1. Introduction

Two-dimensional (2D) Janus materials are a kind of special 2D materials with distinct components and properties on two sides [1–4]. The structural asymmetry in the out-of-plane direction leads to built-in electric fields across the monolayer and an intrinsic electric dipole perpendicular to the material plane, which makes Janus materials a promising platform for both electrical and optical applications [3]. For example, researchers have investigated excitonic dynamics in Janus MoSSe and WSSe monolayers/stacks [5–8]. They have found that the excitons in Janus structures form faster than those in pristine transition metal dichalcogenides and have significantly longer radiative recombination lifetimes. All these features are due to such built-in electric dipole, which can lead to the enhancement of the electron-phonon interactions and the separation of the electron and hole wavefunctions inside the material [8]. When touching with other materials, 2D Janus materials can even lead to doping levels about hundreds of meV just by contact, without applying, for example, any back-gate voltages [9], making 2D Janus materials convenient in doping engineering. Due to all these advantages, there has been increasing interest in the study of Janus 2D materials, especially their optical properties.

Hyperbolic polaritons have attracted much attention in recent years. In contrast to elliptic polaritons, hyperbolic polaritons possess extremely large momentum showing excellent optical field confinements [10–13]. The iso-frequency surface of hyperbolic polaritons is a hyperboloid [10]. In three dimensions, depending on the number of principle axes (assuming that x, y and z are three principal axes) with negative permittivities, there are two kinds of hyperbolic plasmon responses: one is called type-I, where the permittivity is negative along one of the three axes; the other is called type-II, where the permittivities are negative along two of the three axes [10]. As for hyperbolic polaritons in two dimensions, there is only one kind of hyperbolic plasmon responses and the iso-frequency contour is a hyperbola [14]. Usually, researchers carefully design and fabricate optical systems with hyperbolic polaritons based on the concept of metamaterials [15]. The electromagnetic waves propagating inside this kind of materials have extraordinary properties, leading to applications such hyperlens, negative refraction and emission enhancement [16–21].

Two-dimensional materials with hyperbolic plasmonic responses are rare in nature, which requires that the total permittivities are negative along one direction, but positive along the other. Recently, few-layer WTe₂ has been discovered to have hyperbolic polaritons in the infrared [22]; the occurrence of hyperbolic polaritons can be ascribed to the strongly anisotropic interband transition of electrons. However, Janus black arsenic phosphorus (bAsP) has not been successfully synthesized into large-scale monolayers yet. In despite of the large electrostatic energy, one example of Janus 2D materials that has been successfully synthesized by using chemical vapor deposition is MoSSe [1,23], and we believe that similar procedures can be applied to synthesize bAsP as well. Here we propose Janus bAsP, a 2D material which has similar structure to black phosphorus (bP) but with one side of phosphorus atoms completely replaced by arsenic atoms [24,25], as a new platform for plasmonic applications. Such atomic replacement preserves the anisotropic properties of this monolayer and creates an intrinsic dipole perpendicular to the plane, leading to richer optical phenomena in the material. bAsP should be much more chemically stable compared with black phosphorus [8], and has electron band structures with a slightly smaller bandgap located at the Γ point. It shows different curvatures at the bottom/top of the conduction/valence band along zigzag and armchair directions, which is of course originated from its puckered structure [25]. Based on these observations, bAsP should have polarization-dependent in-plane optical responses in the infrared [26–30]; similar plasmonic devices proposed in recent years based on black phosphorus nano- and micro-structures, such as waveguides, absorbers and so on [31–46], can also use bAsP as the building material [8]. Compared with bP, the intrinsic dipole in bAsP can change the Fermi energy significantly through the electrostatic fields when other two-dimensional materials are nearby, and also the strain required to trigger the band inversion as well as the direct-to-indirect bandgap transition is much smaller than that in bP. Thus, using Janus bAsP especially their functionalities in doping may bring new opportunities in plasmonic applications. To thoroughly explore the plasmonic responses of bAsP and the heterostructures with other 2D materials, simple approach such as effective mass approximation is not applicable since it is only suitable for lightly doped semiconductors under excitation near the band edge; with high doping levels which lead to metallic bAsP, the Fermi energy may go deep into the conduction/valence bands, causing failure of such simple approximation. To avoid this issue, a good option is to apply density functional theory (DFT).

In this paper, we have systematically investigated the plasmonic responses in bAsP based on density functional theory. In Section 2, we give the details about our DFT calculations. In Section 3, plasmonic responses in free-standing monolayer bAsP are investigated. The permittivities include both the intraband and interband contributions have been calculated. Due to the electron-hole asymmetry which means the difference in a material’s electronic properties upon doping with electrons versus holes, only electron doping can lead to exotic plasmonic phenomena especially when strains are applied; thus only electron doping has been investigated in this paper. With large enough doping concentrations, bAsP is no longer a semiconductor; the Fermi level shifts upwards and meets the conduction bands, leading to non-zero Drude plasma frequencies along zigzag and armchair directions. Independent particle approximation (IPA) is used throughout this paper. For metallic materials, the electron-hole interaction would be screened by the conduction electrons, which is exactly the case for metallic bAsP; thus, IPA should be accurate enough. We have further derived the permittivities based on the optimal basis functions. The contribution from the nonlocal parts of the pseudopotentials has also been included which may give us more accurate results. The interband and intraband parts of the permittivities have been separately calculated and the Drude plasma frequencies have been extracted at various Fermi levels, giving us details about the plasmonic responses in bAsP under doping. Then, the permittivities are converted into 2D conductivities using a simple model, based on which the plasmonic dispersion relations can be derived analytically. At specific frequency, elliptic-to-hyperbolic transition of the k-surfaces has been observed, offering us a way to control the topology of the plasmonic responses in bAsP. At low frequencies, the interband transition part of the permittivity is weak, giving us a simple but effective model for applications. In Section 4, strains have been applied along either the zigzag or the armchair direction. It is found that strains along the zigzag direction can change the bands drastically. With more than 3% strain along this direction, bAsP becomes an indirect-bandgap material. The corresponding projected band structures are used to give explanations about the changes of the band dispersions against strains. Since plasmonic responses are only related to the bands near the Fermi level and the Drude plasma frequency is totally determined by the intraband transition of electrons, indirect bandgap would not be a problem. The Drude plasma frequencies along two directions can be significantly modified by strains. Under specific condition, the k-surfaces in the hyperbolic regime can be reversed, where the permittivity along the zigzag direction is negative. In Section 5, a bAsP-graphene heterostructure has been investigated. As bAsP is a Janus material, graphene has been slightly doped due to the electrostatic fields and the Fermi level is shifted away from the Dirac point making such heterostructure support strong plasmons without extra doping. The anisotropy of the plasmonic responses as well as the Drude plasma frequencies of the heterostructure can easily be modified by changing the Fermi energy. In Section 6, a conclusion has been given. Our investigations suggest that bAsP is a promising Janus material platform for plasmonic applications.

2. Methods

In this section, we give details about our DFT calculations. Quantum espresso (QE) software package was used to perform the calculations with optimized norm-conserving pseudopotentials and Perdew-Burke-Ernzerhof (PBE) exchange-correlation functionals [47–49]. In the calculations of the band structures, the cut-off energies of the wavefunctions and densities were respectively 120 Ry and 480 Ry, and the k-mesh was 21 × 15 × 1. The van del Waals correction was DFT-D2. Of course, dipole correction was applied to deal with the out-of-plane structure asymmetry. In the calculations of the permittivities under electron doping, the cut-off energies of the wavefunctions and densities were increased to 150 Ry and 600 Ry, and the k-mesh was 38 × 28 × 1. The calculations based on optimal basis function approach were performed using the QE package [50]. 20 valence bands and 30 conduction bands were involved in the construction of the functions and the threshold used to control the precision of these functions was 0.005. A smearing of 50 meV was used to determine the Fermi energy and the Drude plasma frequencies. The broadenings for the interband and intraband permittivities were both 20 meV. Slightly changing of these parameters would not affect our main conclusions in this paper. In the calculations of the bAsP-graphene heterostructure, the whole structure was relaxed until the total force was 0.007 Ry/a.u., the cut-off energies of the wavefunctions and densities were respectively 80 Ry and 320 Ry, and the k-mesh was 10 × 18 × 1. 420 valence bands and 210 conduction bands were involved in the construction of the optimal basis functions and the threshold used to control the precision of these functions was 0.02. The Drude plasma frequencies were calculated with a denser mesh of 30 × 54 × 1 and a smearing of 100 meV was used. The phonon spectra were calculated using ultrasoft pseudopotentials on a 7 × 5 × 1 q-mesh, which were mainly used to verify the stability of bAsP even at strains as large as 10%.

3. Plasmonic responses of free-standing monolayer bAsP

In this section, we investigate the plasmonic responses in free-standing monolayer bAsP. Since bAsP is a Janus material, there exists an intrinsic dipole perpendicular to the plane. Dipole correction has been used in the DFT calculations. The horizontally averaged potential has been plotted in Fig. 1(a), where the regions with constant potential correspond to vacuum. Right inset in Fig. 1(a) shows the structure of bAsP, where purple and orange spheres denote respectively As and P atoms. The unit cell has been enlarged and atoms are plotted with small dots. Left inset illustrates the same structure with small dots representing the atoms [51,52]. The x/y axis is along the zigzag/armchair direction. After relaxation, traditional crystallographic constants are $\textrm{a} = 3.477$ Å, $\textrm{b} = 4.717$ Å, $\textrm{c} = 40$ Å, and $\mathrm{\alpha } = \mathrm{\beta } = \mathrm{\gamma } = {90^\textrm{o}}$. The total thickness of the cell is 4 nm which is large enough to avoid interactions with its replicas along c axis. The calculated cell parameters and the atomic positions are consistent with those in the literature [25]. Due to the built-in electric field, a potential step about 0.6 eV occurs across the monolayer. Following investigations would show that such static electric field or potential step has little effect on the electron bands unless contacting with other materials which could cause electron or hole doping as well as charge transfer between the layers. To explore the plasmonic responses of bAsP, extra electrons have been added into the primitive cell representing doping. The band structures are similar except for the Fermi energy E_F. The highest two valence bands and the lowest two conduction bands under various doping levels are plotted in Fig. 1(b), where the Fermi energy has been set to zero. The band structures slightly depend on, for example, the type of pseudopotentials and exchange-correlation functionals. Our choice of these conditions and parameters can already give reasonable results. Using more advanced techniques such as hybrid functionals as in Ref. [25] can improve the band structures slightly, but could drastically increase the amount of work making the following calculations impossible in reality. With increasing level of electron doping, the Fermi energy moves upwards and eventually meets the conduction bands. Then, bAsP can be regarded as a metal with non-zero Drude plasma frequencies along two in-plane directions. As one can see, increasing doping does not change the shapes of the bands but only moves the Fermi energy which is exactly the results of standard DFT. With more than 10⁻² electrons in the unit cell, the Fermi energy begins to meet the lowest conduction band, and bAsP becomes metallic. We have further calculated the permittivity along two in-plane directions without the consideration of the nonlocal parts of the pseudopotentials. Figures 2(a) and 2(b) are respectively the real and imaginary parts of the permittivities as functions of the photon energy along the x (solid curves, denoted as ${\varepsilon _{xx}}$) and y (dashed curves, denoted as ${\varepsilon _{yy}}$) directions, where extra charges -0.05 and -0.1 have been added in the cell. The Fermi energy of the charged system has been determined from self-consistent calculations. After proper electron doping, bAsP becomes metallic; the total permittivities as shown in Fig. 2 are clearly Drude-type. At low frequencies (small photon energies), ${\varepsilon _{xx}} < 0$ and ${\varepsilon _{yy}} < 0$, bAsP is optically metallic along both two directions. As the photon energy increases, ${\varepsilon _{xx}}$ firstly becomes positive, while ${\varepsilon _{yy}}$ is still negative, i.e. ${\varepsilon _{xx}} > 0$ and ${\varepsilon _{yy}} < 0$, which means that bAsP is optically metallic only along the x direction; hyperbolic plasmonic responses occur under this condition. The shaded regions in Fig. 2 correspond to the photon energy ranges supporting hyperbolic plasmonic responses under two doping levels. Increasing doping could move the hyperbolic regions to higher frequencies.

 

Fig. 1. (a) Horizontally averaged potential of bAsP. Right inset illustrates the structure where purple and orange spheres represent As and P atoms, respectively, and the cell has been enlarged and atoms are plotted with small dots. Left inset illustrates the same structure with small dots representing the atoms. The zigzag and armchair directions are set along x and y axes. There is a potential step about 0.6 eV between two sides of bAsP. (b) The highest two valence and lowest two conduction bands of bAsP under different levels of doping. The numbers in the legends are the extra electrons added in the unit cell. Inset is the phonon spectrum. With heavy enough electron doping, bAsP becomes metallic supporting surface plasmons.

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Fig. 2. The (a) real and (b) imaginary parts of the permittivities along x and y axes calculated using independent particle approximation without consideration of the nonlocal parts of the pseudopotentials. Numbers in the legends correspond to the extra charges added in the unit cell. The shaded areas denote the hyperbolic regions. With increasing doping, the hyperbolic regions move to higher frequencies.

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To give an in-depth study of the plasmonic responses at various doping levels, we have further applied the optimal basis function approach to calculate the permittivities, where the contribution from the nonlocal parts of the pseudopotentials has been included. The intraband part of the permittivity is defined as ${\varepsilon _{intra}} ={-} \omega _D^2/[{\omega ({\omega + i\gamma } )} ]$, where γ is the band broadening [50]. The expression of the Drude contribution to the permittivity might be different from those in other literatures and it is just a matter of definition. $\omega _D^2$ denotes the square of the Drude plasma frequency written as $\omega _D^2({\hat{q}} )= ({4\pi /V} )\mathop \sum \limits_{nk} |{ < {\varphi_{nk}}} |\hat{q}\cdot {\boldsymbol v}{|{{\varphi_{nk}} > } |^2}({ - \partial f/\partial E} )$, where V denotes the volume of the cell, f is the Fermi-Dirac function, ${\varphi _{nk}}$ is the wavefunction with band index n and momentum k, $\hat{q}$ is the unit vector indicating the direction, v is the velocity operator; the interband part can be written as ${\varepsilon _{inter}} = 1 - ({4\pi /V} )\mathop \sum \limits_{k,n^{\prime} \ne n} [|{ < {\varphi_{n^{\prime}k}}} |\hat{q}\cdot {\boldsymbol v}{|{{\varphi_{nk}} > } |^2}({{f_{nk}} - {f_{n^{\prime}k}}} )]/[{{{({{E_{n^{\prime}k}} - {E_{nk}}} )}^2}\ast ({\omega - ({{E_{n^{\prime}k}} - {E_{nk}}} )+ i\eta } )} ]$, where the new symbol η is the band broadening [50]. The square of the Drude plasma frequency is generally a tensor, each component of which can be calculated based on the formula as shown above, and then the Drude plasma frequency itself is derived; due to the symmetry of bAsP, only ${\omega _{D,xx}}$ and ${\omega _{D,yy}}$ are non-zero. The permittivities calculated are local and without any momentum transfer, which means that they are functions of $\omega $ only, i.e. $\varepsilon (\omega )$. The obtained values are reasonable.

The intraband parts of the permittivities under various levels of doping are plotted in Fig. 3(a). For clarity, only the real parts have been plotted. Solid and dashed curves correspond to $\textrm{Re}\{ {\varepsilon _{xx}}\} $ and $\textrm{Re}\{ {\varepsilon _{yy}}\} $, respectively. 0.5 eV above the highest occupied state in the undoped bAsP determined from ground state calculations has been regarded as the zero-level for the Fermi energy. As the Fermi energy moves towards the conduction bands, intraband transition of electrons happens and the permittivities start to show Drude-type signatures. Higher Drude plasma frequencies have been observed along the y direction as shown in Fig. 3(a) for all cases. Figure 3(b) shows the corresponding total permittivities. The interband parts of the permittivities can thus be regarded as positive background. At low frequencies, bAsP possesses negative total permittivities along both in-plane directions, indicating elliptic plasmonic responses. With positive interband part of the permittivity as background, bAsP possesses positive/negative total permittivities along the x/y direction within specific frequency ranges. Hyperbolic surface plasmons occur under this condition. Increasing the Fermi energy could move the hyperbolic region to higher frequencies, leading to topological transition of the plasmonic responses at specific photon energy. A small back-gate voltage, for example 0.5 eV, is enough to trigger this phenomenon which is easily achievable in experiments.

 

Fig. 3. (a) The intraband and (b) the total permittivities under various Fermi energies. Only the real parts are plotted. The contributions from the nonlocal parts of the pseudopotentials have been included here. From this figure, it is clear that permittivities due to the interband transition of electrons play a vital role in the hyperbolic regime. Numbers in the legends are the Fermi energies in the unit of eV.

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The total permittivities can be further converted into two-dimensional conductivities, from which the k-surfaces can be easily derived. A simple conductive model has been used here: ${\varepsilon _{tot}} = 1 + i\sigma /\omega {\varepsilon _0}t$, where ${\varepsilon _{tot}}$ is the total permittivity, $\sigma $ is the two-dimensional conductivity, $\omega $ is the angular frequency, ${\varepsilon _0}$ is the permittivity of vacuum, and t is the thickness of cell of bAsP used during the DFT calculations, i.e. $t = 4{\; }$nm. The imaginary parts of the conductivities correspond to the permittivities plotted in Fig. 3(b) have been derived and plotted in Fig. 4. Solid and dashed curves denote $\textrm{Im}\{ {\sigma _{xx}}\} $ and $\textrm{Im}\{ {\sigma _{yy}}\} $, respectively. Since the losses are small and only phenomenological, the real parts of the conductivities have been neglected for simplicity. The corresponding k-surfaces can be calculated from the following equation [53]:

 

Fig. 4. Imaginary parts of the two dimensional conductivities along x (solid curves) and y (dashed curves) axes calculated from the permittivities based on the model. Inset shows the analytically calculated k-surfaces of the surface plasmon polaritons, of which the Fermi energies are 0.45 eV and 0.5 eV. The photon energy indicated by the black arrow is selected in order to observe the topological transition.

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We are only interested in the plasmonic responses at low frequencies. The interband part of the permittivity is in most cases weak, thus it is reasonable to apply the Drude dispersion model for single frequency electromagnetic simulations. The total permittivity of bAsP thus can be separated into two parts: one is the constant background permittivity and the other is the intraband permittivity determined by the Drude plasma frequencies alone. Thus, ${\varepsilon _{total}}(\omega )= {\varepsilon _{r,i}} - \omega _{D,i}^2/[{\omega ({\omega + i\gamma } )} ]$, where i = xx or yy, and ${\varepsilon _{r,i}}$ is the constant representing the background permittivity, the exact value of which can be derived based on the fitting of the total permittivities of the model to those from DFT. The Drude plasma frequencies have already been calculated using the optimal basis function approach within QE, then ${\varepsilon _{r,i}} = \left\langle Re\{{{\varepsilon_{total}}(\omega )} \}+ \omega _{D,i}^2/({{\omega^2} + {\gamma^2}} )\right\rangle$, where $Re\{{{\varepsilon_{total}}(\omega )} \}$ is the real part of the total permittivity and $\left\langle\right\rangle$ means averaging over the data with photon energies from 0.1 eV to 0.9 eV. We plot the Drude plasma frequencies along the x (denoted as ${\omega _{D,xx}}$) and the y (denoted as ${\omega _{D,yy}}$) directions as functions of the Fermi energy in Fig. 5(a), where the inset shows the results on a wider Fermi energy range with open circles denoting ${\omega _{D,xx}}$ and ${\omega _{D,yy}}$ calculated with small smearing (25 meV) and broadening (5 meV) parameters and solid curves denoting ${\omega _{D,xx}}$ and ${\omega _{D,yy}}$ calculated with large smearing (50 meV) and broadening (20 meV) parameters. Relatively large smearings should be applied to remove the oscillations due to numerical noise and get more realistic results. One can find that the Drude plasma frequencies are significantly larger along the y direction than the ones along the x direction, which corresponds to the band structures plotted in Fig. 1(b) showing smaller electron masses along the y direction near the Γ point. In reality, the electron doping caused by, for example, applying back-gate voltages can rarely move the Fermi energy up to 1 eV; here, we focus on the cases where ${E_F} < 0.8\; \textrm{eV}$. The background permittivities along the x (denoted as ${\varepsilon _{r,xx}}$) and y (denoted as ${\varepsilon _{r,yy}}$) directions derived through such fitting are shown in Fig. 5(b) as functions of the Fermi energy. One can notice that with relatively low Fermi energy (usually below 0.3 eV), the background permittivities do not change with doping, meaning that the simple Drude model works very well at low doping levels. The physical reason behind this is that the photon energy is below the bandgap and no significant vertical excitations occur within this material at low doping levels. As the Fermi energy goes up, ${\varepsilon _{r,i}}$ may change. Inset in Fig. 5(b) shows the background permittivities on a wider range. Open circles denote the ones derived based on the DFT results with small smearing (25 meV) and broadening (5 meV) parameters, while the ones derived with large smearing (50 meV) and broadening (20 meV) parameters are plotted with solid (averaging over 0.1∼0.9 eV) curves. One might notice that ${\varepsilon _{r,i}}$ may sometimes become negative, for example, ${\varepsilon _{r,xx}}$ near 1.0 eV as shown by the inset in Fig. 5(b). The reason is that the plasmonic responses are so strong that ${\varepsilon _{total}}(\omega )$ from the DFT is completely negative in the photon energy range 0.1∼0.9 eV. We have also applied the averaging in the photon energy range 0∼0.9 eV as shown by the dashed curves in the inset in Fig. 5(b), and the results are all positive. Such negativity would not affect the Drude dispersion model since we intend to fit the permittivities at low frequencies with low doping levels.

 

Fig. 5. (a) The Drude plasma frequencies along the x (${\omega _{D,xx}}$, with subscript xx) and y (${\omega _{D,yy}}$, with subscript yy) axes. (b) The constants ${\varepsilon _{r,xx}}$ and ${\varepsilon _{r,yy}}$. The inset in (a) shows the Drude plasma frequencies on a wider range. Open circles denote ${\omega _{D,xx}}$ and ${\omega _{D,yy}}$ calculated with small smearing (25 meV) and broadening (5 meV) parameters, while solid curves with large smearing (50 meV) and broadening (20 meV) parameters. Inset in (b) shows the background permittivities on a wider range. Open circles denote the background permittivities derived from the DFT results with small smearing (25 meV) and broadening (5 meV) parameters, while the ones derived from the DFT results with large smearing (50 meV) and broadening (20 meV) parameters are plotted with solid (averaging over 0.1∼0.9 eV) and dashed (averaging over 0∼0.9 eV) curves.

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4. Strain engineering of the plasmonic responses

In this section, we investigate the effects of simple strains on the plasmonic responses. Strains can seriously modify the band structures. Figures 6(a) and 6(b) respectively show the bands under strains up to 4% along x and y directions, where insets are the corresponding phonon spectra with strains as large as 10%. All positive phonon frequencies indicate the stability of bAsP even under such extreme conditions. For clarity, only the lowest two conduction bands and highest two valence bands have been plotted. Strains along x direction (along the zigzag edge) can significantly modify the band dispersions, which is due to the crystal structure of bAsP. Direct-to-indirect bandgap transition happens under 3% strain along the x direction as shown by the orange curve in Fig. 6(a). For plasmonic applications, the type of the bandgap, direct or indirect, has little effect on the permittivities which are mainly determined by the Drude plasma frequencies and only related to the intraband transition of electrons. Under more than 3% strain along the x direction, bAsP is an indirect-bandgap material. The part of the band structures that contributes most to the Drude plasma frequencies under doping has been changed leading to the possibility of strain engineering of the plasmonic responses in bAsP. The direct-to-indirect bandgap transition can be understood from the projected band structures as shown in Fig. 7. Figures 7(a) and 7(b) are the bands under strains along the x direction from 1% to 4% (from left to right) projected on the P_z and P_y orbitals of P atom, respectively; while Figs. 7(c) and 7(d) are respectively the same bands projected on the P_z and P_y orbitals of A_s atoms. Near the $\mathrm{\Gamma }$ point in the Brillouin zone, the lowest conduction band is mainly formed by the bonding states of the P_z orbitals; thus increasing the strain can push the energy upwards as shown by the red dashed lines in Figs. 7(a) and 7(c). However, along the $\mathrm{\Gamma } - \textrm{Y}$ direction the second lowest conduction band is mainly formed by the anti-bonding states of the P_y orbitals; thus increasing the strain would lower the energy, as shown by the blue dashed lines in Figs. 7(b) and 7(d). As the strain increases, “band inversion” may happen, where the first/second lowest conduction band now becomes the second/first one, leading to direct-to-indirect bandgap transition and the change of the conduction band minimum (CBM). Our calculations indicate that 3% strain along the zigzag direction is enough for causing such transition. The plasmonic responses are related to the bands near the Fermi level. In the low doping limit, the Drude plasma frequency is mostly determined by the states near CBM. Clearly, such transition induced by applying strains can seriously modify the plasmonic responses in bAsP.

 

Fig. 6. The highest two valence and lowest two conduction bands under strains along (a) the x and (b) the y axes. Strains up to 4% have been applied. With 3% and 4% strains along the x axis (the zigzag direction), bAsP becomes an indirect-bandgap material. Insets in (a) and (b) show the phonon spectra under positive strains as large as 10% along x and y directions, respectively. All positive phonon frequencies indicate the stability of bAsP under such large strains.

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Fig. 7. (a) Projected bands on the P_z orbital of P atom. (b) Projected bands on the P_y orbital of P atom. (c) Projected bands on the P_z orbital of A_s atom. (d) Projected bands on the P_y orbital of As atom. From left to right, four figures in each case correspond to strains along the x axis from 1% to 4%. Red and blue dashed lines indicate the variations of conduction band minimums.

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Figures 8(a) and 8(b) are the Drude plasma frequencies ${\omega _{D,xx}}$ (plotted by the solid curves) as well as ${\omega _{D,yy}}$ (plotted by the dashed curves), and the fitted background permittivities ${\varepsilon _{r,xx}}$ (plotted by the solid curves) as well as ${\varepsilon _{r,yy}}$ (plotted by the dashed curves) as functions of the Fermi energy under 1%-4% strains along x direction. The inset in Fig. 8(a) shows the same Drude plasma frequencies calculated using small smearing (25 meV) and broadening (5 meV) parameters but a much denser k-mesh (152 × 112 × 1), which is put here only for comparison. In our calculations, the Fermi level has been increased by a step of 0.01 eV. Although the smearing used during the DFT calculations can slightly affect the values of ${\omega _{D,xx}}$ and ${\omega _{D,yy}}$, the tendency of how they change as functions of the Fermi energy under various strains is similar, which would not affect the main conclusions in this paper. Figures 8(c) and 8(d) are the Drude plasma frequencies ${\omega _{D,xx}}$ (plotted by the solid curves) as well as ${\omega _{D,yy}}$ (plotted by the dashed curves) and the fitted background permittivities ${\varepsilon _{r,xx}}$ (plotted by the solid curves) as well as ${\varepsilon _{r,yy}}$ (plotted by the dashed curves) as functions of the Fermi energy under 1%-4% strains along y direction. Comparing Figs. 8(a) with 8(c), one can find that 3% and 4% strains along the x direction can lead to ${\omega _{D,xx}} > {\omega _{D,yy}}$, which is opposite to the usual cases. We explicitly plot the permittivities with E_F = 0.6 eV under 1% and 4% strains along the x axis in Figs. 9(a) and 9(b), respectively. Blue and red colors denote the permittivities in the x and y directions, respectively. Both the results from DFT (open circles) and the model (solid curves) are shown, and they are almost identical. In the hyperbolic region, the permittivity along y direction is negative under 1% strain as shown in Fig. 9(a), while under 4% strain it is the permittivity along the x direction that is negative as shown in Fig. 9(b). The propagation direction of the surface plasmons in the hyperbolic regime has been changed.

 

Fig. 8. (a) The Drude plasma frequencies and (b) the background permittivities under strains along the x axis (zigzag direction). (c) The Drude plasma frequencies and (d) the background permittivities under strains along the y axis (armchair direction). Insets in (a) and (c) are the Drude plasma frequencies calculated with small smearing (25 meV) and broadening (5 meV) parameters but a much denser k-mesh (152 × 112 × 1), and the results show no qualitative differences.

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Fig. 9. (a) The permittivities under (a) 1% and (b) 4% strains along the x axis with E_F = 0.6 eV.

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To explicitly show this, we respectively plot the k-surfaces at specific photon energies in Figs. 10(a) and 10(b) corresponding to the permittivities in Figs. 9(a) and 9(b) both in the elliptic and hyperbolic regimes. Red and blue curves respectively correspond to ${E_{photon}} = 0.3\; \textrm{eV}$ and ${E_{photon}} = 0.5\; \textrm{eV}$. Losses have been neglected here for simplicity. In the elliptic regime, the surface plasmons can propagate along all directions with anisotropic wave number, while in the hyperbolic regime they can only propagate along the direction with negative permittivity. It is clear that strains along the zigzag direction can modify the anisotropy of the k-surfaces and even the propagating direction of the surface plasmons in the hyperbolic regime. At specific photon energy, changing the doping can lead to elliptic-to-hyperbolic transition of the k-surfaces; when combined with proper strains, we can also change the anisotropy leading to the so-called reversed hyperbolicity.

 

Fig. 10. k-surfaces both in the elliptic and hyperbolic regimes under (a) 1% and (b) 4% strains along the x axis. Red and blue curves respectively correspond to 0.3 eV and 0.5 eV photon energies. Losses have been neglected here for simplicity.

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At the end of this section, we show that the interband permittivities are rather small compared with the total ones, which is the reason that the Drude dispersion model works well. The interband permittivities corresponding to Fig. 9 have been fitted with the Lorentz model at low frequencies. Figures 11(a) and 11(b) are respectively the total and interband permittivities corresponding to Fig. 9(a) along x and y directions plotted together. It is clear that the interband permittivities (solid blue and red curves) are rather small compared with the total ones (dashed blue and red curves), thus they can be neglected for simplicity. Figures 11(c) and 11(d) are respectively the total and interband permittivities corresponding to Fig. 9(b) along x and y directions plotted together. Similar phenomena can be observed. We have also fitted the interband permittivities with the Lorentz model: ${\varepsilon _{inter}} = 1 - \mathop \sum \limits_j \frac{{{f_j}\omega _p^2}}{{{\omega ^2} - \omega _j^2 + i{\gamma _j}\omega }}$, where ${\omega _j}$ is the resonant frequency of the oscillator j (an integer), ${\gamma _j}$ denotes the damping, and ${f_j}$ describes the contribution made by oscillator j. Each oscillator has three parameters that needed to be determined during the fitting, i.e. [${\omega _j},\; {\gamma _j},\; {f_j}$]. Our results are (in the unit of eV, except for ${f_j}$): for Fig. 11(a), [0.0201, 0.0401, 0.0285]; for Fig. 11(b), [0.1061, 0.0698, 0.0017] and [0.2, 0.0561, 0.0025]; for Fig. 11(c), [0.0413, 0.0458, 0.0082] and [0.1346, 0.0362, 0.0008]; for Fig. 11(d), [0.0368, 0.0524, 0.0205] and [0.1387,0.0366, 0.0022]. It is obvious that ${f_j}$ are small. For time-domain electromagnetic simulations, such fitting might be necessary; otherwise, the Drude model is enough.

 

Fig. 11. The interband and total permittivities along (a) x and (b) y directions corresponding to Fig. 9(a) are plotted together for comparison. Both real and imaginary parts are shown. The interband and total permittivities along (c) x and (d) y directions corresponding to Fig. 9(b) are plotted together for comparison. Both real and imaginary parts are shown. Insets are the enlarged plots of the imaginary parts of the interband permittivities (open circles), where the red solid curves denote the results from the Lorentz model.

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5. Plasmonic responses in bAsP-graphene heterostructures

Since bAsP is a Janus material, it can cause doping when making contact with other two-dimensional materials. The mechanism behind this is just significant charge transfer triggered by the intrinsic perpendicular dipole or the electrostatic fields near bAsP, just like applying back-gate voltages. As an example, we investigate the plasmonic responses in bAsP-graphene heterostructures. DFT calculations of a large system could be rather computationally expensive. We have searched the twist angle between graphene and bAsP looking for supercells with no more than 6 replicas along a and b axes and minimized strains, and finally found a relatively small heterostructure with total 132 atoms and an angle about $44^\circ$ between their zigzag directions, which is enough for the justification of our idea [54,55], as shown in Fig. 12(a). Other bAsP-graphene heterostructures should lead to similar conclusions. Large heterostructures may have less deformations but the DFT calculations would be extremely time-consuming, and small ones may possess significant deformations or strains which may lead to failure during structure optimization. Figure 12(b) is the horizontally averaged potential of the bAsP-graphene heterostructure, where the inset shows the top view. A large smearing of 100 meV has been used to calculate the Drude plasma frequencies under different levels of doping. The square of the Drude plasma frequencies as functions of the Fermi energy are shown in Fig. 12(c). Blue, red and green curves respectively denote $\omega _{D,xx}^2$, $\omega _{D,yy}^2$ and $\omega _{D,xy}^2$. $\omega _{D,xy}^2$ occurs because x and y are no longer the principal axes. A large smearing used here may reduce the discrepancy caused by residual strains, relatively low cut-off energies and a small k-mesh used during the DFT calculations. This is the reason that none of the square of the Drude plasma frequencies in Fig. 12(c) are below 0.01 eV². Contacting with Janus materials would cause significant charge transfer, which is the reason that the bAsP-graphene heterostructure supports plasmonic responses, i.e. non-zero Drude plasma frequencies, without applying any back-gate voltages (non-zero $\omega _{D,xx}^2$ and $\omega _{D,yy}^2$ at E_F = 0 eV).

 

Fig. 12. (a) Illustrations of the heterostructures. (b) Horizontally averaged potential of the bAsP-graphene heterostructrue.Inset is the top view. (c) The square of the Drude plasma frequencies as functions of the Fermi energy calculated with a relatively large smearing.

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6. Conclusion

In this paper, we have investigated the plasmonic responses in bAsP based on density functional theory. It is found that this material can support both elliptic and hyperbolic plasmons in the infrared under electron doping. At specific photon energy, the isofrequency contour or the k-surface can change its topology when varying Fermi energy, leading to elliptic-to-hyperbolic transition of the plasmonic responses. At low frequencies, the plasmonic responses of bAsP are well captured by the Drude model where the interband transition part of the permittivity is just neglected. Strains along the zigzag direction have been found to distort the band structures significantly and lead to direct-to-indirect bandgap transition in bAsP. Such distortions of the band structures would change the part of the bands that contributes most to the Drude plasma frequencies and thus seriously modify the plasmonic responses. With more than 3% strains along the zigzag direction, the total permittivity along the zigzag/armchair direction is negative/positive after doping, which is opposite to the normal case; the k-surface can change its orientation leading to a special case of hyperbolicity. In the end, the Drude plasma frequencies in a bAsP-graphene heterostructure have been calculated. Since bAsP is a Janus material, significant charge transfer could occur causing non-zero Drude plasma frequencies in the heterostructure without extra doping. Our investigations indicate that bAsP is a promising Janus material for plasmonic applications.