The second-order nonlinear Hall effect in 8-*Pmmn* monolayer borophene under the influence of an out-of-plane electric field and intrinsic spin–orbit interaction is reported. This unconventional response sensitive to the breaking of discrete and crystal symmetries can be tuned by the applied electric field, which can vary the bandgap induced by spin–orbit coupling. It is described by a Hall conductivity tensor that depends quadratically on the applied electric field. We find that the nonlinear Hall effect strongly depends on the spin polarization. In particular, it exhibits out of the phase character for spin-up and spin-down states. Remarkably, it undergoes a phase flip in the spin-up state at a large out-of-plane electric field that generates a staggered sublattice potential greater than the spin–orbit interaction strength. It is shown that the nonlinear Hall effect in the system originates from the broken inversion symmetry that plays an indispensable role in developing finite Berry curvature and its relevant dipole moment. It is found that at zero temperature, the nonlinear Hall response is maximal when the Fermi energy is twice the bandgap parameter and vanishes at large Fermi energies. Notably, the peak of nonlinear Hall response shifts to lower Fermi energies at finite temperature.

## I. INTRODUCTION

The nonlinear Hall effect (NLHE) has garnered significant attention in condensed matter physics due to its intriguing properties and potential applications.^{1–6} It is characterized by a Hall conductivity tensor that depends quadratically on the applied electric field and represents an atypical reaction observed in specific materials under certain experimental conditions. It deviates from the conventional Hall effect, which produces a voltage when a current flows at right angle to a magnetic field. The nonlinear Hall effect in a Hall-bar measurement setup involves the generation of a transverse voltage due to the interaction of two perpendicular currents.^{7} The NLHE originates from finite Berry curvature dipole moment, which is a measure of the distribution of the Berry curvature in momentum space and can be nonzero in systems lacking inversion symmetry. This phenomenon is complex and typically related to distinctive quantum attributes and material properties, offering valuable insights into the intricate interactions of charge carriers, symmetry, and topology in these materials. Understanding and manipulating these effects provide a platform for understanding the unique electronic behavior of materials and hold potential for applications in emerging areas of electronics and quantum technologies. In the nonlinear Hall effect, higher-order terms lead to the appearance of higher harmonics in the Hall voltage, which provide valuable insights into the complex nonlinear properties of materials and open up new possibilities for advanced technological applications in electronic and optoelectronic devices.^{1} The nonlinear Hall effect involves additional contributions that depend on higher powers of the electric field or current density. Such a nonlinear effect can arise from various mechanisms, including higher-order contributions in the electronic band structure, interactions between electrons, and the presence of nontrivial electronic states, such as those in topological materials. Nonlinear terms may become significant under certain conditions, such as high current densities or strong magnetic fields. The nonlinear Hall effect (NLHE) offers a wide range of mechanisms and applications in condensed matter physics and advanced technology. For instance, topological phase transitions in Moiré systems can be probed using the nonlinear anomalous (NLA) Hall effect.^{8} The nonlinear Hall effect in noncentrosymmetric quantum materials provides a promising avenue for developing broadband long-wavelength photodetectors.^{9} Interesting aspects of the nonlinear Hall effect have been reported. These include the long-lived valley polarization based nonlinear Hall Effect,^{10} nonlinear Hall effect realized in magnetic topological insulator flakes,^{11} the nonlinear Hall effect in systems with both inversion and time-reversal symmetry that generate a valley Hall current,^{12} NLHE in bulk $ T d\u2212 WTe 2$ realized as a quadratic response to an applied DC electric field,^{13} NLHE in magnetic insulator attached to a multiferroic material,^{14} NLHE observed in a spin-valley locked Dirac system,^{15} and NLHE at room-temperature reported in thin films of bismuth.^{16}. It has been shown that NLHE arises in systems that preserve time-reversal symmetry. However, it is very sensitive to the broken spatial inversion and crystal symmetry. Fundamental theoretical research on the nonlinear Hall effect has been reported.^{1,3} The finite Berry curvature dipole (BCD) induced NLHE has been reported in bilayer and few-layer $ WTe 2$.^{4,5} Recently, nonlinear Hall effects have also been reported in monolayer phosphorene,^{17} $ Bi 2 Se 3$.^{6,18}

On the other hand, borophene, a single layer of boron atoms arranged in a pattern of a two-dimensional (2D) lattice structure, has garnered significant interest due to its exceptional properties^{19,20} and potential applications.^{21} It is a new elemental single-layer, which hosts anisotropic and tilted massless Dirac fermions. It exhibits interesting electronic and transport properties; for instance, it undergoes a topological insulator phase transition^{22} like in silicene.^{23} The specific arrangement of boron atoms within the unit cell of the 8-*Pmmn* monolayer borophene would follow the constraints imposed by this space group symmetry. Different borophene structures can exhibit varied electronic properties, such as metallic, semiconducting, or topological behavior, depending on their atomic arrangement and bonding configuration.^{24} 8-*Pmmn* monolayer borophene exhibits a buckled crystal structure, where there is a vertical displacement between the two sublattices, resulting in a non-planar configuration. This structure typically retains inversion symmetry within each layer but breaks mirror symmetry perpendicular to the layer. This unique structure is a significant factor contributing to its interesting physical properties, such as its electronic and optical behavior. The buckling can enhance the spin–orbit coupling effects, making 8-*Pmmn* monolayer borophene an interesting candidate for spintronic applications. In systems with intrinsic spin–orbit coupling, breaking inversion symmetry can open a bandgap at the Dirac points. This effect is crucial for the transition of the material from a semi-metallic to an insulating state.

The application of an out-of-plane electric field to 8-*Pmmn* monolayer borophene can have significant effects on its effective charges and overall electronic properties. It has a buckled crystal structure that can exhibit inversion symmetry in the absence of an external electric field. Applying an out-of-plane electric field breaks this inversion symmetry that can lead to an asymmetric distribution of electronic charge density. Electrons may accumulate more on one side of the borophene layer, resulting in an effective polarization that leads to anisotropies in electronic and transport properties. In addition, the asymmetry introduced by the electric field can induce an electric dipole moment in the monolayer. This dipole moment affects the interaction of borophene with external electromagnetic fields,^{19} influencing its optical and electronic properties. Moreover, the Berry curvature, which plays a crucial role in anomalous and nonlinear Hall effects and other topological phenomena, can exhibit asymmetry when inversion symmetry is broken.

Several methods are used to enhance spin–orbit coupling (SOC) in 8-*Pmmn* monolayer borophene including adatom functionalization, substrate engineering, electric field application, and strain engineering. These effects can be harnessed to tune the electronic, optical, and transport properties of borophene, making it a versatile material for advanced applications in topological insulators, spintronics, optoelectronics, and quantum computing. Borophene is characterized by a tilt in its band structure, a characteristic feature for the existence of second-order nonlinear Hall effect. That is why it can be considered a prospective candidate for observing the nonlinear Hall effect. The nonlinear Hall effect in monolayer borophene is important due to its potential to unlock new functionalities in electronic devices, provide insights into fundamental electronic properties, and enhance performance in applications ranging from flexible electronics to quantum devices.

Breaking inversion symmetry in 8-*Pmmn* monolayer borophene is essential for observing the nonlinear Hall effect (NLHE). Inversion symmetry can be broken through several methods, including structural modifications, substrate interactions, and external field applications.^{25–27}

In this paper, the nonlinear Hall effect in 8-*Pmmn* monolayer borophene under the influence of applied transverse electric field in addition to spin–orbit interaction is investigated. Particular emphasis is focused on the spin-polarization effects in the nonlinear Hall effect. We consider the system that relies on inversion symmetry breaking and survives in time-reversal invariance. We find that the Hall response shows pronounced nonlinear effects inherited from the interplay between disorder scattering and inversion symmetry breaking effects in the presence of finite tilt in the Dirac cone of the band structure. The paper is organized as follows: Theory and model for the electronic transport of Dirac fermions in 8-*Pmmn* monolayer borophene is presented in Sec. II. The eigenstates and eigenvalues of the Hamiltonian are determined, followed by the evaluation of the Berry curvature. In Sec. III, the relaxation time and nonlinear Hall response function are computed. The nonlinear Hall effects at zero and finite temperature are analyzed. Finally, the results are concluded in Sec. IV.

## II. METHODOLOGY

In this section, we present the model and related theoretical background of the second-order nonlinear Hall effect in 8-*Pmmn* monolayer borophene under the influence of an out-of-plane electric field in addition to the spin–orbit interaction.

### A. Theory and model

^{22,28}

^{28}The first two-terms in Eq. (1) describe the Dirac fermions in borophene. The intrinsic spin–orbit coupling is described by the Kane–Mele term [see the third term in Eq. (1)], which plays a crucial role in opening a bandgap, $ \Delta SO$, in the energy spectrum. The SOC strength in 8-

*Pmmn*monolayer borophene is generally weak to moderate, given that boron atoms have a relatively low atomic number (Z = 5), resulting in weaker intrinsic SOC compared to heavier elements. However, it can be enhanced through similar strategies used for enhancing SOI in graphene.

^{29–33}The fourth term, $ \Delta z=el E z$ with $l=2.26\xc5$ in Eq. (1), includes the effects of an applied transverse electric field $ E z$. This term modifies the potential energy of the electrons, introducing an additional asymmetry in the system. It essentially breaks the inversion symmetry of a single-layer borophene through similar mechanism as used for phosphorene.

^{27}In this case, the Berry curvature can develop asymmetries in the $k$-space distribution, which can result in a non-zero integrated Berry curvature over the Brillouin zone. Moreover, in 8-

*Pmmn*monolayer borophene, the two types of inequivalent boron atoms, creating the buckled crystal structure, with opposite effective charges,

^{26}lead to significant modifications in the material’s response to an applied electric field. These modifications include induced polarization, asymmetric Berry curvature, and changes in the electronic band structure. These effects are crucial for understanding and exploiting the nonlinear Hall effect and other electronic phenomena in borophene, potentially leading to novel applications in optoelectronics and quantum devices. In particular, the applied electric field enhances asymmetry in the buckled structure and band structure, resulting in a more pronounced Berry curvature. As a consequence, the polarization of opposite effective charges on the inequivalent boron atoms under the influence of an electric field create a Berry curvature dipole that leads to a finite nonlinear Hall effect. Moreover, $ s z=\xb1$ characterizes the spin direction such that $ s z=+$ denotes the spin-up state and $ s z=\u2212$ stands for the spin-down state. $\sigma = ( \sigma x , \sigma y , \sigma z )$ represent the Pauli matrices, while $ 1$ is used for a unit matrix. The Hamiltonian in Eq. (1) can alternatively be described as

*Pmmn*monolayer borophene is characterized by the energy offset in Eq. (3). Moreover, the normalized eigenstates obtained by diagonalizing the Hamiltonian in Eq. (2) are given by

*Pmmn*monolayer borophene means that the Berry curvature will also exhibit anisotropy. The application of an out-of-plane electric field can lead to an enhanced Berry curvature in certain directions, depending on the field orientation relative to the borophene lattice. The Berry curvature tends to be concentrated around the Dirac points, and with the broken inversion symmetry, this concentration can become asymmetric, leading to distinct regions in momentum space with high Berry curvature. Moreover, the Berry curvature in a crystalline structure of 8-

*Pmmn*monolayer borophene exhibits specific symmetry properties that are determined by the underlying symmetries of the crystal lattice. These include the following symmetry properties: (i) It exhibits the character of an odd function in momentum-space, i.e., $ \Omega \lambda ( \u2212 k )=\u2212 \Omega \lambda ( k )$, implying that the system retains time-reversal symmetry. This ensures that any non-zero Berry curvature induced by the electric field will be antisymmetric in momentum space. (ii) For spin-polarized bands, the Berry curvature is opposite in sign for spin-up and spin-down states. (iii) In an inversion symmetry broken crystal structure, it remains finite.

## III. TRANSPORT PROPERTIES

*Pmmn*monolayer borophene subjected to an out-of-plane electric field in addition to intrinsic spin–orbit interaction are investigated. For better insight, we show the energy dispersion of 8-

*Pmmn*monolayer borophene around the $\Gamma $ symmetry point in Fig. 1. It can be seen that all bands are spin split for spin-up and spin-down states, which is evident from the comparison of the black dashed and blue solid curves in each panel. However, the bandgap for the spin-down state is larger than the one for the spin-up state. Comparison of Figs. 1(a) and Fig. 1(b) reveals that the bandgap increases with the increase in the electric field strength. To gain insight into the electronic transport in 8-

*Pmmn*monolayer borophene, we determine the density of states (DOS) described by

### A. Berry curvature

For understanding the transport characteristics in 8-*Pmmn* monolayer borophene, we analyze the nonlinear Hall response function and the related conductivity tensor. The buckled structure of 8-*Pmmn* monolayer borophene inherently creates an asymmetric environment for the boron atoms, which is further enhanced under an out-of-plane electric field. This results in a more pronounced Berry curvature and Berry curvature dipole. These effects are crucial for understanding and exploiting various transport phenomena, such as nonlinear Hall effect. These effects have been taken into account for determining the nonlinear Hall effect in this paper. For simplicity, the Fermi level is taken to be in the conduction band. Figure 3 shows the Berry curvature $\Omega $ for 8-*Pmmn* monolayer borophene. In particular, Fig. 3(a) shows the Berry curvature for the spin-up state and Fig. 3(b) shows that for the spin-down state using $ \delta z<1$. Likewise, Fig. 3(c) shows the Berry curvature for the spin-up state and Fig. 3(d) shows that for the spin-down state using $ \delta z>1$. Obviously, this figure exhibits the symmetry properties illustrated above. It is symmetric about $ k x=0$ and $ k y=0$. For further insight, we show the Berry curvature $\Omega $ in Fig. 4 as a function of $ k y$, keeping $ k x$ fixed and vice versa. In particular, Fig. 4(a) has been plotted as a function of $ k y$ for $ \delta z<1$ [see Fig. 4(a)] and $ \delta z>1$ [see Fig. 4(b)]. In each panel, the blue solid curve shows the Berry curvature for the spin-up state and the black dashed curve is used for the spin-down state. This figure shows that the Berry curvature is well pronounced in the region, $\u22120.3/a< k y<0.3/a$ with maximal around $ k y=0$. It, however, vanishes for large values of the crystal momentum $ k y$. It is obvious from this figure that the Berry curvature exhibits Gaussian-type behavior as a function of $ k y$ for fixed values of $ k x$, revealing reflection symmetry about $ k y=0$. Moreover, Fig. 4(a) shows that the Berry curvatures are in the opposite phase for spin-up and spin-down states. However, the Berry curvature for the spin-up state changes its phase at a large applied electric field, i.e., $ \delta z>1$ [see Fig. 4(b)]. In this case, the amplitude of the Berry curvature for the spin-up state is larger than the one for the spin-down state.

### B. Berry curvature dipole density

It is shown that the 8-*Pmmn* monolayer borophene exhibits a finite Berry curvature dipole (BCD) density, leading to finite nonlinear Hall response in the system. The Berry curvature dipole (BCD) density exhibits specific symmetry properties that are crucial for understanding the nonlinear Hall effect (NLHE). (i) It is an odd function of $ k y$, i.e., $ \u2202 \u2202 k y\Omega ( k x , \u2212 k y )=\u2212 \u2202 \u2202 k y\Omega ( k x , k y )$. (ii) It exhibits the character of an even function for $ k x$, i.e., $ \u2202 \u2202 k y\Omega ( \u2212 k x , k y )= \u2202 \u2202 k y\Omega ( k x , k y )$. For further insight, we demonstrate the Berry curvature dipole density in Fig. 5 obtained by evaluating numerically the first-order derivative of Eq. (5) with respect to $ k y$ for the spin-up state [Fig. 5(a)] and the spin-down state [Fig. 5(b)] using $ \delta z<1$, where spin-up state [Fig. 5(c)] and spin-down state [Fig. 5(d)] using $ \delta z>1$. It shows that the BCD density is symmetric about $ k x=0$ and asymmetric about $ k y=0$. Hence, the integration of BCD over the entire $k$-space leads to a finite nonlinear Hall response. For further understanding, we demonstrate the Berry curvature dipole density $ \u2202 k y\Omega $ as a function of $ k y$, keeping $ k x$ fixed in Fig. 6 for $ \delta z<1$ [Fig. 6(a)] and $ \delta z>1$ [Fig. 6(b)], whereas as a function of $ k x$, keeping $ k y$ fixed, for $ \delta z<1$ [Fig. 6(c)] and $ \delta z>1$ [Fig. 6(d)]. This figure reveals that the Berry curvature dipole density exhibits interesting symmetry properties. In particular, Figs. 6(a) and 6(b) reveal that it exhibits reflection asymmetry about $ k y=0$, whereas symmetry about $ k x=0$. It has shown that the crystal structure of 8-*Pmmn* monolayer borophene consists of two sublattices, with Dirac cones formed by $ p z$ orbitals of each sublattice.^{26} The tilt in the Dirac cone originates from anisotropic second-neighbor hopping on a buckled honeycomb lattice and can be tuned by atomic manipulations.^{34} This tilted structure leads to oblique and asymmetric Klein tunneling across NP and NPN junctions^{35} and asymmetric Berry curvature dipole. This asymmetry arises in the direction in which the tilt is taken. Moreover, it is well pronounced in the region, $\u22120.3/a< k y$ $<0.3/a$ and $\u22120.3/a< k x<0.3/a$ and remains minimal in the regions of large crystal momentum [see Figs. 6(a), 6(b), 6(c), and 6(d)]. Moreover, Figs. 6(b) and 6(d) reveal that the Berry curvature dipole density changes its phase for the spin-up state at a large electric field ( $ \delta z>1$); however, its amplitude is smaller than that for the spin-down state. Interestingly, BCD exhibits a maximum at $ k x=0$ [see Figs. 6(b) and 6(d)] and at $a k y\u2248\u22120.3$ and $a k y\u22480.3$ [see Figs. 6(a) and 6(b)].

### C. Nonlinear Hall response

^{36}

### D. Nonlinear Hall effect at zero temperature

*Pmmn*monolayer borophene are consistent with those investigated in tilted 2D Dirac quantum systems.

^{1,2}

### E. Nonlinear Hall effect at finite temperature

We have studied the nonlinear Hall effect at the temperature of $100$ and $200 K$. However, it can also be determined at $4$ or $77 K$, cryogenic temperatures where the experimental measurements of the nonlinear Hall effect can be performed.

It is indicated that the nonlinear Hall effect (NLHE) was studied in other tilted 2D Dirac quantum systems.^{1,2} However, our approach is different where we have focused on the spin-polarized nonlinear Hall effect. Additionally, we have considered the impacts of temperature that are not included in the cited references.

Likewise, the NLHE in monolayer borophene and monolayer phosphorene^{17} arises from the interplay of their crystal structure, electronic properties, and symmetry characteristics. Borophene’s NLHE is highly dependent on its phase and directional properties, whereas phosphorene’s NLHE is driven by its anisotropic effective mass and semiconductor nature. Both materials exhibit significant temperature dependence in their NLHE, although the mechanisms and specific effects differ due to their distinct band structures and Berry curvature distributions based on the different models for their description.

Moreover, the NLHE in 8-*Pmmn* monolayer borophene and topological insulators with hexagonal warping^{18} arises from different mechanisms rooted in their unique crystal structures, electronic properties, and Berry curvature distributions. In particular, the intrinsic anisotropy in borophene due to its buckled structure and tilted Dirac cones leads to the NLHE in the presence of an out-of-plane electric field, leading to the formation of anisotropic Berry curvature distribution. However, the hexagonal warping term in TIs introduces higher-order anisotropic terms in the electronic structure, contributing to the NLHE that can be tuned by adjusting the Fermi level, which affects the contributions from the warped surface states. Furthermore, we have taken into account the effects of spin polarization on the nonlinear Hall effect, which have not been considered in the previous studies.

## IV. CONCLUSIONS

In summary, the nonlinear Hall effect of Dirac fermions in two-dimensional 8-*Pmmn* monolayer borophene has been investigated. It is subjected to an out-of-plane electric field that breaks inversion symmetry of the system. In addition, the intrinsic spin–orbit interaction has also been included in the Hamiltonian. It has been found that the Berry curvature in 8-*Pmmn* monolayer borophene exhibits out-of-phase behavior for spin-up and spin-down states at low strength of the electric field, i.e., $ \delta z<1$. However, the Berry curvature for spin-up state flips its phase at a large electric field, i.e., $ \delta z>1$. Moreover, the amplitude of the Berry curvature for the spin-up state is larger than the one for the spin-down state. It has been shown that the Berry curvature exhibits important symmetry properties. In particular, it is symmetric about $ k x=0$ and $ k y=0$. Likewise, we have found that the 8-*Pmmn* monolayer borophene also shows the finite Berry curvature dipole (BCD) density, leading to a finite nonlinear Hall response. It has been shown that the BCD density is symmetric about $ k x=0$ and asymmetric about $ k y=0$, resulting in a finite Berry curvature dipole moment after integrating over the entire Brillouin zone. Notably, the Berry curvature dipole moment also undergoes a flip of phase for the spin-up state at a large electric field, where its amplitude is smaller than that for the spin-down state. It is found that $ D x$ vanishes identically, whereas $ D y$ remains finite at zero temperature. $ D y$ also vanishes for $ E F= \Delta SO , s z$. It means that the integral over the $k$-space vanishes in the absence of tilt in the Dirac cone. It occurs because the Berry curvature remains constant on the Fermi surface, while the group velocity in the $x$-direction is equal in magnitude but with opposite sign on the opposite sides of the Fermi surface, which leads to a zero Berry curvature dipole moment (BCDM). Remarkably, the BCDM is maximal around $ E F=2 \Delta SO , s z$ and vanishes at large Fermi energies. It has been demonstrated that in spin-up and spin-down states, the second-order nonlinear Hall response shows an out-of-phase character. However, it makes a phase flip at a large transverse electric field, i.e., $ \delta z>1$. At finite temperature, the Berry curvature dipole moment for the spin-up state decreases with increasing temperature, accompanied by a level broadening effect. This behavior can be understood through the interplay between the thermal excitation of electrons and the broadening of energy levels due to finite temperature effects. The smoothening of the Fermi–Dirac distribution reduces the sharpness of the contribution from states near the Fermi energy, and the broadening of energy levels further dilutes the effect. The Berry curvature dipole moment for the spin-down state decreases with increasing temperature, similar to the spin-up state. However, instead of a level broadening effect, the width of the Berry curvature peak decreases with increasing temperature. This difference in behavior can be attributed to the specific interactions and distribution of the Berry curvature for spin-down states, leading to a more concentrated contribution near the Fermi energy as the temperature rises. This nuanced understanding is crucial for optimizing the nonlinear Hall effect in materials with distinct spin states, particularly in spintronic applications where temperature control is essential.

It is illustrated that sensitivity to the spin polarization of carriers makes the nonlinear Hall effect a powerful tool for manipulating and detecting spin-related phenomena, with potential applications in emerging fields of science and technology. These findings are expected to hold promise for a variety of technological advancements, particularly in the fields of sensing, computing, and quantum technologies such as advanced electronic and spintronic devices, magnetic sensors.

Finally, it is demonstrated that our calculations are based on a free-standing 8-*Pmmn* monolayer borophene sheet. However, for the experimental measurement of the nonlinear Hall effect in monolayer borophene, it would be more realizable to grow the sample on a metallic substrate. This may need more meticulous theoretical modeling, which we leave for future study.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Abdullah Yar:** Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead). **Sumayya:** Conceptualization (supporting); Investigation (equal); Writing – original draft (supporting); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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