Rashba-type spin splitting and transport properties of novel Janus XWGeN₂ (X = O, S, Se, Te) monolayers

Abstract

We discuss and examine the stability, electronic properties, and transport characteristics of asymmetric monolayers XWGeN₂ (X = O, S, Se, Te) using ab initio density functional theory. All four monolayers of quintuple-layer atomic Janus XWGeN₂ are predicted to be stable and they are all indirect semiconductors in the ground state. When the spin–orbit coupling (SOC) is included, a large spin splitting at the K point is found in XWGeN₂ monolayers, particularly, a giant Rashba-type spin splitting is observed around the Γ point in three structures SWGeN₂, SeWGeN₂, and TeWGeN₂. The Rashba parameters in these structures are directionally isotropic along the high-symmetry directions Γ–K and Γ–M and the Rashba constant α_R increases as the X element moves from S to Te. TeWGeN₂ has the largest Rashba energy up to 37.4 meV (36.6 meV) in the Γ–K (Γ–M) direction. Via the deformation potential method, we calculate the carrier mobility of all four XWGeN₂ monolayers. It is found that the electron mobilities of OWGeN₂ and SWGeN₂ monolayers exceed 200 cm² V⁻¹ s⁻¹, which are suitable for applications in nanoelectronic devices.

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1 Introduction

The successful exfoliation of graphene from graphite¹ is an important step forward in the studies of two-dimensional (2D) nanomaterials. Since its experimental fabrication, graphene has rapidly become the most comprehensively studied object in over a decade.^2,3 In parallel with the effort on graphene, many other 2D graphene-like nanostructures have been continuously discovered, including silicene,⁴ germanene,⁵ hexagonal boron nitride,⁶ post-transition metal monochalcogenides^7,8 or transition metal dichalcogenides.^9,10 In particular, the successful synthesis of 2D asymmetric Janus structures has created a breakthrough in the studies of 2D layered nanomaterials.^11,12 Many novel Janus 2D systems have been proposed and examined in recent times.^13–17 Along with the monolayers, van der Waals heterostructures, which are constructed by stacking different monolayers on top of each other, have also received special attention recently from the scientific community, both theoretically and experimentally.^18–21 With diverse atomic structures and outstanding physical properties, 2D layered nanomaterials have great application prospects in various fields of nanotechnology and devices.

Recently, transition metal nitride (TMN) structures have been studied extensively.^22–24 TMN monolayers are reported to have outstanding mechanical properties^22,25 and they have a lot of potential for applications in the field of photocatalysis.^26,27 Recently, 2D TMN MoSi₂N₄ and WSi₂N₄ have been experimentally fabricated²⁸ and quickly gained intense interest in the scientific community.^29–31 MoSi₂N₄ (and also WSi₂N₄) has a seven atomic layer structure with a MoN₂ (WN₂) monolayer sandwiched between two Si₂N₂ monolayers. Here, MoN₂ has the same structure as MoS₂ while Si₂N₂ has the InS-type of structure. Both MoSi₂N₄ and WSi₂N₄ were found to be indirect semiconductors with many outstanding physical and mechanical properties.^32–34 A giant spin splitting was reported in WSi₂N₄.^28,33 Recently, Wang et al. proposed a series of layered van der Waals systems (54 compounds) with a MoSi₂N₄-like structure known as the MA₂Z₄ family (M = group IVB, VB, and VIB transition metals; A = Si and Ge; and Z = N, P, and As).³² MA₂Z₄ monolayers exhibit superior electronic and transport properties. Previous studies indicated that the electron mobility of MA₂Z₄ is much faster than that of 2D transition metal dichalcogenide structures.^28,32 Particularly, the mobility of the carriers of the MoSi₂N₄ monolayer at the K point is up to 6 times higher than those of the transition metal dichalcogenide MoS₂.²⁸

The WGe₂N₄ monolayer, a member of the MA₂Z₄ family, was calculated to be an indirect semiconductor with a bandgap of 1.99 eV by using the hybrid functional.³² The WGe₂N₄ monolayer was found to be a stable and non-magnetic material.³² Stimulated by the successes in both theoretical and experimental studies of materials in the MA₂Z₄ family²⁸ and also 2D Janus structures,^11,12 we propose and study the asymmetric structures XWGeN₂ (X = O, S, Se, and Te), which can be constructed from the WGe₂N₄ monolayer. We theoretically studied the stability, structural characteristics, electronic states, and transport properties of XWGeN₂ monolayers using ab initio density functional theory (DFT) calculations. The influence of strain engineering on the electronic characteristics of XWGeN₂ is also investigated in this paper. Particularly, we focus on the Rashba spin–orbit splitting (SOC) effect in the XWGeN₂ monolayers.

2 Computational methods

We investigated the atomic structure, electronic properties, and transport characteristics of XWGeN₂ monolayers by using the DFT method as performed in the Quantum Espresso simulation code.³⁵ Projector augmented-wave potentials were utilized in our calculations. The exchange–correlation interaction is determined by a generalized gradient approximation (GGA) with the Perdew, Burke, and Ernzerhof (PBE) functional. To consider the weak van der Waals interactions in the layered structures, we use the Grimme DFT-D2 approach.³⁶ In the electronic structure calculations, SOC was added in the self-consistent calculations.³⁷ Furthermore, we use the hybrid functional by Heyd, Scuseria, and Ernzerhof (HSE06)³⁸ to correct the bandgap values of the considered systems. For the vertical asymmetric Janus structures, the dipole correction was used to correct the errors introduced under the periodic boundary condition.³⁹ The Brillouin zone of the studied monolayers is sampled with a (15 × 15 × 1) k-point mesh. The cut-off energy is selected to be 50 Ry for the plane-wave expansion of PAWs. The monolayers are optimized when the Hellmann–Feynman forces on each atom are smaller than 10⁻³ eV Å⁻¹. The optimization of structures is performed based on the PBE functional. A vacuum distance is set to 30 Å in the vertical direction (the z-axis) to reduce the interactions between adjacent layers.

After full optimization, the structural stabilities are examined by evaluating the phonon spectra using the PHONOPY code⁴⁰ in a 6 × 6 × 1 supercell. Also, the ab initio molecular dynamics (AIMD) simulation is performed at 300 K within 6 ps (each time step of 1 fs) to confirm the thermal stability of the studied monolayers. We use the criterion for mechanical stability suggested by Born and Huang to examine the mechanical stability.^41,42 Finally, we use the deformation potential approximation⁴³ to calculate the mobility of the studied monolayers, in which the transport parameters are obtained by the DFT calculations.

3 Structure and stability

We can construct XWGeN₂ from WGe₂N₄ by simultaneously removing the two layers of atoms Ge and N on the same side and replacing the remaining N layer on this side with the X atoms. The atomic structures of the quintuple atomic XWGeN₂ (X = O, S, Se, and Te) monolayers are depicted in Fig. 1. XWGeN₂ has an asymmetric structure (so-called Janus structure) with five layers of atoms stacked in the order N–Ge–N–W–X. The optimized lattice constant a of XWGeN₂ is from 2.96 Å to 3.16 Å as the X element varies from O to Te. The lattice constant a of XWGeN₂ is proportional to the W–X length and the W–X bond length d_W–X follows a similar trend. Meanwhile, the difference in other bond lengths, including d_W–N and d_Ge–N, between monolayers is very small. The calculated results for the structural parameters of the quintuple atomic XWGeN₂ structures are listed in Table 1. We can see that the thickness of the monolayer, which is defined as the distance between the bottom N atom (N2) and the X atom as illustrated in Fig. 1(b), also increases when the X constituent moves from O to Te, from 4.96 to 5.56 Å.

Fig. 1 (a) Side views of monolayers WGe₂N₄ (left panel) and XWGeN₂ (right panel); (b) top view of the atomic structure of XWGeN₂ (X = O, S, Se, and Te) monolayers; and (c) electron localization function of XWGeN₂. 1 and 2 in (a) indicate the N atoms in the middle and bottom N sublayers, respectively. The red and blue regions in (c) correspond to fully localized electrons and very low charge density, respectively.

Table 1 Lattice constant a (Å), bond length d (Å), bond angle Θ (deg), thickness Δh (Å), and cohesive energy E_coh (eV per atom) of the XWGeN₂ (X = O, S, Se, and Te) monolayers. N1 (N2) indicates the N atoms in the upper (lower) sublayer as illustrated in Fig. 1

	a	d W–X	d W–N	d Ge–N1	d Ge–N2	Θ ∠X–W–X	Θ ∠X–N–Ge	Θ ∠N1–Ge–N2	Θ ∠N2–Ge–N2	Δh	E coh
OWGeN2	2.96	2.09	2.11	1.87	1.82	90.27	126.03	110.67	108.25	4.96	8.30
SWGeN2	3.06	2.39	2.13	1.87	1.86	79.40	124.17	108.44	110.48	5.28	7.83
SeWGeN2	3.09	2.51	2.14	1.88	1.87	75.93	123.42	107.74	111.14	5.39	7.61
TeWGeN2	3.16	2.70	2.15	1.89	1.90	76.61	122.13	106.46	112.31	5.56	7.28

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We next estimate the cohesive energy E_coh to examine the strength of chemical bonds in considered structures. The E_coh value of XWGeN₂ can be calculated according to the following formula:

where E_ξ (ξ = X, W, Ge, and N) is the single-atom energy and of the ξ element; E_tot is the total energy of the XWGeN₂ monolayer; and N_ξ is the number atom ξ in the unit-cell.

In Table 1, we list the calculated cohesive energies of XWGeN₂. It is indicated that the four investigated monolayers of XWGeN₂ are energetically favorable. The cohesive energy of XWGeN₂ varies from 7.28 to 8.30 eV per atom. The OWGeN₂ monolayer is the most energetically stable system with a cohesive energy of 8.30 eV per atom.

To get insights into the bonding nature of the investigated structures, the electron localization function (ELF) of XWGeN₂ monolayers is evaluated and presented in Fig. 1(c). The blue (0.0), green (0.5), and red (1.0) domains refer to the very low density of charge, maximum delocalized electron, and maximum localized electron, respectively. It is found that the high electron localization is around the X and N atoms. For covalent bonds, the electrons which constitute the bonds are delocalized. This suggests that W–X or Ge–N bonds are covalent bonds. X and N atoms are positively charged while W and Ge are negatively charged. Our Bader charge analysis of atoms for XWGeN₂, as listed in Table 2, indicates that the charge of X reduces from 1.03e to 0.01e as the X element varies from O to Te. This is related to the charge redistribution, which is dependent on the electronegativity difference between the X and W constituents.

Table 2 Bader charge (in unit of |e|) of atoms for XWGeN₂ monolayers. N1 and N2 refer to the N atoms located in the middle and bottom sublayers, respectively, as shown in Fig. 1(b)

	q X	q W	q Ge	q N1	q N2
OWGeN2	1.03	−1.94	−1.61	1.37	1.15
SWGeN2	0.30	−1.08	−1.63	1.27	1.14
SeWGeN2	0.22	−1.12	−1.63	1.36	1.17
TeWGeN2	0.01	−0.90	−1.63	1.35	1.17

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Next, we analyze the phonon dispersions of XWGeN₂ to examine their dynamical stability. In principle, the structure of materials is confirmed to be dynamically stable when its vibrational spectrum contains only positive frequencies. The restoring force will be gone if the phonon dispersions contain soft modes. The calculated phonon dispersions of XWGeN₂ monolayers are shown in Fig. 2(a). The unit-cell of XWGeN₂ contains five atoms. Therefore, there are 15 phonon modes (three acoustic and 12 optical modes) in its vibrational spectrum. At the Γ point, there are both doubly degenerate phonon modes (E modes) and four non-degenerate phonon modes (A₁ modes) in the optical vibrational region. The A₁ modes relate to the out-of-plane optical (ZO) branches and the E modes refer to a combination of the in-plane longitudinal (LO) and transverse (TO) optical vibrational branches at the Γ point. In all four vibrational spectra, we can see that no phonon gap is found between acoustic and optical phonon branches. It is observed that there is a range of frequencies where both phonon modes coexist in XWGeN₂ monolayers. This may lead to robust optical–acoustic scattering, and then the materials will have low thermal conductivity. The structure of all four considered monolayers of XWGeN₂ is confirmed to be dynamically stable because of the absence of the negative frequency in its phonon dispersions. Besides the evaluation of the phonon dispersions, we also perform the AIMD simulation to analyze the thermal stability of the considered structures. Fig. 2(b) shows the AIMD calculations for the total energy and temperature fluctuations at 300 K (room temperature) within 6 ps with each step of 1 fs. Our simulated results reveal that the total energy of the systems fluctuates only by about 0.5 eV and their temperature fluctuates at around 300 K during the whole simulated time. No chemical bonds break nor structural phase transitions were found. The atomic structure of the considered systems remains robust after heating within 6 ps, suggesting that all four structures of XWGeN₂ are thermally stable at 300 K.

Fig. 2 (a) Phonon dispersions and (b) AIMD simulations for the total energy E_tot and temperature T fluctuations at room temperature of XWGeN₂.

Based on the analysis of stiffness coefficients, we also examine the mechanical stability of the considered structures. Generally, there are four independent stiffness coefficients C_ij, including C₁₁, C₂₂, C₁₂, and C₆₆, and should be calculated. However, with the centered-hexagonal symmetry of XWGeN₂ as shown in Fig. 1(a), only C₁₁ and C₁₂ should be calculated because C₂₂ = C₁₁ and C₆₆ is calculated by C₆₆ = (C₁₁ − C₁₂)/2. The elastic coefficients are obtained by parabolic fitting the strain-dependent elastic energy of the considered systems. In our calculations, we calculate the elastic energy of the monolayers at different values of the uniaxial strain ε_x (ε_y) in the range from −0.02 to 0.02 (each step being 0.005) along the two in-plane directions. This approach has previously been performed to calculate the stiffness coefficients of the other 2D nanosheets.^44–46

In Table 3, we present the calculated results for the elastic stiffness coefficients of XWGeN₂. It is calculated that both C₁₁ and C₁₂ decrease when the X constituent moves from O to Te. More importantly, XWGeN₂ monolayers have the elastic stiffness coefficients satisfying Born–Huang's criteria for mechanical stability for the hexagonal structures: C₁₁ > 0 and C₁₁² > C₁₂².⁴² Then we can conclude that XWGeN₂ monolayers are mechanically stable.

Table 3 Caculated elastic constant C_ij, Young's modulus Y_2D, and Poisson's ratio ν of XWGeN₂

	C 11	C 12	C 66	Y 2D	ν
OWGeN2	359.72	114.65	122.54	323.18	0.32
SWGeN2	307.65	79.34	114.15	287.19	0.26
SeWGeN2	295.32	68.26	113.53	279.54	0.23
TeWGeN2	277.02	54.01	111.51	266.49	0.20

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Young's modulus Y_2D and Poisson's ratio ν, which can depend on the investigated direction (angle-dependent Y_2D(α) and ν(α)), are calculated based on the elastic constants C_ij by the following formulas:^47,48

where A = sin α, B = cos α, and Π = (C₁₁C₂₂ − C₁₂²)/C₆₆. α refers to the angle between the armchair and studied directions.

Fig. 3 presents the polar diagrams of Y_2D(α) and ν(α) for XWGeN₂ structures. It is calculated that the polar diagrams of Y_2D(α) and ν(α) are perfectly circles, implying both Y_2D and ν do not depend on the angle α. This is due to the in-plane isotropic structures of XWGeN₂. A similar angle-dependence of Y_2D(α) and ν(α) is found in other isotropic 2D materials.^14,49–51 Our calculated results indicate that Y_2D of XWGeN₂ is in the range from 266.49 N m⁻¹ (TeWGeN₂) to 323.18 N m⁻¹ (OWGeN₂), which is comparable with graphene (340 N m⁻¹).⁵²

Fig. 3 Angle-dependent Young's modulus Y_2D(α) (a) and Poisson's ratio ν(α) (b) of XWGeN₂.

4 Electronic properties and the Rashba spin splitting effect

In this section, we show the calculated results for the electronic characteristics of XWGeN₂ monolayers by using the DFT method. It is revealed that all four studied systems of the XWGeN₂ structure are indirect semiconductors as shown in Fig. 4(a). It is shown that the calculated PBE and HSE06 band structures of XWGeN₂ are similar profiles. However, the PBE method is known as a method that underestimates the bandgap value and the bandgap of semiconductors calculated by the HSE06 method is considered to have an acceptable accuracy. It is calculated that the HSE06 bandgap of XWGeN₂ is from 1.54 to 2.44 eV, which is comparable to the bandgap of WGe₂N₄ (1.69 eV).²⁸ The HSE06 bandgaps of SWGeN₂ and SeWGeN₂ are 2.44 and 2.43 eV, respectively. The OWGeN₂ monolayer has the smallest bandgap of 1.54 eV (1.07 eV) at the HSE06 (PBE) level. For comparison, both the HSE06 and PBE bandgaps of XWGeN₂ are presented in Table 4 and also shown in Fig. 4(b). As depicted in Fig. 4(a), it is found that the valence band maximum (VBM) and conduction band minimum (CBM) of OWGeN₂ locate at the Γ and K points in the first Brillouin zone, respectively. Meanwhile, the VBM of the other three structures is shifted away from the Γ point and lies on the KΓ path (near the Γ point). The band structures of SWGeN₂ and SeWGeN₂ near the Fermi level are quite similar. For TeWGeN₂, the CBM is on the KΓ path, i.e., it is different from the other structures. The electronic states of 2D structures are extremely sensitive to their atomic structure and structural perfection. Hence, the great difference in the atomic structure (lattice constant a and the W–X bond length d_W–X) between OWGeN₂ (TeWGeN₂) and SWGeN₂ or SeWGeN₂ is one of the reasons for their band structure to be different. To evaluate the contribution of the orbitals of atoms in the compound to the band structure, we calculate the HSE06 weighted bands of XWGeN₂ monolayers as presented in Fig. 5. As shown in Fig. 5, we can see that both the CBM and VBM of the studied monolayers are mainly contributed by the W-d orbitals. The p-orbital of the N atoms also makes a significant contribution to the VBM, in particular, in the cases of SWGeN₂ and SeWGeN₂. The X-p orbitals have a large contribution to the formation of the valence band far from the Fermi level, around −3 eV. Note that the contributions of atomic orbitals to the electronic bands calculated by the PBE and HSE06 functionals are the same.

Fig. 4 (a) PBE (solid curves) and HSE06 (dashed curves) electronic energy band diagrams and (b) band gaps of the XWGeN₂ monolayers.

Table 4 The calculated band gaps E_g (eV) (using the PBE, PBE + SOC, and HSE functionals), spin splitting energy in the valence band at the K point λ_v, Rashba energy E_R (meV), momentum offset k_R (Å⁻¹), and Rashba constant α_R (eV Å). M and K refer to the Γ–M and Γ–K directions, respectively

	E ^PBEg	E ^HSE06g	E ^PBE+SOCg	λ v	E ^KR	E ^MR	k ^KR	k ^MR	α ^KR	α ^MR
OWGeN2	1.07	1.54	1.04	0.44	—	—	—	—	—	—
SWGeN2	1.90	2.44	1.83	0.42	15.9	15.6	0.087	0.086	366	359
SeWGeN2	1.90	2.43	1.77	0.41	15.1	15.0	0.082	0.083	368	361
TeWGeN2	1.29	1.81	1.15	0.33	37.4	36.6	0.127	0.121	589	605

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Fig. 5 Weighted bands calculated by the HSE06 method of (a) OWGeN₂, (b) SWGeN₂, (c) SeWGeN₂, and (d) TeWGeN₂ monolayers.

For heavy-constituent-based compounds, the effect of the SOC on the electronic characteristics is significant. The PBE + SOC band diagrams of XWGeN₂ calculated in the M–K–Γ–M direction of the Brillouin zone are shown in Fig. 6. It is indicated that, when the SOC is included, the bandgaps of XWGeN₂ significantly reduces, particularly in TeWGeN₂. The PBE + SOC bandgap of TeWGeN₂ is calculated to be 1.15 eV, which is much smaller than its PBE bandgap (1.29 eV) as presented in Table 4. When the SOC effect was introduced, the degeneracy of the spin at the CBM and VBM is removed and a large spin splitting energy λ_v at the K point in the valence band is found for all the four studied monolayers. The spin splitting energy λ_v is from 0.33 to 0.44 eV as presented in Table 4. Such a large spin-splitting energy was also previously found in the Janus WSeTe monolayer.⁵³ More interestingly, the Rashba-type SOC splitting is found around the Γ point in the conduction band of three structures, including SWGeN₂, SeWGeN₂, and TeWGeN₂. The magnified view of the Rashba splitting of these monolayers is also shown in Fig. 6. The Rashba effect strength can be described through the three key quantities, including the Rashba energy E_R, Rashba momentum offset k_R, and Rashba constant α_R. E_R and k_R can be directly calculated from the band diagrams as depicted in Fig. 6 while the Rashba constant α_R is calculated by the relationship α_R = 2E_R/k_R.^54–56 From Fig. 6, we can see that the Rashba parameters in different directions (i.e. the Γ–K and Γ–M directions) can be different. The Rashba energy E_R is defined by the energy difference between the degenerate energy at the Γ point and the two peaks as shown in Fig. 6 (E^K_R for the peak in the Γ–M direction and E^M_R for the peak in the Γ–M direction).

Fig. 6 Band diagrams of XWGeN₂ monolayers with spin–orbit coupling. The small figures at the top are the magnified view of the lowest conduction bands around the Γ point.

We show the obtained results for the Rashba parameters in the Γ–M and Γ–K high-symmetry directions in Table 4. It is revealed that the Rashba parameters in the Γ–K direction (E^K_R, k^K_R, α^K_R) are very close to those in the Γ–M direction (E^M_R, k^M_R, α^M_R). The maximum value of the Rashba energy is found to be 37.4 meV for TeWGeN₂ in the Γ–K direction. Previously, a giant Rashba energy of 52.0 meV was reported in WSeTe.⁵³ The Rashba constants α^K_R (α^M_R) are found to be 369(366), 360(354), and 589(605) meV Å for SWGeN₂, SeWGeN₂, and TeWGeN₂, respectively. The difference between α^K_R and α^M_R is quite small. Therefore, we roughly conclude that the Rashba parameters of XWGeN₂ are directionally isotropic in the Brillouin zone. It is found that, as listed in Table 4, α_R increases when the X constituent of the compound moves from S to Te. This is due to the increase in the SOC strength of the chalcogen constituent. However, there is no significant difference in the Rashba constant α_R between the SWGeN₂ and SWGeN₂ monolayers. Similar trends in the Rashba constant have also been found previously in Janus transition metal dichalcogenides and the Rashba constant of WSeTe was reported to be up to 524 meV Å.⁵⁵

Strain engineering is considered an effective way to control the electronic characteristics of 2D nanomaterials. Here, we examine the effect of a biaxial strain ε_b on the electronic characteristics of XWGeN₂ by using the PBE approach. The biaxial strain can be defined as ε_b = (t − t₀)/t₀, where t₀ and t are the lattice parameters of the unstrained and strained monolayers. In our calculations, the biaxial strain ranges from 0 to ±10%. Here, the “±” sign indicates the compressive/tensile strain. It is indicated that the indirect semiconductor behaviors of all four XWGeN₂ monolayers are preserved in the presence of the biaxial strain with a strain range of −10 to +10% (not shown). No phase transitions are found in the strained XWGeN₂. However, the bandgap of XWGeN₂ dramatically changes when ε_b is applied. Fig. 7 presents the calculated results for the dependence of the PBE bandgaps of XWGeN₂ on ε_b. It is shown that the tensile strain rapidly reduces the bandgap of XWGeN₂ monolayers. The graphs showing the dependence of the bandgap on the biaxial strain for monolayers containing chalcogen atoms S, Se, and Te (i.e. SWGeN₂, SeWGeN₂, TeWGeN₂) are similar. This characteristic is consistent with similar 2D compounds based on chalcogenides.⁵⁷ The bandgap of these monolayers increases slightly and then decreases under the compressive strain. Meanwhile, the bandgap of OWGeN₂ increases under the compressive strain.

Fig. 7 Dependence of the PBE bandgaps of XWGeN₂ monolayers on the biaxial strain ε_b.

5 Transport properties

In the last part, we show our predictions for the transport properties of the four XWGeN₂ structures. We focus on the carrier mobility and related transport parameters of the considered systems. The mobilities of carriers can be evaluated by using the deformation potential approximation.⁴³ For 2D nanomaterials, the mobility of the carrier μ_2D is calculated as follows:⁵⁸where C_2D is the elastic modulus; E_d is the deformation potential constant;, e, ħ, and k_B are the elementary charge, reduced Planck constant, and Boltzmann constant, respectively; m* is the effective mass of carriers; and is the average effective mass. In our calculations, the temperature T is selected to be 300 K (room temperature).

The carrier effective mass, according to eqn (4), is one of the most important parameters, which strongly influences the carrier mobility. The carriers will respond quickly to the external field if they have low effective mass and vice versa. We can obtain the carrier mass m* by fitting the parabolic function to the band edge positions (the VBM for holes and the CBM for electrons) through the formula as follows:

where k is the wave-number, and E (k) is the energy at the edge-position (in the k_xk_y plane).

The effective masses of carriers are dependent on the band structure around the CBB/VBM. We can see that, as shown in eqn (5), the carrier effective mass m* is inversely proportional to the second derivative ∂²E(k)/∂k². Then, m* will be high if the band structure around the CBB/VBM is flat (large radius of curvature). The calculated results for the effective mass of electrons and holes along the transport directions m_x* and m_y* are tabled in Table 4. It is shown that the effective masses of carriers are directionally isotropic (m_x* = m_y*). However, a large difference in the effective mass between electrons m_e* and holes m_h* is found, particularly in the case of the OWGeN₂ monolayer. This is due to the difference in the radius of the curvature around the CMB (for electrons) and VBM (for holes) as shown in Fig. 4. For example, the band diagram around the VBM for OWGeN₂ has a larger radius of curvature (flatter) than that around the CBM. Consequently, the m_h* of OWGeN₂ is higher than its m_e*. It is calculated that m_e* of XWGeN₂ is from 0.39m₀ to 0.80m₀, where m₀ is the free electron mass. The m_e* of OWGeN₂, SWGeN₂, and SeWGeN₂ monolayers are smaller than m_h*. This suggests that the electrons in these structures will respond to the external fields more quickly than holes. Meanwhile, in the case of TeWGeN₂, our calculated results indicate that m_e* < m_h*. This greatly affects the electron mobility in comparison to the hole mobility. However, as presented in eqn (4), the carrier mobility of 2D nanostructures depends also on C_2D and E_d.

The elastic modulus C_2D of 2D structures can be calculated by the following expression:

where Ω₀ is the optimized unitcell area, E_tot is the total energy, and ε_uni is the applied uniaxial strain along the x/y direction. The deformation potential constant E_d can be obtained bywhere ΔE_edge is the energy changing of the band-edges referencing to the vacuum level.

In our calculations, the strain-dependence of energies was investigated in the applied strain range from −1 to 1% (with each step being 0.5%) along the transport directions x and y. We fit the strain-dependence of energies to obtain C_2D and E_d as introduced in previous studies.^58,59 In Fig. 8, we present the energy changing and CBM/VBM positions as a function of ε_uni. The calculated results for both C_2D and E_d of XWGeN₂ are listed in Table 4. The C_2D of XWGeN₂ is high directionally isotropic. However, a directional anisotropy of E_d is found in TeWGeN₂. The ratio of E_d for electrons along transport directions in XWGeN₂ is calculated to be |E^x_d/E^y_d| = 1.81. Recently, the anisotropy of E_d in isotropic 2D structures, such as Al₂Se₃ or Ga₂Te₃ monolayers, was reported.⁶⁰ A large ratio of |E^x_d/E^y_d| = 3.87 was observed in Ga₂Te₃.⁶⁰ Such a directional anisotropy will lead to the anisotropy in the carrier mobility of the considered structures (Table 5).

Fig. 8 The uniaxial strain ε^x/y_uni dependent energy shifting (a) CBM/VBM positions (b) of XWGeN₂ along the two transport directions x and y. The fitting curves are presented by the solid lines.

Table 5 Effective mass of the carrier m* (m₀), elastic modulus C_2D (N m⁻¹), deformation potential constant E_d (eV), and the mobility of the carrier μ (cm² V⁻¹ s⁻¹) along the x (y) transport direction of XWGeN₂. m₀ is the free mass of electrons

		m x *	m y *	C ^x2D	C ^y2D	E ^xd	E ^yd	μ x	μ y
OWGeN2	Electron	0.39	0.39	208.99	209.05	−9.97	−10.01	311.70	309.04
	Hole	3.10	3.10	208.99	209.05	−2.96	−2.96	52.74	52.76
SWGeN2	Electron	0.41	0.41	181.38	181.34	−10.67	−10.77	204.84	201.01
	Hole	1.45	1.45	181.38	181.34	−3.03	−3.16	151.55	139.67
SeWGeN2	Electron	0.61	0.61	175.03	175.35	−8.51	−7.52	34.43	44.20
	Hole	0.89	0.89	175.03	175.35	−3.63	−3.60	241.69	246.45
TeWGeN2	Electron	0.80	0.80	165.34	165.47	−7.25	−4.00	58.55	192.89
	Hole	0.58	0.58	165.34	165.47	−5.52	−4.74	220.92	300.29

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The obtained results for the carrier mobility μ of XWGeN₂ are presented in Table 4. It is calculated that the electron mobility of XWGeN₂ is high, up to 311.70 cm² V⁻¹ s⁻¹ for OWGeN₂. The carrier mobilities of OWGeN₂, SWGeN₂, and SeWGeN₂ are quite directionally isotropic. The difference in the carrier mobility between the different directions μ_x,y in these structures is small. However, there is a large difference between μ_x and μ_y for electrons in the case of TeWGeN₂. The electron mobility of TeWGeN₂ is calculated to be μ_x = 58.55 cm² V⁻¹ s⁻¹ and μ_y = 192.89 cm² V⁻¹ s⁻¹. This is due to the directional anisotropy of the deformation potential constant E_d in TeWGeN₂ as above-mentioned. It is also found that, in SeWGeN₂ and TeWGeN₂ monolayers, the mobilities of holes are higher than those of electrons. This is different from the other two structures, i.e., OWGeN₂ and SWGeN₂, which have an electron mobility much larger than the hole mobility. With a high electron mobility, the XWGeN₂ monolayers, such as OWGeN₂ and SWGeN₂, are suitable for applications in high-speed electronic nanodevices.

6 Conclusions

In conclusion, we have suggested new stable monolayers of the Janus XWGeN₂ which have a quintuple-layer atomic structure. Our DFT calculations indicated that these monolayers are indirect semiconductors and their bandgap can be controlled by applied strains. The Rashba-type SOC with a large Rashba constant is found around the Γ point of SWGeN₂, SeWGeN₂, and TeWGeN₂. The obtained calculations indicate that the Rashba parameters of these three monolayers are not too sensitive to the investigated directions in the first Brillouin zone. The existence of the giant Rashba SOC effect together with the large bandgap makes these materials promising for applications in highly efficient spintronic and optoelectronic devices. OWGeN₂ has the highest electron mobility of 311.70 and 309.04 cm² V⁻¹ s⁻¹ along the transport directions x and y, respectively. Our findings not only provide important physical characteristics of XWGeN₂ systems but also guide further investigations on quintuple-layer atomic Janus materials formed from the symmetric structures of the MA₂Z₄ family.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

Tuan V. Vu would like to thank the Van Lang University for funding this work.

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