expect values in excess of ${\mu}=0.3{\times}10^{3}$ cm2V-1s-1,
which has been reported for phosphorene [6]. These results are very promising
for the realization of 2D semiconducting devices with high ON/OFF ratios and
on-state currents.

## Summary and Conclusions

In summary, we have introduced previously unexplored 2D ternary compounds BNP2
and C2SiS as isoelectronic counterparts of 2D PC structures. Using ab initio
density functional theory, we have identified four stable allotropes of each
compound and confirmed their stability by calculated phonon spectra and
molecular dynamics simulations. All allotropes display structural and elastic
anisotropy due to structural ridges in their geometry, which are caused by
coexisting $sp^{2}$ and $sp^{3}$ hybridization. Whereas all BNP2 allotropes
are semiconducting, we find only two allotropes of C2SiS to be semiconducting.
The other two allotropes of C2SiS are semimetallic and show anisotropic Dirac
cones at $E_{F}$. The fundamental band gaps of the semiconducting allotropes
we study range from $0.5$ eV to $1.4$ eV at the PBE level and $1.4$ eV to
$2.2$ eV at the HSE06 level of DFT. These 2D systems display carrier
mobilities as high as $1.5{\times}10^{5}$ cm2V-1s-1. Such high mobilities,
with two orders of magnitude in anisotropy ratio, are desirable, but quite
uncommon in semiconductors with so wide band gaps. Combination of wide band
gaps with high and anisotropic carrier mobilities offer great promise for
applications of 2D BNP2 and C2SiS structures in electronics and
optoelectronics.

###### Acknowledgements.

This study was supported by the National Natural Science Foundation of China
(NSFC) under Grant No. 61704110, the Fundamental Research Fund for the Central
Universities, the Shuangchuang Doctoral Program of the Jiangsu Province, and
by the Zhongying Young Scholar Program of Southeast University. D.T.
acknowledges financial support by the NSF/AFOSR EFRI 2-DARE grant number
EFMA-1433459. We thank the Big Data Computing Center of Southeast University
for providing facility support for performing calculations presented in this
manuscript.

## Appendix

### Metastable structures of 2D BNP2 and C2SiS

Figure A1: Ball-and-stick model of metastable structures related to the
$\alpha_{1}$ and $\beta_{1}$ allotropes of BNP2, shown in top view. Cohesive
energies $E_{coh}$ are presented below each structure, with the value for the
most stable configuration highlighted in red.  Figure A2: Ball-and-stick model
of metastable structures related to the $\alpha_{1}$ and $\beta_{1}$
allotropes of C2SiS, shown in top view. Cohesive energies $E_{coh}$ are
presented below each structure, with the value for the most stable
configuration highlighted in red.  Figure A3: Ball-and-stick model of
metastable structures related to the $\alpha_{2}$ and $\beta_{2}$ allotropes
of BNP2, shown in top view. Cohesive energies $E_{coh}$ are presented below
each structure, with the value for the most stable configuration highlighted
in red.  Figure A4: Phonon spectra for monolayers of (a) $\alpha_{1}$-BNP2,
(b) $\beta_{1}$-BNP2, (c) $\alpha_{2}$-BNP2, (d) $\beta_{2}$-BNP2, (e)
$\alpha_{1}$-C2SiS, (f) $\beta_{1}$-C2SiS, (g) $\alpha_{2}$-C2SiS, and (h)
$\beta_{2}$-C2SiS.  Figure A5: Fluctuations of the total potential energy
(left panels) and structural snap shots after $15$ ps (right panels) for (a)
$\alpha_{1}$-BNP2, (b) $\alpha_{2}$-BNP2, (c) $\beta_{1}$-BNP2, and (d)
$\beta_{2}$-BNP2 monolayers during canonical MD simulations at $300$ K, $500$
K and $1000$ K.  Figure A6: Fluctuations of the total potential energy (left
panels) and structural snap shots after $15$ ps (right panels) for (a)
$\alpha_{1}$-C2SiS, (b) $\alpha_{2}$-C2SiS, (c) $\beta_{1}$-C2SiS, and (d)
$\beta_{2}$-C2SiS monolayers during canonical MD simulations at $300$ K, $500$
K and $1000$ K.  Figure A7:  Carrier mobility versus band gap for different 2D
materials. Results for semiconducting allotropes predicted in this study,
shown by the red symbols, are compared to previously reported values for
selected 2D materials, shown by the black symbols. All results have been
obtained at the PBE level of DFT.

As seen in Fig. 1 of the main text, $\alpha_{1}$ and $\beta_{1}$ allotropes of
BNP2 contain four polar B-N dimers aligned with the $x$-axis in each unit
cell. Each of these dimer bonds may be rotated by 180∘. This gives rise to
four metastable structures for $\alpha_{1}$-BNP2 and another four structures
for $\beta_{1}$-BNP2, shown in Fig. A1. The most stable structures among
these, with the cohesive energy value highlighted in red, have been considered
in the main text.

Quite different is the arrangement of B and N atoms in zigzag chains in the
$\alpha_{2}$ and $\beta_{2}$ allotropes of BNP2 shown in Fig. 1 of the main
text. Also in this case, each bond may be rotated by 180∘. This gives rise to
two metastable structures for $\alpha_{2}$-BNP2 and another two structures for
$\beta_{2}$-BNP2, shown in Fig. A2. Only the most stable among these
allotropes have been considered in the main text.

Next we turn to the $\alpha_{1}$ and $\beta_{1}$ allotropes of C2SiS,
displayed in Fig. 1(e) and (f) of the main text. Unlike in BNP2, we can find
only three inequivalent metastable structures for these systems, displayed in
Fig. A3. For the $\alpha_{2}$ and $\beta_{2}$ allotropes of C2SiS,
optimization leads to only one stable structure. As for the other systems
discussed above, we only consider only the most stable allotropes in the main
text.

### Phonon spectra of 2D BNP2 and C2SiS allotropes

A real confirmation of structural stability, more important than a high
cohesive energy, comes from the phonon spectra. Structures can be considered
stable if no imaginary frequencies can be found in the phonon spectra. The
calculated phonon spectra of all BNP2 and C2SiS allotropes discussed in this
study are displayed in Fig. A4.

We should note at this point that phonon calculations for 2D structures with
the uniquely soft flexural ZA mode are very demanding on precision [46]. We
found our phonon spectra to be essentially free of imaginary frequencies
associated with decay modes. The highest imaginary frequency value
${\omega}{\lesssim}12i$ cm-1 in our spectra belongs to the ZA mode near the
$\Gamma$ point in the Brillouin zone and is an artifact related to numerical
precision [46].

### Thermodynamic stability of 2D BNP2 and C2SiS allotropes

Whereas phonon spectra tell about structural stability in the harmonic regime,
they cannot determine if a structure will or will not fall apart at a given
finite temperature. To study the thermodynamic stability of the ternary
structures in this study, we performed a set of canonical ab initio molecular
dynamics (MD) simulations at $300$ K, $500$ K and $1000$ K and present our
results in Fig. A5 for 2D BNP2 and Fig. A6 for C2SiS structures. For $15$ ps
long MD runs, we plotted both the fluctuations of the total potential energy
and snapshots of the structures after $15$ ps.

Our results in Fig. A5 indicate that all four allotropes of BNP2 maintained
their geometries up $1000$ K, indicating a high thermodynamic stability. The
corresponding results for C2SiS in Fig. A6 indicate all allotropes to be
stable at $300$ K and $500$ K. Further increase of temperature to $1000$ K
causes a dramatic degradation of the $\alpha_{1}$ and $\beta_{1}$ allotropes
of C2SiS after $3$ ps. The $\alpha_{2}$ and $\beta_{2}$ allotropes appear more
stable at $1000$ K, but their significant distortion indicates onset of
degradation. The thermodynamic stability of all the BNP2 and C2SiS allotropes
is either comparable to or superior to other similar 2D structures including
GeP3 [16], which was shown to be stable at $T{\lesssim}500$ K, as well as PC
[11] and binary V-V compounds [47], which were shown to be stable only up to
${\approx}300$ K. Our MD results show good consistency with the conclusions
obtained from the formation energy calculations discussed in the main text.

###  Suitability of BNP2 and C2SiS allotropes for electronic applications

Combination of high carrier mobility and wide band gap is very desirable, as
it results in a high ON/OFF ratio and a high on-state current in semiconductor
devices. The 2D materials we introduce appear particularly suitable for
electronic applications from that viewpoint. This is evidenced in Fig. A7,
where we compare mobilities and band gaps of BNP2 and C2SiS to those of 2D-PC,
GeP3 [16], SnP3 [17], black phosphorene, and 2D GeSe [48]. There is a general
trade-off relation between carrier mobility and band gap. We find most of the
BNP2 and C2SiS allotropes to be near the upper limit of the trade-off.

## References

  * Novoselov  _et al._ [2004] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric field effect in atomically thin carbon films, Science 306, 666 (2004).
  * Novoselov  _et al._ [2005] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature 438, 197 (2005).
  * Zhang  _et al._ [2005] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Experimental observation of the quantum Hall effect and Berry’s phase in graphene, Nature 438, 201 (2005).
  * Radisavljevic  _et al._ [2011] B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, and A. Kis, Single-layer MoS2 transistors, Nature Nanotech. 6, 147 (2011).
  * Fuhrer and Hone [2013] M. S. Fuhrer and J. Hone, Measurement of mobility in dual-gated MoS2 transistors, Nature Nano 8, 146 (2013).
  * Liu  _et al._ [2014] H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, Phosphorene: An unexplored 2D semiconductor with a high hole mobility, ACS Nano 8, 4033 (2014).
  * Li  _et al._ [2014] L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, Black phosphorus field-effect transistors, Nature Nanotech. 9, 372 (2014).
  * Wang  _et al._ [2015] L. Wang, A. Kutana, X. Zou, and B. I. Yakobson, Electro-mechanical anisotropy of phosphorene, Nanoscale 7, 9746 (2015).
  * Wood  _et al._ [2014] J. D. Wood, S. A. Wells, D. Jariwala, K.-S. Chen, E. Cho, V. K. Sangwan, X. Liu, L. J. Lauhon, T. J. Marks, and M. C. Hersam, Effective passivation of exfoliated black phosphorus transistors against ambient degradation, Nano Lett. 14, 6964 (2014).
  * Guan  _et al._ [2016] J. Guan, D. Liu, Z. Zhu, and D. Tománek, Two-dimensional phosphorus carbide: Competition between $sp^{2}$ and $sp^{3}$ bonding, Nano Lett. 16, 3247 (2016).
  * Wang  _et al._ [2016] G. Wang, R. Pandey, and S. P. Karna, Carbon phosphide monolayers with superior carrier mobility, Nanoscale 8, 8819 (2016).
  * Tan  _et al._ [2017] W. C. Tan, Y. Cai, R. J. Ng, L. Huang, X. Feng, G. Zhang, Y.-W. Zhang, C. A. Nijhuis, X. Liu, and K.-W. Ang, Few-layer black phosphorus carbide field-effect transistor via carbon doping, Adv. Mater. 29, 1700503 (2017).
  * Tan  _et al._ [2018] W. C. Tan, L. Huang, R. J. Ng, L. Wang, D. M. N. Hasan, T. J. Duffin, K. S. Kumar, C. A. Nijhuis, C. Lee, and K.-W. Ang, A black phosphorus carbide infrared phototransistor, Adv. Mater. 30, 1705039 (2018).
  * Huang  _et al._ [2018] X. Huang, Y. Cai, X. Feng, W. C. Tan, D. M. N. Hasan, L. Chen, N. Chen, L. Wang, L. Huang, T. J. Duffin, C. A. Nijhuis, Y.-W. Zhang, C. Lee, and K.-W. Ang, Black phosphorus carbide as a tunable anisotropic plasmonic metasurface, ACS Photon. 5, 3116 (2018).
  * Li  _et al._ [2018] L. Li, W. Wang, P. Gong, X. Zhu, B. Deng, X. Shi, G. Gao, H. Li, and T. Zhai, 2D GeP: An unexploited low-symmetry semiconductor with strong in-plane anisotropy, Adv. Mater. 30, 1706771 (2018).
  * Jing  _et al._ [2017] Y. Jing, Y. Ma, Y. Li, and T. Heine, GeP3: A small indirect band gap 2D crystal with high carrier mobility and strong interlayer quantum confinement, Nano Lett. 17, 1833 (2017).
  * Ghosh  _et al._ [2018] B. Ghosh, S. Puri, A. Agarwal, and S. Bhowmick, SnP3: A previously unexplored two-dimensional material, J. Phys. Chem. C 122, 18185 (2018).
  * Yu  _et al._ [2019] T. Yu, Z. Zhao, Y. Sun, A. Bergara, J. Lin, S. Zhang, H. Xu, L. Zhang, G. Yang, and Y. Liu, Two-dimensional PC6 with direct band gap and anisotropic carrier mobility, J. Am. Chem. Soc. 141, 1599 (2019).
  * Yang  _et al._ [2012] J.-H. Yang, Y. Zhai, H. Liu, H. Xiang, X. Gong, and S.-H. Wei, Si3AlP: A new promising material for solar cell absorber, J. Am. Chem. Soc. 134, 12653 (2012).
  * Zhu  _et al._ [2015] Z. Zhu, J. Guan, D. Liu, and D. Tománek, Designing isoelectronic counterparts to layered group V semiconductors, ACS Nano 9, 8284 (2015).
  * Yang  _et al._ [2016] J.-H. Yang, Y. Zhang, W.-J. Yin, X. G. Gong, B. I. Yakobson, and S.-H. Wei, Two-dimensional SiS layers with promising electronic and optoelectronic properties: Theoretical prediction, Nano Lett. 16, 1110 (2016).
  * Gomes and Carvalho [2015] L. C. Gomes and A. Carvalho, Phosphorene analogues: Isoelectronic two-dimensional group-IV monochalcogenides with orthorhombic structure, Phys. Rev. B 92, 085406 (2015).
  * Gomes  _et al._ [2015] L. C. Gomes, A. Carvalho, and A. H. Castro Neto, Enhanced piezoelectricity and modified dielectric screening of two-dimensional group-IV monochalcogenides, Phys. Rev. B 92, 214103 (2015).
  * Fei  _et al._ [2016] R. Fei, W. Kang, and L. Yang, Ferroelectricity and phase transitions in monolayer group-IV monochalcogenides, Phys. Rev. Lett. 117, 097601 (2016).
  * Haleoot  _et al._ [2017] R. Haleoot, C. Paillard, T. P. Kaloni, M. Mehboudi, B. Xu, L. Bellaiche, and S. Barraza-Lopez, Photostrictive two-dimensional materials in the monochalcogenide family, Phys. Rev. Lett. 118, 227401 (2017).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  * Kresse and Hafner [1993] G. Kresse and J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev. B 47, 558 (1993).
  * Kresse and Hafner [1994] G. Kresse and J. Hafner, Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium, Phys. Rev. B 49, 14251 (1994).
  * Monkhorst and Pack [1976] H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976).
  * Kresse and Joubert [1999] G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Heyd  _et al._ [2003] J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid functionals based on a screened Coulomb potential, J. Chem. Phys. 118, 8207 (2003).
  * Heyd  _et al._ [2006] J. Heyd, G. E. Scuseria, and M. Ernzerhof, Erratum: “Hybrid functionals based on a screened Coulomb potential” [J. Chem. Phys. 118, 8207 (2003)], J. Chem. Phys. 124, 219906 (2006).
  * Grimme [2006] S. Grimme, Semiempirical GGA-type density functional constructed with a long-range dispersion correction, J. Comp. Chem. 27, 1787 (2006).
  * Hestenes and Stiefel [1952] M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Natl. Bur. Stand. 49, 409 (1952).
  * Baroni  _et al._ [1987] S. Baroni, P. Giannozzi, and A. Testa, Green’s-function approach to linear response in solids, Phys. Rev. Lett. 58, 1861 (1987).
  * Gonze [1995a] X. Gonze, Adiabatic density-functional perturbation theory, Phys. Rev. A 52, 1096 (1995a).
  * Gonze [1995b] X. Gonze, Perturbation expansion of variational principles at arbitrary order, Phys. Rev. A 52, 1086 (1995b).
  * Togo and Tanaka [2015] A. Togo and I. Tanaka, First principles phonon calculations in materials science, Scr. Mater. 108, 1 (2015).
  * Zhu and Tománek [2014] Z. Zhu and D. Tománek, Semiconducting layered blue phosphorus: A computational study, Phys. Rev. Lett. 112, 176802 (2014).
  * Sugie  _et al._ [1979] M. Sugie, H. Takeo, and C. Matsumura, Microwave spectrum of aminoborane, BH2NH2, Chem. Phys. Lett. 64, 573 (1979).
  * Şahin  _et al._ [2009] H. Şahin, S. Cahangirov, M. Topsakal, E. Bekaroglu, E. Akturk, R. T. Senger, and S. Ciraci, Monolayer honeycomb structures of group-IV elements and III-V binary compounds: First-principles calculations, Phys. Rev. B 80, 155453 (2009).
  * Liu  _et al._ [2016] D. Liu, A. G. Every, and D. Tománek, Continuum approach for long-wavelength acoustic phonons in quasi-two-dimensional structures, Phys. Rev. B 94, 165432 (2016).
  * Bardeen and Shockley [1950] J. Bardeen and W. Shockley, Deformation potentials and mobilities in non-polar crystals, Phys. Rev. 80, 72 (1950).
  * Qiao  _et al._ [2014] J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus, Nature Commun. 5, 4475 (2014).
  * Zólyomi  _et al._ [2014] V. Zólyomi, N. D. Drummond, and V. I. Fal’ko, Electrons and phonons in single layers of hexagonal indium chalcogenides from ab initio calculations, Phys. Rev. B 89, 205416 (2014).
  * Yu  _et al._ [2016] W. Yu, C.-Y. Niu, Z. Zhu, X. Wang, and W.-B. Zhang, Atomically thin binary V-V compound semiconductor: A first-principles study, J. Mater. Chem. C 4, 6581 (2016).
  * Xu  _et al._ [2017] Y. Xu, H. Zhang, H. Shao, G. Ni, J. Li, H. Lu, R. Zhang, B. Peng, Y. Zhu, H. Zhu, and C. M. Soukoulis, First-principles study on the electronic, optical, and transport properties of monolayer $\alpha$\- and $\beta$-GeSe, Phys. Rev. B 96, 245421 (2017).