A density functional theory study of two-dimensional bismuth selenite: layer-dependent electronic, transport and optical properties with spin–orbit coupling

Abstract

Recently, atomic-thickness van der Waals (vdW) layered bismuth selenite (Bi₂SeO₅) has been successfully synthesized, not only expanding the family of two-dimensional (2D) materials, but also playing a pivotal role in the advancement of 2D electronics as a high-κ dielectric. In this work, we systematically study the basic properties of 2D Bi₂SeO₅ through first-principles calculations, focusing on the spin–orbit coupling (SOC) effect and layer-dependent behaviors. The results show that SOC can adjust the bandgap of bulk/2D Bi₂SeO₅ from direct to indirect, with the bandgap decreasing upon increasing the thickness due to quantum confinement. Importantly, we observe that SOC has a negligible effect on the valence band edge but significantly impacts the conduction band edge, due to the specific distribution of the Bi-p orbital. We also explore the vdW magnetic tunneling junction based on 2D Bi₂SeO₅, which can exhibit significant tunneling magnetoresistance between the parallel and antiparallel magnetic alignments of electrodes, e.g. 900% for 1L and 1800% for 2L. As for the optical properties, strong layer dependence is also verified, and the large absorption coefficient is determined to be ∼10⁶ cm⁻¹. At last, we also explore the piezoelectric properties. Overall, layered Bi₂SeO₅ is a potential candidate material for electronic device and optoelectronic applications, as well as nano-spintronic applications.

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1. Introduction

Spin–orbit coupling (SOC), a fundamental quantum mechanical interaction between the spin and orbital angular momentums, plays a pivotal role in determining the electronic structure, optical properties, and transport phenomena in materials.^1–4 Specifically, SOC can induce profound modifications in band structures, manifesting as phenomena such as spin splitting, spin scattering and band inversion, etc.,^5–7 rendering materials with strong SOC particularly promising for next-generation optoelectronic, spintronic, and quantum computing applications.^8,9 Van der Waals (vdW) layered materials have emerged as a revolutionary platform for materials science research in recent decades.^10–14 A distinctive advantage of vdW materials lies in their atomically precise surfaces characterized by the absence of dangling bonds, exceptional thickness uniformity, and atomically smooth interfaces-properties that markedly distinguish them from conventional three-dimensional materials.¹⁵ Remarkably, vdW systems always show the layer-dependent properties when the thickness is confined to sub-nanometer (few-layer). The interplay between SOC and quantum confinement effects in ultrathin vdW materials offers unprecedented opportunities to engineer their electronic structure and transport properties with atomic-layer precision, which establishes a versatile platform for designing next-generation multifunctional materials with precisely tailored properties, enabling transformative advances in quantum technology, optoelectronics, and spintronics.^16–18

Very recently, large-scale two-dimensional (2D) bismuth selenite (Bi₂SeO₅) with a high dielectric constant has been achieved through layer-by-layer oxidation or exfoliation.^19–21 Layered Bi₂SeO₅ was reported to have a large bandgap and as a high-κ dielectric, it can show improved electronic performance compared to 2D materials such as Bi₂O₂Se, MoS₂, and graphene due to suppression of the current leakage.²² SOC plays a pivotal role in the electronic structure, and thus, it may be important for Bi-based system. In addition, as a vdW material, the layer-dependent properties have not been studied yet, including the electronic, transport, optical, and piezoelectric properties, etc. Therefore, an in-depth study is necessary.

In this paper, the geometric, electronic, transport, optical, and piezoelectric properties of bulk/2D Bi₂SeO₅ are investigated using density functional theory (DFT). The cleavage energy is 31.87 meV Å⁻², indicating the experimental exfoliation feasibility. The band calculations exhibit that the bandgap decreases from 3.06 to 2.79 eV with an exponential tendency, and SOC will further reduce the bandgap by ∼0.3 eV. Interestingly, it is found that the valence band (VB) edge is dominated by the O-p orbital, and Bi-p contributes obviously to the conduction band (CB) edge. As a result, SOC mainly affects the CBs while leaves the VBs intact, which is quite different from the normal metal compounds. Next, we construct a vdW FeS₂/Bi₂SeO₅/FeS₂ magnetic tunneling junction (MTJ) and it shows significant tunneling magnetoresistance (TMR). The optical properties of 2D Bi₂SeO₅ show obvious layer dependence and SOC modulation, and large absorption coefficients (∼10⁶ cm⁻¹) are also observed. Finally, we show that bulk/2D Bi₂SeO₅ has broken inversion symmetry, and possibly has potential for piezoelectric applications. Our study not only reveals that SOC is both intriguing and significant for Bi₂SeO₅, but also highlights the layer-dependent electronic, optical, and piezoelectric properties of 2D Bi₂SeO₅, emphasizing its potential applications in spintronics.

2. Computational methods

We performed first-principles calculations using the Vienna Ab initio Simulation Package (VASP) to describe the exchange–correlation potential by the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE).^23–25 The interactions between the nuclear and valence electrons were described by the projected augmented wave (PAW) potential, with an energy-cutoff of 520 eV for the plane-wave basis. A 3 × 2 × 6/2 × 6 × 1 k-point mesh was used to sample the Brillouin zone (BZ) of primitive bulk/2D Bi₂SeO₅. To avoid the interactions between the periodic images perpendicular to the basal plane, we included a vacuum space of 20 Å. We tested different vdW interactions and chose the optB86b exchange functional (optB86b-vdW).²⁶ The systems were optimized with a force threshold of 10⁻³ eV Å⁻¹ and the energy convergence criterion was set to be 10⁻⁶ eV. SOC was included due to the heavy element Bi.²⁷ A hybrid-exchange functional (HSE06)²⁸ with the PBE functional was used to compare the band structure.

In order to obtain more information about the transport properties of 2D Bi₂SeO₅, we investigated its relaxation time (τ) based on the Bardeen–Shockley deformation potential (DP) theory for 2D materials:^29,30

where m*, ħ, k_B, C, T and E₁ are the effective mass, Planck constant, Boltzmann constant, elastic constant, temperature and DP constant, respectively.

We used double-zeta polarization function (DZP) to initialize the Kohn–Sham states and adopted the nonequilibrium Green's function (NEGF-DFT) to calculate the transmission properties. The corresponding energy cutoff was set to be 80 Hartee. Then, the k-point meshes of the scattering region and the electrode were set to 4 × 11 × 1 and 4 × 11 × 100, respectively. The energy convergence criteria for both self-consistent calculations and transmission spectrum calculations were consistently set to 10⁻⁵ eV. Transmission spectra were calculated using Nanodcal³¹ and Atomistic ToolKit^32,33 software.

The linear optical properties can be derived from its complex dielectric function, providing crucial insights into its applicability in advanced optoelectronic technologies:

ε(ω) = ε₁(ω) + iε₂(ω), (2)

where ω represents the photon frequency. The imaginary part ε₂(ω) can be computed from the momentum matrix elements, while the real part ε₁(ω) can be derived from ε₂(ω) utilizing the Kramer–Kronig relationship:where p is the principal value of integral. The refractive spectrum n(ω) and optical absorption coefficients α(ω) are calculated using the formula:

3. Results and discussion

3.1 Geometric and electronic properties of bulk/2D Bi₂SeO₅

3.1.1 Bulk Bi₂SeO₅. Firstly, bulk Bi₂SeO₅ is studied as depicted in Fig. 1a. Bulk Bi₂SeO₅ belongs to the orthogonal lattice with a space group of ABM2. Along the c-direction, covalently saturated Bi₂SeO₅ layers are AA-stacked, reflecting the vdW gap between layers. To study the vdW interactions, we utilize different vdW methods to optimize the bulk structure and then compare the lattice constants with experimental ones.^19,21,34 The corresponding results are summarized in Table S1 (ESI†). As shown in Table 1, Tables S1 and S2 (ESI†), we find that DFT-D3,³⁵ dDsC,^36,37 optB88-vdW,³⁸ optB86b-vdW,²⁶ and rev-vdW-DF2³⁹ methods can yield better results than others, e.g., relative differences are within 1%. We finally adopt the optB86b-vdW functional, since it gives the smallest variation of the volume^19,21,34 and the corresponding lattice constants are a = 11.33, b = 16.37, and c = 5.52 Å, respectively.

Fig. 1 (a) Crystal structure of bulk Bi₂SeO₅. The vdW gap is marked with a gray plane, reflecting the cleavage position. Electronic band structures of (b) bulk, (c) 1L and (d) 2L Bi₂SeO₅ with SOC (orange dashed lines). For easy comparison, we plot the bands without SOC together (blue solid lines). Green and black pentagrams represent the CBM and VBM, respectively. Inset: The Brillouin zones of (b) bulk and (c) 2D Bi₂SeO₅ with high symmetry points and lines.

Table 1 Lattice constants and the volume of bulk Bi₂SeO₅ calculated by using different computational methods

System	a (Å)	b (Å)	c (Å)	V (Å^3)
Exp.	11.44	16.28	5.49	1022.48
DFT-D3BJ	11.37	16.41	5.52	1029.93
dDsC	11.38	16.43	5.52	1032.09
optB88-vdW	11.37	16.44	5.54	1035.55
optB86b-vdW	11.33	16.37	5.52	1023.81
rev-vdW-DF2	11.35	16.40	5.53	1029.35

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As shown in Fig. 1b, we find that bulk Bi₂SeO₅ is a semiconductor with a direct bandgap of 2.79 eV at the Z point, which is consistent with the previous calculations²¹ and also suitable for ultraviolet optoelectronic devices.^40,41 We also use the HSE06 functional to compare the band structure, which also has a direct bandgap (of 3.87 eV), as shown in Fig. S1a (ESI†). Near the conduction band minimum (CBM) and valence band maximum (VBM), i.e. the Z point, the effective mass is calculated to be 0.397 m₀ [electron effective mass (m_e)]/5.345 m₀ [hole effective mass (m_h)] along the Z–T direction, reflecting that the top VB is weakly dispersed (see Fig. 1b). Here m₀ represents the electron mass. In addition, the calculated values of effective mass are 1.431 m₀ (m_e)/0.998 m₀ (m_h) along the Z–Γ direction, and 0.776 m₀ (m_e)/1.580 m₀ (m_h) along the Z–R direction, respectively. These results are comparable to those of some typical 2D materials, such as MoS₂ (0.48 m₀)⁴² and h-BN (0.99 m₀).⁴³

We also plot the local density of states (LDOS) in Fig. 2a. It is found that the VBs are dominated by the O-p orbital, and the contribution of the Bi-p orbital becomes the largest near the CBM. Such a behavior is different from that of normal metal compounds, in which the s orbital of the metal is not negligible near the CBM.⁴⁴ In Fig. 2a, we present a detailed plot of the partial density of states (PDOS), revealing that the p_y and p_z orbitals of Bi have nearly equal contribution, both of which are significantly larger than the Bi-p_x orbital. This suggests that the interlayer interaction in this compound effectively mixes the out-of-plane and in-plane orbitals [see the spatial distribution of wavefunction for the lowest CBs in Fig. S2a–c (ESI†)].

Fig. 2 (a)–(c) The LDOS and PDOS of bulk/1L/2L Bi₂SeO₅, respectively. The p orbitals of Bi, Se and O are projected in the LDOS and the p_x, p_y, and p_z orbitals of Bi are projected in the PDOS. Note that the directions of p_x, p_y, and p_z for the bulk system correspond to the b, c, and a directions (see Fig. 1a).

3.1.2 Two-dimensional Bi₂SeO₅. The exfoliation energy is firstly estimated using the equation below:⁴⁵where E_Bulk and E_1L denote the energy of the optimized bulk and 1L Bi₂SeO₅; A is the area of the basal plane. The calculated exfoliation energy is 31.87 meV Å⁻², which is comparable to 28.35 meV Å⁻² (h-BN)⁴⁶ and ∼21 meV Å⁻² (MoS₂).⁴⁵ Thus, we believe bulk Bi₂SeO₅ can be exfoliated to the 2D form. Indeed, previous experiments have successfully exfoliated 2D Bi₂SeO₅.²²

Typically, the thickness has a significant influence on the basic properties of layered systems due to the quantum confinement.⁴⁷ For instance, single-layer MoS₂/MoSe₂ has a direct bandgap while multilayer systems have indirect bandgaps.^48,49 Therefore, we further investigate the layer-dependent electronic properties of 2D Bi₂SeO₅, and several results are summarized in Table 2. For example, the in-plane lattice constants gradually increase by 0.04 Å when reducing the thickness from the bulk to 1L Bi₂SeO₅, which can be ascribed to the decreased interlayer interaction. The bandgap variation under the quantum confinement effect is also discussed and verified [see Note S1 and Fig. S3 (ESI†)].

Table 2 Lattice constants and bandgap values of bulk/2D Bi₂SeO₅

System	a (Å)	b (Å)	Bandgap (eV)	Type	Bandgap-SOC (eV)	Type
1L	16.41	5.48	3.06	Indirect	2.73	Indirect
2L	16.39	5.49	2.91	Direct	2.62	Indirect
3L	16.38	5.51	2.85	Direct	2.59	Indirect
Bulk	16.37	5.52	2.79	Direct	2.55	Indirect

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Fig. 1c and d display the band structures of 1L and 2L Bi₂SeO₅ [see 3L in Fig. S4 (ESI†)]. In the 2D limit, the initial Z point of 3D BZ folds back to the Γ point. The VBM for 1L Bi₂SeO₅ is 27.21 meV above the Γ point, which is located in the Y–Γ direction; while the CBM is much close to the Γ point, leading to an indirect bandgap of 3.06 eV [4.18 eV using the HSE06 functional (see Fig. S1b ESI†)]. When interlayer interactions are naturally included for the multilayer Bi₂SeO₅, the bandgap becomes direct and the values range from 2.91 to 2.85 eV for 2L and 3L systems (see Table 2).

To visualize the interlayer interaction, the LDOS and PDOS of 1L and 2L Bi₂SeO₅ are also calculated as shown in Fig. 2b and c. Similar to the bulk case, VBs are dominated by the O-p orbital. We also found that the Bi-p orbital has obvious contribution to CBs. For 1L Bi₂SeO₅, the p_z-orbital dominates while for 2L Bi₂SeO₅, p_z and p_y orbitals have nearly equal contributions near the CBM mimicking the bulk case. This orbital redistribution phenomenon is directly related to the interlayer interaction, which directly entangles the in-plane orbitals with the out-of-plane p_z orbital. As a result, the bonding and antibonding like charge distribution is formed and the proportion of the p_z orbital decreases near the CBM [see Fig. S2d–f (ESI†)], demonstrating the orbital mixing and redistribution.

3.1.3 SOC effect on the electronic properties. SOC is a relativistic interaction that entangles the spin and orbital degrees of freedom, and has an important effect on the electronic properties,^50,51 magnetism,^52,53 and band topology.^54,55 Considering the heavy element of Bi, one must include SOC for the electronic properties. As depicted in Fig. 1b for bulk Bi₂SeO₅, it is clearly observed that SOC can significantly reduce the bandgap from 2.79 to 2.55 eV [also see DOS in Fig. 2a-bottom]. Besides, it is observed that SOC moves the CBM slightly away from Γ to a general point along Z–R while leaves the VBM at Γ, leading to a bandgap direct-to-indirect transition. Moreover, we find that SOC only pulls down the CBs, while leaving the VBs nearly intact. This interesting behavior can be ascribed to the large/small contribution of the Bi-p orbital to CBs/VBs (see Fig. 2a).

Subsequently, we explore the SOC effect on the electronic properties of 2D Bi₂SeO₅. The results of the band structures are plotted in Fig. 1b–d. It can be seen that SOC also leads to the reduction of the bandgap. For example, the bandgap exhibits a strong layer-dependent evolution, decreasing from 3.06 to 2.73 eV for 1L Bi₂SeO₅, from 2.91 to 2.62 eV for 2L Bi₂SeO₅, and from 2.85 to 2.59 eV for 3L Bi₂SeO₅ [also see Fig. S4 (ESI†)]. In the absence of SOC, 2D Bi₂SeO₅ systems are direct bandgap semiconductors except for 1L Bi₂SeO₅. However, under the effect of SOC, the CBM is lightly shifted from Γ, and 2D Bi₂SeO₅ becomes an indirect bandgap semiconductor regardless of the layer number (see Table 2). Furthermore, as shown in Fig. 2b and c, the Bi-p orbital in 2D Bi₂SeO₅ also has a significant contribution to the CBs. Therefore, under the influence of SOC, the CBs are pulled down obviously, leading to a decrease of the bandgap.

Besides, the introduction of SOC will lead to a change in effective mass, especially the m_e since SOC mainly affects CBs, as shown in Fig. S5 (ESI†). Taking 1L Bi₂SeO₅ as an example, the magnitude of m_e is 0.447 m₀ along the Γ–X direction and 0.458 m₀ along the Γ–Y direction without the inclusion of SOC, exhibiting a relatively isotropic behavior. With SOC, m_e increases significantly along both the Γ–X and Γ–Y directions. For 2L/3L Bi₂SeO₅, when SOC is considered, the magnitude of m_e also increases along the Γ–X and Γ–Y directions. As the layer number increases from 1L to 3L Bi₂SeO₅, the in-plane anisotropy of m_e becomes more pronounced, probably due to the interlayer interaction, which mixes the in-plane and out-of-plane p orbitals. As for m_h, the impact of SOC is generally minimal [see Fig. S5b and d (ESI†)]. All results above demonstrate that SOC has a crucial influence on the electronic properties of Bi₂SeO₅.

With a moderate m_e, we then estimate the electrical conductivity (σ) of 1L Bi₂SeO₅ for electrons, as shown in Fig. S6 (ESI†). Due to the relaxation time approximation, we need to determine the τ of electrons. Using the DP theory, τ values are determined to be 63.56 and 114.07 fs along the x and y directions, respectively, which manifesting the strong anisotropy.⁵⁶ Therefore, σ of electron for 1L Bi₂SeO₅ is about 0.90 and 1.62 × 10³ Ω⁻¹ m⁻¹ along the x and y directions with a carrier concentration of 10¹¹ cm⁻², which are comparable to those of some 2D materials.^57,58 To demonstrate the anisotropy in 2D Bi₂SeO₅, we also estimate the τ of h along the x and y directions, which are determined to be 502.10 and 770.97 fs, respectively.

3.2 Bi₂SeO₅ based tunneling junction

After that, we turn to the transport properties of Bi₂SeO₅ based tunneling junctions because of the uniform surface of vdW layers and strong SOC.^52,59,60 The transport device model is depicted in Fig. 3a and a nonmagnetic electrode is used firstly. We calculate the transmission properties of 1L/2D Bi₂SeO₅ (details in methods), and the zero-bias transmission spectra are shown in Fig. 3b and S7a (ESI†). One can find a large transmission gap near the Fermi energy due to the bandgap of 2D Bi₂SeO₅, indicating that the transmission in this region is forbidden with zero-bias. Beyond this region, the transmission probability gradually increases. Fig. 3b and S7a (ESI†) also illustrates the impact of SOC on the transmission rate. With SOC, the scattering probability near the CBM is substantially modulated, which is consistent with the band structure (see Fig. 1). Fig. 3b and S7a (ESI†) also exhibit the trend of transmission with different thickness of scattering region. For example, stacking one more layer on single-layer Bi₂SeO₅ will obviously enhance the scattering probability, thus reducing the transmission rate. All these results suggest that SOC is also important in the Bi₂SeO₅ based tunneling junctions and thus, one can make other kinds of tunneling junctions, where SOC is crucial. Besides, Bi₂SeO₅ based tunneling junction can be fabricated in future experiments to study the layer-dependent transmission phenomenon.

Fig. 3 (a) and (b) Schematic diagram and transmission spectra of the Al/1L Bi₂SeO₅/Al device. (c) and (d) Schematic diagram and transmission spectra of the FeS₂/1L Bi₂SeO₅/FeS₂ MTJ.

Since SOC also plays a significant role in the MTJs,^61,62 we then make a device using FeS₂/Bi₂SeO₅/FeS₂ layers, as shown in Fig. 3c, which can be used to explore the magnetic tunneling properties. Typically, MTJs exhibit high/low conductance when two ferromagnetic electrodes have the parallel/antiparallel magnetic configurations (PC/APC), due to the relative alignment of magnetization directions. The relative difference between the PC and APC configurations can be defined as the TMR, which is calculated using the formula:

where E represents the chemical potential, while T_PC(E) and T_APC(E) denote the transmission coefficients in the PC and APC states, respectively. As illustrated in Fig. S8 and S9 (ESI†), the k-resolved transmission coefficients for both PC and APC states of FeS₂/1L Bi₂SeO₅/FeS₂ and FeS₂/2L Bi₂SeO₅/FeS₂ MTJs are plotted, which reveal a significantly higher transmission around the Γ point and BZ boundary in the PC configuration compared to the APC state. Therefore, the calculated TMR should be positive. For 1L Bi₂SeO₅, the TMR at E = 0 eV reaches 137% (Fig. 3d), which is comparable to those of some typical vdW MTJs such as Fe₃GeTe₂/h-BN/Fe₃GeTe₂ (160%)⁶³ and VSe₂/CrI₃/VSe₂ (178%).⁶⁴ If chemical potential is tuned away from E = 0 eV, MTJ will exhibit a significant change in the transmission properties. For example, it shows a tremendous TMR value of 900% around E = −0.26 eV, as shown in Fig. S10a (ESI†). Similar results are observed for 2L case, as plotted in Fig. S7b and S10b (ESI†). Moreover, the TMR demonstrates a clear dependence on the layer number. For 2L Bi₂SeO₅, the TMR at E = 0 eV increases to 244%. Deviations in chemical potential further amplify the TMR, reaching an impressive 1800% at E = −0.28 eV. Therefore, Bi₂SeO₅ based vdW MTJ exhibits tremendous potential for multi-state magnetic storage in spintronics, and the layer-dependent TMR effect provides a practical strategy for optimizing the device performance.

3.3 Optical properties of bulk/2D Bi₂SeO₅

The electronic structures of bulk/2D Bi₂SeO₅, which feature a significant contribution from p-orbitals in the conduction bands of bismuth, play a crucial role in defining their distinctive optical properties. This unique electronic configuration allows for the manipulation of optical behaviors in Bi₂SeO₅, minimizing the dependence on sp-hybridization. As a result, Bi₂SeO₅ emerges as a promising candidate for both optoelectronic and electronic applications.⁶⁵ The dielectric properties of Bi₂SeO₅ have been calculated across an energy range from 0 to 12 eV.

As shown in Fig. 4a–h, the optical properties of bulk/2D Bi₂SeO₅ are essentially equivalent along the xx and yy directions. Therefore, subsequent analyses focus solely on the optical properties along the xx direction. The calculated spectra of ε₁(ω) are presented in Fig. 4a and the static dielectric constants, i.e. ε(0), are provided by ε₁(ω) at ω = 0. As shown in Table 3, the static dielectric constant, ε(0), exhibits a dependence on the layer number. Specifically, ε(0) increases monotonically with the addition of more layers while the differences between ε(0) values diminish as layers are added. This behavior inversely correlates with the variation in the bandgap, a phenomenon attributed to interlayer vdW interactions.

Fig. 4 (a)–(d) The real and imaginary parts of the dielectric function, as well as the absorption coefficient and refractive index, along the xx direction. (e)–(h) The real and imaginary parts of the dielectric function, as well as the absorption coefficient and refractive index, along the yy direction. We replace the unit of energy by eV, according to E = ħω.

Table 3 The static dielectric constants ε(0), and static refractive constants n(0) of bulk/2D Bi₂SeO₅

Value	1L	2L	3L	Bulk
ε(0)	2.69	3.50	3.96	5.62
ε(0)-SOC	2.82	3.65	4.16	5.90
n(0)	1.64	1.87	1.99	2.37
n(0)-SOC	1.68	1.91	2.04	2.43

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Fig. 4a also illustrates the variation in the magnitude of the real part of dielectric function, i.e. ε₁(ω), as it changes with photon energy (ħω). When ω increases, ε₁(ω) of 1L Bi₂SeO₅ firstly increases, and reaches a peak at 3.80 eV. The peak shifts to a low energy region (red shift) and increases in magnitude with the increase of the layer number. Notably, bulk Bi₂SeO₅ exhibits a region where ε₁(ω) becomes negative. According to the wavevector equation , ε₁(ω) < 0 implies that the wavevector K becomes imaginary, indicating that light cannot propagate within this frequency range. These findings suggest that bulk Bi₂SeO₅ displays metallic characteristics under photon irradiation at these frequencies, whereas 2D Bi₂SeO₅ retains its intrinsic semiconductor properties.

ε ₂(ω) of the dielectric function is closely related to the electronic band structure and reveals the absorption characteristics of materials.⁶⁶ The absorption edge of ε₂(ω) closely aligns with the bandgap values reported in Table 2, indicating the optical bandgap of Bi₂SeO₅. At this energy level, direct optical transitions occur between the highest occupied VB and the lowest unoccupied CB, attributed to interband transitions between the occupied and unoccupied states. As the energy surpasses this threshold, the curve rises sharply, indicating the presence of additional interband transitions. As illustrated in Fig. 4b, ε₂(ω) of 1L Bi₂SeO₅ increases from the edge and exhibits a prominent peak at 4.70 eV. This may correspond to the electronic transitions from the O-p orbital in VBs to the Bi-p orbital in CBs, with the magnitude of the peak being related to the maximum density of transition states. With the increase of the layer number, such a peak has a red shift like ε₁(ω) and shows a significant enhancement in both the visible and ultraviolet regions, as also observed in 2D MoS₂,⁶⁷ ultimately providing validation for the absorption spectra presented in Fig. 4c. The imaginary part of the dielectric function for both bulk/2D Bi₂SeO₅ consistently exhibits a peak in the ultraviolet region, which renders Bi₂SeO₅ particularly advantageous for applications in ultraviolet optoelectronic devices.

The optical absorption coefficient α(ω) is crucial for assessing the ability to absorb light energy, offering valuable insights into its efficiency for solar energy conversion.^68,69 The α(ω) curves are shown in Fig. 4c. One can find that α(ω) of bulk/2D Bi₂SeO₅ can reach ∼10⁶ cm⁻¹ near the visible-light range, and a significantly higher absorption is observed in the ultraviolet region, demonstrating excellent absorption properties. The increase of thickness will enhance the absorption coefficient, and the absorption peak also has a red shift, reflecting the layer dependence. These characteristics make Bi₂SeO₅ as an exceptionally competitive candidate for optoelectronic device applications.

The refractive index n(ω) quantifies the speed at which light propagates through a crystal, providing insights into the material's stability for device applications. Additionally, it can be used to determine the phase velocity of electromagnetic waves within the medium, making it a critical parameter in the study and design of optical materials.^70,71Fig. 4d shows the refractive index of Bi₂SeO₅. The static refractive index of 1L Bi₂SeO₅ is n(0) = 1.64, much higher than those of other typical 2D materials such as MgCl₂⁷² and CaF₂.⁷³ The static refractive indices also increase with the layer number, and the difference between the layers becomes smaller and smaller, as shown in Table 3. The n(ω) of 1L Bi₂SeO₅ increases progressively with larger photon energy, reaching a maximum value of 2.02 at 3.89 eV. As the layer number increases, the peak value of n(ω) continues to increase and undergoes a red shift, which attains a maximum value of 2.98 at 3.82 eV for bulk Bi₂SeO₅. A higher refractive index effectively focuses light, enhancing the resolution and imaging quality of optical systems. Therefore, bulk/2D Bi₂SeO₅ is well-suited for the design of high-precision optical devices.

The results presented in Fig. 4a–d and Table 3 clearly demonstrate the impact of SOC on the optical properties of Bi₂SeO₅. Under the influence of SOC, the static dielectric constant ε(0) increases significantly, with the extent of this increase varying across different layer numbers. This observation indicates an enhanced SOC effect as the material's thickness increases. SOC causes both ε₁(ω) and ε₂(ω) shift to lower energy regions, accompanied by a reduction in peak values. These shifts align with the SOC-induced bandgap reduction observed in the band structure, resulting in altered interband transition energies and modified optical absorption characteristics. Similarly, in the presence of SOC, the static refractive index n(0) increases, and both the α(ω) and n(ω) exhibit a red shift, along with a reduction in their peak values.

3.4 Piezoelectric properties

The piezoelectricity phenomenon was firstly discovered in 1880 by Pierre et al.,⁷⁴ and is explained as transformation of mechanical strain and vibration energy into electrical energy. Therefore, piezoelectricity offers the opportunities for implementing renewable energy through power harvesting.⁷⁵ According to the inversion asymmetric space group ABM2 for bulk Bi₂SeO₅, we reasonably believe the existence of piezoelectricity in Bi₂SeO₅. By employing the first-principles-based density functional perturbation theory (DFPT), we calculate the piezoelectric stress coefficients and the calculated e₃₃ component of the piezoelectric stress tensor is 0.54 C m⁻² for bulk Bi₂SeO₅, which is comparable to those of some typical three-dimensional materials.^76,77 We also calculate the e₁₁ components for 1L and 2L Bi₂SeO₅, which are 8 × 10⁻¹⁴ and 4 × 10⁻¹⁴ C m⁻¹, respectively. The piezoelectric stress coefficient of 2L is twice that of 1L Bi₂SeO₅, indicating the layer dependence and tunability.

4. Conclusions

In brief, we systematically study the layer-dependent properties of Bi₂SeO₅ through DFT calculations. 2D Bi₂SeO₅ follows the standard exponential relationship between the bandgap and the thickness. The SOC effect leads to a decrease of the bandgap and an increase of the effective mass. Importantly, the specific orbital distribution induces a significant difference of the SOC effect on VBs and CBs. Additionally, the vdW MTJ based on FeS₂/Bi₂SeO₅/FeS₂ exhibits strong TMR, which can be tuned to ∼1800%. The optical properties of 2D Bi₂SeO₅ are explored, which show an obvious red shift as the layer number increases. The computed piezoelectric coefficient also shows the layer dependence. Overall, vdW Bi₂SeO₅ provides a superior opportunity for new electronic and optical device applications.

Data availability

The data that support the findings of this study are available from the corresponding author, [Chenqiang Hua], upon reasonable request.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors thank Shengyuan A. Yang (from University of Macau) and Shenbo Yang (from HZWTECH) for help and discussions regarding this study. The authors acknowledge financial support from the Natural Science Foundation of Zhejiang Province (No. LY23E020010 and Q23A040023), the National Natural Science Foundation of China (Grants No. 12204029), and the funding of “Leading Innovative and Entrepreneur Team Introduction Program of Zhejiang” (Grant No. 2020R01002). Yao Wang acknowledges the funding from China Postdoctoral Science Foundation (2023M743098). Chenqiang Hua acknowledges the start-up funding from Hangzhou International Innovation Institute of Beihang University (Grant No. 2024KQ009).

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4cp03782k

‡ Yao Wang and Jinsen Zhang contributed equally to this work.

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