Intrinsic spin Hall and Rashba effects in metal nitride bromide monolayer for spin-orbitronics

The conceptual framework of relativistic spin–orbit coupling, emerging from the relativistic quantum-mechanical Dirac equation, stands as a foundational element contributing to various extraordinary phenomena. These include the anomalous Hall effect (AHE),¹ anisotropic magnetoresistance (AMR),² spin Hall effect (SHE), Rashba–Edelstein effect,³ and a plethora of others. This paper specifically directs attention toward the SHE and the Rashba effect. Notably, the SHE, initially envisioned by Mikhail I. Dyakonov and Vladimir I. Perel in 1971, involves the spin-splitting phenomenon induced by a longitudinal electric field.⁴ This results in the accumulation of different spins at opposite lateral ends of the sample, making the SHE a fundamental approach to generate pure spin currents. Its significance lies in its pivotal role in inducing spin–orbit torque (SOT) at the interface between normal metal (NM) and magnetic material (M) in SHE-SOT-based devices.^5,6 Moreover, SOT can also emerge from the Rashba effect, where an external electric field generates the non-equilibrium spin density at the interface characterized by broken inversion symmetry and out-of-plane mirror symmetry.^7–10 Consequently, it exerts a torque on the magnetization of a neighboring magnetic layer through exchange coupling. The realization of mechanisms such as the SHE and the Rashba effect in inducing SOT at interfaces has paved the way for a new paradigm in electronics, commonly referred to as spin-orbitronics.

The landscape of materials research has witnessed a transformative shift since the mechanical exfoliation of graphene, with a profound focus on 2D materials and miniaturization of spintronics devices. Beyond conventional charge-based applications, the next generation of devices explores additional degrees of freedom, including spin and valley. The intrinsic spin property associated with charge carriers, tunable via an external magnetic field, holds promise for next-generation high-speed spintronic devices, enabling the storage of large data for extended periods.^11–15 While many bulk materials have been explored for the SHE and the Rashba effect, the demand for promising spintronic properties in nanodevices necessitates the exploration of 2D materials. These materials exhibit strong coupling with feasible external perturbations such as strain and electric fields,^16,17 presenting heightened responsiveness compared to bulk counterparts. It underscores the significance of identifying ideal 2D materials for practical device realization.

In this context, our work delves into the exploration of the SnNBr monolayer, aiming to unravel its potential in spintronic-based applications. A tin (Sn) and nitrogen (N)-based system is anticipated to hold promise among the emerging 2D materials owing to its eco-friendly characteristics and intriguing chemical properties. While bromine (Br) is considered a higher-risk element than Sn and N, it is still less hazardous than elements like lead (Pb) when assessed from a sustainability perspective.¹⁸ The SnNBr monolayer is extensively studied theoretically for applications in piezoelectricity and ferroelasticity,^19,20 while the transformation of electrical charge into intrinsic spin and manipulation of spin transport remain relatively unexplored for the SnNBr monolayer. Our study employs first-principles calculations to investigate its charge-spin conversion, magnetization switching efficiency, and its applicability in spin-orbitronic-based devices, positioning the SnNBr system as a prospective material for futuristic nanoelectronics.

The manipulation of spin-related characteristics in two-dimensional materials emerges as a captivating avenue, effortlessly susceptible to control and tuning through strain, owing to the inherent flexibility of these materials. Strain, acting as a potent force, holds the capacity to reduce spatial symmetries within a system, even inducing non-centrosymmetry in initially inversion-symmetric materials. This induces spin splitting, Berry curvature dipole, and modulation in spin polarizations within the realm of two-dimensional compounds. The Rashba effect and the spin Hall effect, intricately derived from the electronic band structure, undergo significant transformations under the influence of strain. As a result, strain engineering emerges as a remarkably effective method for both enhancing and achieving the desired Rashba spin splittings and spin Hall effect. Also, it is noteworthy that the manufacturing processes of the materials inherently impose strain on the lattice, prompting a compelling exploration into the profound effects of strain on their physical properties. A study by Son et al.²¹ illustrated the tunability of valley magnetization and Berry curvature dipole in a MoS $2$ monolayer through strain. Further contributing to this discourse, Slawinska et al.²² delved into strain impact on the spin Hall conductivities for group-IV monochalcogenides, shedding light on the intricate interplay between strain and the spin-related characteristics of 2D materials.

The consequent sections of this paper are structured as outlined. The following Sec. II presents a concise overview of computational details. Section III, designating results and discussion section, is subdivided into four subsections. Section III A presents the analysis of structural and electronic properties, while Sec. III B focuses on describing and elucidating the tunability of Rashba physics in the SnNBr monolayer. In Sec. III C, a schematic analysis of the k-resolved spin Berry curvature is presented, accompanied by a discussion of the description and tunability of large spin Hall conductivity (SHC). Finally, in Sec. IV, conclusions drawn from all analyses and descriptions are summarized.

First-principles calculations utilizing the density functional theory (DFT)²³ have been conducted employing the Vienna ab initio Simulation Package (VASP version 5.3.4)^24,25 and the open-source software package for quantum simulations QUANTUM ESPRESSO.²⁶ The projector-augmented wave (PAW) scheme is adopted to describe the electron–ion interaction. The exchange-correlation functional is implemented by the generalized gradient approximation (GGA), in the form of Perdew–Burke–Ernzerhof (PBE) functional.²⁷ Both the self-consistent field (SCF) and non-self-consistent field (NSCF) calculations are carried out using fully relativistic ultra soft pseudopotentials.²⁸ For band structure calculation, a uniform k-point mesh of $20×20×1$ is employed to sample integrations over the Brillouin zone (BZ). An open-source package Wannier90 is utilized for generating the maximally localized Wannier functions (MLWF), and subsequently, MLWF advanced transport and Berry phase properties are calculated.^29–31 Post-processing using the Wannier90 code is conducted to calculate the k-resolved spin Berry curvature in the BZ, as implemented in the transport module of Wannier90. Moreover, the spin Hall effect or spin Hall conductivity calculations are based on Wannier interpolation. This approach to calculate a desired property is composed of two parts: the construction of the MLWF and the calculation of property based on the MLWF. Also, the intrinsic spin Hall conductivity based on Wannier interpolation is determined within the Kubo–Greenwood formula as considered in the Wannier90 package.^32,33

The monolayer configuration of SnNBr crystallizes in a trigonal crystal system, adopting the ABC prototype and falling under the space group P3m1 (space group number - $156$⁠). Intricacies of its crystal structure are visually represented in Figs. 1(a) and 1(b), delineating three distinct atomic layers. Notably, the central metal layer, featuring Sn atoms, is nestled between the nonmetal (N) and halogen (Br) atomic layers. Key structural parameters include lattice constants $a=b=3.47Å$ and a layer thickness/height of $h=2.79Å$⁠. The average bond lengths, denoted as $d$⁠, for different bonds are $d Sn − Br=2.904Å$ and $d Sn − N=2.115Å$⁠. The corresponding bond angle between Br, N, and Sn atoms is $∠( Br− N− Sn)= 84.78 °$⁠. The stability of the SnNBr monolayer is validated and confirmed through cohesive energies (⁠ $eV$⁠), phonon dispersions, and AIMD (Fig. S1 in the supplementary material).

The symmetries inherent in the crystal lattice are pivotal in predicting and understanding various properties of the material. In the case of the SnNBr monolayer, its non-centrosymmetric nature coupled with time-reversal invariance imparts four distinct time-reversal invariant momenta (TRIM) points within the BZ. The determination of various physical observables for the monolayer relies on eigenvalue analysis conducted at these four TRIM points. For the specific electronic property calculations in this study, emphasis is placed on the TRIM point $Γ$⁠, which maintains time-reversal invariance (⁠ $T$ invariant). Also, the $Γ$ point adheres to all symmetry operations of the $C 3 ν$ space group. It is noteworthy that at other points in the BZ, the group of wave vectors represents a subgroup of the symmetry group present at point $Γ$⁠, making accidental degeneracies possible only at these specific k-points.

Depiction of the electronic band structure for the SnNBr monolayer configuration, using only the PBE and HSE06 correlation functionals, is illustrated in Fig. 1(c). Subsequently, the spin–orbit coupling (SOC) effect is incorporated with the HSE06 functional in Fig. 1(d). Notably, the SnNBr monolayer manifests a distinctive direct bandgap nature with a magnitude of $1.81 eV$ without SOC (⁠ $1.53 eV$ with SOC) utilizing the HSE06 functional. Both the valence band maximum (VBM) and conduction band minimum (CBM) are precisely located at the TRIM point $Γ$ within the 2D BZ. The prominence of direct bandgap materials arises from their efficacy in light emission, robust optical absorption capabilities, broader wavelength range, and heightened device performance. Upon examining the orbital-projected band structure as depicted in Fig. 1(d), it becomes evident that the VBM of the SnNBr monolayer predominantly originates from the in-plane orbitals (⁠ $p x+ p y≈48.7%+48.7%≈97%$⁠) associated with the Br atom. In contrast, the CBM is influenced by the spherically symmetric $s$-orbital (non-directional) and the out-of-plane $p z$-orbital with lobes orientation (⁠ $s+ p z≈70.3%+28.2%≈98.5%$⁠) contributed by all three constituent atoms as explicitly depicted in Table S1 in the supplementary material. This intricate interplay of orbital contributions underscores the nuanced electronic properties of the SnNBr monolayer, holding significant implications for its potential applications in optoelectronic devices.

The inversion asymmetry and out-of-plane mirror asymmetry coupled with the influence of SOC engender an intrinsic out-of-plane electric field within the SnNBr system. This built-in electric field, in turn, plays a pivotal role in inducing Rashba spin splitting at the TRIM point $Γ$ of the conduction band in the SnNBr monolayer. To visually elucidate the manifestation of Rashba spin splitting within the SnNBr system, spin-projected electronic band structures have been plotted in Fig. 2, considering the influence of SOC along the high-symmetry lines denoted as $M−K−Γ− K ′− M ′$⁠. The inequivalence of the $K$ and $K ′$ points arises from the three-fold rotational symmetry (⁠ $C 3 ν$⁠) inherent in the SnNBr system. It is imperative to note that at the TRIM point $Γ$⁠, the bands exhibit Kramers degeneracy, i.e., each energy level or band in the electronic band structure is doubly degenerate. However, the presence of SOC and the out-of-plane electric field leads to the lifting of the Kramers degeneracy along the designated high-symmetry path $K−Γ− K ′$⁠. This intricate interplay of broken symmetries and their consequential effects on the electronic band structure underscores the rich and distinctive electronic properties of the SnNBr system.

The spin-projected band structure for in-plane spin components is illustrated in Figs. 2(a) and 2(b), with an enlarged view of the CBM shown in Fig. 2(c). It distinctly reveals the lifting of spin degeneracy in the vicinity of the TRIM point $Γ$⁠. The calculated value of $α R$ is $340meVÅ$⁠, significantly higher or comparable to the conventional monolayers such as MoSSe (⁠ $α R≈77meVÅ$⁠) and WSTe (⁠ $α R≈514meVÅ$⁠),^34,35 as well as other Rashba materials like interfaces of InGaS/InAlAs (⁠ $α R≈70meVÅ$⁠),³⁶ Au (111) (⁠ $α R≈330meVÅ$⁠),³⁷ and Bi (111) (⁠ $α R≈550meVÅ$⁠).³⁸ The recently reported ultrathick hetero-intercalated MoSi $2$N $4$(WN) monolayer with a metallic nature, has an even stronger Rashba coupling (⁠ $α R≈4300meVÅ$⁠).³⁹ Nevertheless, due to the presence of a semiconducting bandgap and non-centrosymmetric structure of the SnNBr monolayer, it demonstrates a significant in-plane and out-of-plane piezoelectricity.¹⁹ When mechanical perturbation is applied, the resulting piezo-potential varies proportionally to the piezoelectric coefficients. Consequently, this piezo-potential can act as a gate voltage to regulate the transport of electronic charge (or spin) in piezo-potential gated field-effect transistors (FETs) for emerging piezo-spintronic applications. Moreover, the spin textures in Fig. 2(d) illustrate that the CBM state exhibits an in-plane spin distribution around the $Γ$ point, with contrasting spin orientations (clockwise and counterclockwise) for the inner and outer rings, indicative of the Rashba spin splitting nature in 2D systems. Furthermore, Fig. 2(d) indicates that spin-momentum locking is more pronounced along the x-axis compared to the y-axis, emphasizing strong spin polarization and prominent Rashba spin splitting in the SnNBr monolayer.

To validate the tunability of the Rashba effect in the SnNBr monolayer, the impact of $−4%$⁠, $−2%$⁠, $+2%$⁠, and $+4%$ biaxial strains on the Rashba parameter and spin textures are explored in Fig. 3 (Table S3 and Fig. S3 in the supplementary material). The results demonstrate Rashba-type splitting at the TRIM point $Γ$ for both tensile (positive) and compressive (negative) biaxial strains. However, the magnitude of $α R$ is drastically influenced by the application of strain. $α R$ decreases to $≈170 meVÅ$ at $4%$ compressive biaxial strain and increases to $≈517 meVÅ$ at $4%$ tensile biaxial strain. Interestingly, these findings suggest a consistent increase (decrease) in the magnitude of $α R$ under tensile (compressive) biaxial strain. These modulations in the Rashba parameter $α R$ under biaxial strain can be interpreted by the change in the strength of the inherent, built-in out-of-plane electric field, visualized in terms of change in the out-of-plane planar average electrostatic potential difference (PAEPD). It enhances (decreases) under tensile (compressive) biaxial strain applications, as depicted in Table S2 and Fig. S2 in the supplementary material. Additionally, the contribution of the out-of-plane $p z$ orbital (by the Br atom) at the $Γ$ point of the CBM increases (decreases) under tensile (compressive) biaxial strain applications, as presented in Table S4 and Fig. S4 in the supplementary material. Consequently, the inherent out-of-plane electric field enhances, resulting in an increment (decrement) of the Rashba spin splitting at the $Γ$ point with tensile (compressive) biaxial strain. Notably, no significant change in the behavior of spin texture is observed under the application of different strains, as shown in Figs. 3(e)–3(h), consistently depicting strong spin polarization and prominent spin-momentum locking, thus affirming the persistent Rashba nature. Such stability is essential for applications requiring precise control over spin properties. Moreover, the mechanical and dynamical stabilities of the unstrained and strained SnNBr monolayers are ascertained, as illustrated in Table S5, Fig. S6, and Fig. S6 in the supplementary material. Overall, these outcomes underscore the reliability and versatility of the SnNBr monolayer for strain-tunable spintronic applications.

The SnNBr monolayer, being time reversal invariant, odd under inversion symmetry, and containing heavy elements, provides an ideal platform for the realization of the ISHE. Moreover, the SHC (⁠ $σ x y z$⁠) is calculated for different Fermi levels using Eq. (3), within the rigid band approximation in the SnNBr monolayer, as shown in Fig. 4(a). The SHC has attained a maximum value of $88.98$ $( h e) S / cm$ at energy $( E F−0.8 eV)$⁠, surpassing the reported value of SHC for GeS (⁠ $43$ $( h e) S / cm$⁠)²² and $PtT e 2$ (⁠ $27.28$ $( h e) S / cm$⁠).⁴⁰ Additionally, from Fig. 4(a), it is ascertained that the SHC is sensitive to the Fermi level. The conductivity diminishes exponentially as energy approaches $E F$⁠, and this behavior is pronounced owing to the absence of electronic states near the Fermi level. Such variations can be utilized in achieving the optimized and tunable charge-spin conversion efficiency. Another noticeable feature is that the significant SHC magnitude is not only resides at the Fermi level but prominent peaks are also observed in a small window of energy around the Fermi level, indicating the contribution of the gapped states to spin conductivity. The first and second valence bands (VB) at the $T$ invariant point $Γ$ lie slightly lower than the Fermi level and show a significant Zeeman-type spin splitting in the vicinity of the TRIM point $Γ$⁠, which is the underlying reason for the presence of a prominent peak below the initial Fermi level. This analysis is further strengthened by the SBC plot as presented in Fig. 4(b), where the SBC attains almost zero value along $K−M$ and $K ′− M ′$ directions, and reaches a maximum value at the $T$ invariant point $Γ$⁠, coinciding with the location of the first and the second VBM.

Moreover, to confirm the tunability of the ISHE in the SnNBr monolayer, the impact of biaxial strain on the SHC and the SBC is investigated and depicted in Figs. 4(c) and 4(d). It is observed that a $2%$ compressive/tensile biaxial strain has a negligible effect on the SHC of the SnNBr monolayer. On the other hand, a prominent effect is evident by the SnNBr monolayer under the application of $4%$ biaxial strain, as shown in Fig. 4(c). Also, the SBC attains its maximum value at the $T$ invariant point $Γ$ for both $+4%$ and $−4%$ biaxial strains and remains negligible at other high-symmetry points, as observed in Fig. 4(d). This finding is consistent with the unstrained case, as presented in Fig. 4(b). Additionally, the magnitude of the SBC decreases substantially due to strain applications. A significant decrease in the magnitude of the SBC under the application of $+4%/−4%$ strain results in a drastic decrease in the magnitude of the SHC. The maximum value of the SHC drops to $36.77$ $( h e) S / cm$ at (⁠ $E F−1.4$ eV) for $4%$ tensile strain and $49.45$ $( h e) S / cm$ at (⁠ $E F−1.6$ eV) for $4%$ compressive strain. The SHC magnitude at $4%$ tensile/compressive strain decreases to nearly half of that in an unstrained SnNBr monolayer, although the overall behavior of the SHC remains consistent with strain. The impact of both $4%$ tensile and compressive biaxial strain on the ISHE of the SnNBr monolayer exhibits similar trends, which is further evident from the biaxial strain effect on the band structure, as depicted in Fig. S2 in the supplementary material. The band structure displays a Zeeman-type spin splitting at the first and the second VB near the $T$ invariant point $Γ$ in both unstrained and strained cases. However, for higher strain values, the system does not exhibit any distinct behavior beyond that observed for $4%$ tensile and compressive biaxial strain. Altering the Fermi level position through electron or hole doping or applying gate voltage can modulate spin Hall conductivity; such methods are effective only if they do not alter the band structure. Nonetheless, within the confines of the rigid band approximation, doping offers a means to significantly modulate spin conductivity. Altogether, the potential tunability and sensitivity of the ISHE, consequently the SHC and the SBC, have significant implications for the applicability and functionality of SnNBr-based spintronic devices.

In conclusion, our calculations unveil significant and tunable Rashba and spin Hall effects in the SnNBr monolayer. The novelty and significance of these findings lie in demonstrating the potential for both charge-spin conversion and charge-based spin density accumulation within the SnNBr monolayer. Additionally, the monolayer offers the possibility of optimizing these effects through in-plane biaxial strain. Furthermore, our investigation highlights the potential of tuning and optimizing the spin Hall effect by making slight adjustments to the Fermi level. To further confirm the possibility of spin-charge conversion, we demonstrate monolayer spin texture, confirming strong spin-momentum locking within the studied system. An analysis of the k-resolved spin Berry curvature sheds light on the substantial spin Hall conductivity and its tunability under applied strain. Overall, this research underscores the promising potential of the SnNBr monolayer in the realms of spintronics and spin-orbitronics. We envision that our research will inspire further experimental investigations in this area.

See the supplementary material for additional discussions of the SnNBr monolayer: Sec. 1—Thermodynamical, dynamical, and thermal stability (Fig. S1); Sec. 2—Atomic orbital contributions projected on band extrema (VBM and CBM) for the SnNBr monolayer (Table S1); Sec. 3—Calculated structural parameters and its strain modulations (Table S2, Fig. S2); Sec. 4—Rashba parameter and its strain modulations (Table S3, Fig. S3); Sec. 5—Atomic orbital contributions of the strained SnNBr monolayer (Table S4, Fig. S4); Sec. 6—Mechanical stability of the biaxially-strained SnNBr monolayer (Table S5, Fig. S5); Sec. 7—Dynamical stability of the biaxially-strained SnNBr monolayer (Fig. S6); Sec. 8—Rashba parameter and its electric field modulations (Table S6, Fig. S7); Sec. 9—Insights into interatomic bonding nature of the SnNBr monolayer (Fig. S8).

The authors express their gratitude to the Institute of Nano Science and Technology (INST), Mohali, India, for providing fellowship support. Computational analyses were conducted utilizing the supercomputing resources available on the in-house High-Performance Computing (HPC) cluster at the Institute of Nano Science and Technology (INST), Mohali, India, as well as on PARAM Smriti, housed at the National Agri-Food Biotechnology Institute (NABI), Mohali, under the National Supercomputing Mission, Government of India. Support from Project No. CRG/2023/001896 of Science and Engineering Research Board—Department of Science & Technology (SERB-DST), Govt. of India is gratefully acknowledged.

The authors have no conflicts to disclose.

Pradip Nandi: Data curation (equal); Software (equal). Shivam Sharma: Data curation (equal); Software (equal). Abir De Sarkar: Conceptualization (equal); Supervision (equal). Pradip Nandi and Shivam Sharma have equally contributed.

The data that support the findings of this study are available from the corresponding author upon reasonable request.