Quantum metric non-linear Hall effect in an antiferromagnetic topological insulator thin-film EuSn₂As₂

First discovered by Edwin Hall in the 19^th century, the Hall effect elucidates the generation of transverse voltage when both a longitudinal current and an external magnetic field are concurrently applied  [1]. Later, Hall identified that this transverse Hall voltage can manifest in ferromagnetic materials even without an external magnetic field, a phenomenon he termed the anomalous Hall effect  [2]. With advancements in research on the geometric phase of electronic systems, we now understand that the linear component of the anomalous Hall effect arises from the Berry curvature induced by the broken time-reversal symmetry in solid-state systems  [3], [4].

The non-linear Hall effect (NLHE) has recently garnered global attention beyond the linear response of the anomalous Hall effect, sparked by experimental observations of a transverse non-linear Hall voltage even under conditions of time-reversal symmetry  [5], [6], [7] affirming the theoretical prediction linking NLHE to the Berry curvature dipole (BCD)  [8]. Subsequent, numerous potential applications leveraging NLHE have been proposed, including terahertz detection and rectification  [9], [10], [11], along with its utility as an indicator of topological transitions  [12].

Recent research efforts have predominantly delved into exploring the NLHE within antiferromagnetic (AFM) systems that uphold the space-time reversal $(PT)$ symmetry  [13], [14], [15], [16], [17]. In such systems, the $PT$ symmetry mandates the vanishing of both the Berry curvature and the BCD across the entire Brillouin zone (BZ). Given the inability of BCD to induce NLHE under $PT$ symmetry, an alternative origin, the quantum metric dipole, has been proposed to evoke NLHE in AFM materials  [18]. Within this framework, the non-linear Hall conductivity can be expressed as:

$\displaystyle\sigma^{\alpha\beta\gamma}=2e^{3}\sum_{n,m}^{E_{n}\neq E_{m}}\int\frac{d^{3}k}{(2\pi)^{3}}$

$\displaystyle\frac{v^{\alpha}_{n}g^{\beta\gamma}_{n}}{E_{n}-E_{m}}\frac{\partial{f(E_{n})}}{\partial{E_{n}}}-(\alpha\leftrightarrow\beta)$

(1)

, where $n$ and $m$ denote indices for different bands, $E_{n}$ represents the energy of the Bloch state, $v^{\alpha}_{n}$ signifies the band velocity, and $g^{\beta\gamma}_{n}$ stands for the quantum metric tensor, the real part of quantum geometry  [19]. The quantum metric tensor $g^{\beta\gamma}_{n}$ measures the distance between neighboring Bloch states in Hilbert space, defined as $g^{\beta\gamma}_{n}=Re\sum_{i,j}^{n\neq m}[A_{ni,mj}^{\beta}A_{mj,ni}^{\gamma}]$, where ${\bf\it A}_{n,m}=i\langle u_{n}|\nabla_{k}|u_{m}\rangle$ denotes the Berry potential with the periodic part of the Bloch state  [13], [18]. Thus, $v^{\alpha}_{n}g^{\beta\gamma}_{n}$ can be interpreted as the quantum metric dipole. Remarkably, this Hall conductivity remains independent of relaxation time, originating from the inherent nature of electronic structure, termed the intrinsic non-linear Hall effect (INHE)  [13], [18]. Recent studies suggest utilizing INHE in rectification devices or generating DC currents within the WiFi frequency range  [11], [20]. Given that device efficiency hinges on the NLHE, the pursuit of materials exhibiting substantial NLHE becomes pivotal, not only in condensed matter physics but also in material sciences. According to Eq. (1), a large INHE can naturally be expected in band structures characterized by small gaps and large Fermi velocities. Moreover, tilted band dispersion is a crucial requisite to prevent quantum metric cancellation in the BZ  [13], [14]. Following this rationale, tilted Dirac surface states in topological materials with in-plane antiferromagnetic spin configuration emerge as promising candidates.

In this work, we propose utilizing the AFM topological insulator EuSn₂As₂  [21] with an even-numbered layer slab thin-film structure as a promising candidate to realize the INHE induced by the quantum metric dipole. This thin-film configuration preserves the $PT$ symmetry, with an additional $k_{x}=0$ mirror plane $M_{[100]}$ protecting a nearly gapless topological surface state on the (001) plane. Our calculations reveal a significant non-linear Hall conductivity around this Type-II surface Dirac point. Specifically, the non-linear Hall conductivity of EuSn₂As₂ exceeds 20 mA/V², surpassing previous findings by an order of magnitude (see Table 1). Furthermore, we demonstrate that the INHE of EuSn₂As₂ can be dynamically tuned by rotating the direction of the spin moment with a period of $2\pi$. This tunable characteristic holds promise for enhancing the read-out information of Neel vectors in AFM spintronics  [22],  [23],  [24],  [25]. The single Dirac band dispersion serves as the simplest model to generate INHE. Hence, the even-numbered layer EuSn₂As₂ slab thin-film can be likened to the hydrogen atom of INHE. Additionally, apart from the INHE stemming from surface Dirac band dispersion, our calculations unveil other substantial non-linear Hall conductivity induced by anti-crossing states at the band-edge-states near the Fermi level. Moreover, introducing a 30 $\%$ doping of phosphorus (P) pushes the non-linear Hall conductivity to approximately $45$ mA/V², markedly enhancing the functionality of EuSn₂As₂.

For the second-order response, the relationship between current and electric field is described by $J\sim\sigma E^{2}$, where $\sigma$, $J$, and $E$ represent the second-order non-linear conductivity, electrical current, and electric field, respectively. Since $E^{2}$ is $P$ and $T$ even and $J$ is $P$ and $T$ odd, the second-order Hall conductivity must be $P$ and/or $T$ odd, necessitating the breaking of spatial inversion and/or time-reversal symmetry within the system. In Eq. (1), $\sigma^{\alpha\beta\gamma}$ denotes the non-linear Hall conductivity, exhibiting antisymmetry under $\alpha\leftrightarrow\beta$ and symmetry under $\beta\leftrightarrow\gamma$ permutations. Thus, in two dimensions (2$D$), only two independent components of the non-linear Hall conductivity, $\sigma^{yxx}$ and $\sigma^{xyy}$, exist. To analyze the symmetry constraints imposed by point group operations, the conductivity tensor in a 2$D$ system can be transformed into an equivalent pseudovector form  [26]:

$$\sigma^{\gamma}\equiv\epsilon^{\alpha\beta}\sigma^{\alpha\beta\gamma}/2$$

(2)

. Here, $\alpha$, $\beta$, and $\gamma$ are spatial indices, and $\epsilon^{\alpha\beta}$ denotes the 2$D$ Levi-Civita symbol. According to this transformation rule, $\sigma^{yxx}$ and $\sigma^{xyy}$ can be rewritten as $\sigma^{x}$ and $\sigma^{y}$ (pseudovectors along the $\hat{x}$ and $\hat{y}$ directions), respectively. The transformation of the pseudovector under point group symmetry operation is expressed as:

$$\sigma^{n}=\eta_{T}det(R)R^{n\gamma}\sigma^{\gamma}$$

(3)

, where $n$ represents the spatial index, $R$ signifies the point group operation, and $\eta_{T}=-1$ denotes the magnetic symmetry operations of the form $PT$  [14]. Thus, the non-linear Hall conductivity induced by the quantum metric dipole is time-reversal odd. The direction of the nonzero pseudovector $\sigma^{n}$ in the 2$D$ system must be orthogonal to the crystalline mirror planes. Consequently, the non-linear conductivity is completely suppressed in a 2$D$ material with two or more mirror planes, thereby establishing one mirror plane as the highest crystalline reflection symmetry permitted in a 2$D$ system.

The crystal structure of EuSn₂As₂ adopts a rhombohedral lattice akin to that of Bi₂Se₃, where Eu atoms interconnect with the Sn-As hexagonal network to form hexagonal layers that stack along the $c$ direction. The space group of EuSn₂As₂ is No. 166. Previous studies have identified the magnetic ground state of EuSn₂As₂ as an in-plane A-type antiferromagnet  [21]. The band structure calculations reveal that EuSn₂As₂ preserves multiple topological invariant numbers  [21],  [27]. The side surface exhibits $S$-symmetry-protected topological surface states. Additionally, a mirror-symmetry-protected gapless Dirac surface state is present away from $\bar{\Gamma}$ on the (001) surface, as confirmed by ARPES  [21],  [27]. Due to the bulk structure’s centrosymmetry, second-order conductivity is forbidden by symmetry constraints. By contrast, a finite-size 2$D$ slab structure breaks this centrosymmetry. We define the number of layers in the EuSn₂As$2$ thin-film by counting the Eu layers in the slab model. For instance, Fig. 1(a) and Fig. 1(b) depict 5-layer and 4-layer slab thin-films in our definition, respectively. Notably, an odd-number of layers maintains inversion symmetry akin to the bulk structure, while an even-number of layers breaks inversion symmetry. The remaining symmetry operators in the even-numbered layer slab geometry of EuSn₂As₂ are $PT$, $C_{2[100]}T$, and $M_{[100]}$. Adhering to this symmetry constraint, only $\sigma^{yxx}$ (pseudovector along the $\hat{x}$ direction) behaves as an even function under $M_{[100]}$ (Fig. 1(d)). Consequently, as depicted in the schematic in Fig. 1(e), a non-linear Hall voltage is anticipated to emerge along the $\hat{y}$ axis in response to an external electric field along $\hat{x}$.

The bulk band structure calculations for EuSn₂As₂ were conducted employing the projected augmented wave method as implemented in the VASP package, utilizing the generalized gradient approximation (GGA)  [28], [29] and GGA plus Hubbard $U$ (GGA+$U$) scheme  [30]. For the Eu 4-$f$ orbitals, an on-site $U$ value of 5 eV was employed. Experimental structural parameters were adopted  [21], [31]. Spin-orbit coupling (SOC) was self-consistently included in the calculations using a Monkhorst-Pack k-point mesh of 23 $\times$ 23 $\times$ 3. Wannier functions were constructed from Eu $d$, $f$, Sn $p$, and As $p$ orbitals without the need for the maximization of localization procedure  [32], [33].

Figure 2(a) depicts the band structure of a 4-layer slab model of EuSn₂As₂ exhibiting in-plane A-type antiferromagnetism. We symmetrically construct a finite-size slab model terminated with As atoms. The band dispersion reveals a mirror symmetry protected fourfold-degenerate Dirac point on (001) along $\overline{M}-\overline{\Gamma}$, denoted as DP in Fig. 2(a). An enlarged view of the band structure (inset of Fig. 2(a)) reveals a Type-II Dirac band dispersion formed by the crossing between the conduction and valence bands. This band feature resembles the topological surface state of bulk EuSn₂As₂  [21], suggesting that the 4-layer slab thickness is sufficient to exhibit surface Dirac band dispersion (see Supplementary material). The corresponding intrinsic non-linear Hall conductivity $\sigma^{yxx}$ is illustrated in Fig. 2(b). Our calculations unveil three significant peaks around the Fermi level, labeled $\alpha$, $\beta$, and $\gamma$. Notably, the maximum conductivity exceeds 20 mA/V² at 20 K, which is an order of magnitude larger than previous studies (see Table 1), suggesting EuSn₂As₂ thin-film as a promising candidate for enhancing the operating efficiency of rectification devices. Non-linear Hall conductivity at different temperatures is presented in the Supplementary material.

To elucidate the origin of the INHE in EuSn₂As₂, we plot the intensity of the quantum metric dipole $D_{qm}=v_{y}g_{xx}-v_{x}g_{yx}$ on the band structure, as depicted in Fig. 2(a). Our calculations reveal high intensity of $D_{qm}$ around the surface Dirac point. Owing to the characteristics of the Type-II Dirac cone, the large slope of band dispersion and small energy gap simultaneously augment and diminish the Fermi velocity $v^{\alpha}_{n}$ and energy gap $(E_{n}-E_{m})$ in Eq. (1), respectively. These concurrent changes amplify the quantum metric $g^{\beta\gamma}_{n}$, resulting in a pronounced intensity quantum metric dipole $D_{qm}$ around the Dirac point  [14]. Further, to identify the source of peak $\alpha$, we plot the constant energy contour at the energy of peak $\alpha$ and the intensity of $D_{qm}$ in Fig. 2(d). Our calculations indicate that the Dirac point is the sole hot spot, with no comparable contribution elsewhere in the BZ, establishing that this significant value of INHE stems entirely from the topological surface state. Crucially, the 2$D$ Dirac band dispersion represents the simplest model for generating INHE. Therefore, EuSn₂As₂ thin-film can be likened to the hydrogen atoms of INHE, providing a practical material platform for studying INHE and related properties. Additionally, the effects of strain and thickness on the INHE are discussed in detail in the Supplementary Material.

Given the high sensitivity of the surface Dirac cone and non-linear Hall conductivity of EuSn₂As₂ to crystalline symmetry, altering the direction of the spin moment presents a viable strategy for controlling the INHE. Figure 3 illustrates the non-linear Hall conductivity $\sigma^{yxx}$ and $\sigma^{xyy}$ originating from the surface Dirac cone with different in-plane spin orientations. Here, $\theta$ in Fig. 3 denotes the angle between the spin moment vector and the $x$ axis. Notably, the non-linear Hall conductivity follows the relationship $\sigma(\theta)=-\sigma(\theta+\pi)$, demonstrating a periodicity of 2$\pi$. Specifically, we observe that $\sigma^{yxx}$ achieves its maximum value while $\sigma^{xyy}$ remains at 0 when $\theta$ = 0 (spin moment aligned with the $x$ axis), owing to the enforcement of $M_{[100]}$ causing $\sigma^{xyy}$ to vanish as per Eq. (3). The illustration of the mirror plane with spin configuration is depicted in Fig. 1(c). As the spin direction deviates from the $x$ axis, the protection of the surface Dirac cone by $M_{[100]}$ is compromised, leading to gap opening and a decrease in $\sigma^{yxx}$ intensity. Conversely, with the destruction of $M_{[100]}$, $\sigma^{xyy}$ is no longer restricted, resulting in an increase in its value. At $\theta=\pi/2$, $\sigma^{yxx}$ vanishes due to enforcement by $M_{[120]}$, while $\sigma^{xyy}$ exhibits significant non-linear Hall conductivity. This tunability not only facilitates the detection of EuSn₂As₂ symmetry but also holds potential for advancing spintronics, such as measuring the Neel vector in AFM systems  [13] [14].

In addition to the $\alpha$ peak in Fig. 2(b) originating from the surface Dirac cone, we observe two additional peaks, $\beta$ and $\gamma$, whose values are comparable to $\alpha$, despite their energies being distant from the Dirac point. To delve into the origins of these two peaks, we plot the constant energy contour with the intensity of $D_{qm}$ corresponding to $\beta$ and $\gamma$ as depicted in Fig. 2(e) and 2(f), respectively. From Fig. 2(a) and 2(e), we discern that the $\beta$ peak is contributed by the $D_{qm}$ accumulated along the $\overline{\Gamma}-\overline{K}$ direction. This heightened $D_{qm}$ strength arises from the small band gap induced by the anti-crossing of band-edge-states, denoted as BES Fig. 2(a). Similarly, the origin of the $\gamma$ peak also stems from the intricate anti-crossing of band-edge-states. However, the $D_{qm}$ is not localized at high symmetry lines but distributed across generic points in the BZ (Fig. 2(f)). The nonlinear Hall conductivity for the $\beta$ and $\gamma$ peaks with different in-plane spin orientations is provided in the Supplementary material.

Apart from EuSn₂As₂, we also examine its counterpart, EuSn₂P₂. Figure 4(c) illustrates the non-linear Hall conductivity $\sigma^{yxx}$ of EuSn₂As₂ and EuSn₂P₂ with a 4-layer slab geometry. Notably, we observe that the intensity of $\sigma^{yxx}$ in EuSn₂P₂ is considerably lower than that in EuSn₂As₂ near the Fermi level. In Fig. 4(a), the band structure of EuSn₂P₂ with a 4-layer slab geometry is depicted. Here, we discern a surface Dirac band with an energy gap of approximately 10 meV in EuSn₂P₂, whereas it remains nearly gapless in EuSn₂As₂ at the same thickness. Moreover, the band-edge-states of EuSn₂P₂ are significantly distant from the Fermi level. These band characteristics hinder EuSn₂P₂ from generating a substantial quantum metric dipole $D_{qm}$, thereby reducing the intrinsic non-linear Hall conductivity.

Given the significant impact of energy band characteristics on INHE, we propose P-doped EuSn₂As₂, denoted as EuSn₂As_2(1-x)P_2x, as an experimental platform for controlling INHE. Specifically, we calculate $\sigma^{yxx}$ of EuSn₂As_2(1-x)P_2x using a linear interpolation of tight-binding model matrix elements of EuSn₂As₂ and EuSn₂P₂. Since tight-binding parameters encapsulate crucial information such as lattice constants and atomic bonding strength, interpolation is expected to capture all systematic changes in the electronic structure between the two endpoints. Thus, this approach proves highly reliable and effective for studying iso-valence substitution/doping  [34]. The calculated $\sigma^{yxx}$ with different doping ratios is presented in Fig. 4(d). Interestingly, we observe that the intensity of the $\alpha$ peak monotonically decreases with P doping, while the behavior of the $\beta$ peak is more intricate. The intensity of the $\beta$ peak gradually increases before reaching 20$\%$ doping. At 30$\%$ doping, $\sigma^{yxx}$ surpasses 40 mA/V², more than twice that of the undoped case. However, upon exceeding 40$\%$ doping, the intensity of $\sigma^{yxx}$ decreases rapidly. These findings underscore the effectiveness of doping as a means to control INHE in EuSn₂As₂.

Nonreciprocal transport within semiconductor $p$-$n$ junctions constitutes a cornerstone of contemporary microelectronics, pivotal in the operation of diodes and transistors. However, traditional semiconductor junctions, particularly diodes, confront inherent limitations at high frequencies and low input voltages. Recent research underscores the potential of quantum geometric effects as a fundamentally novel operational principle for diodes. Consequently, materials exhibiting substantial intrinsic nonreciprocity can serve as diodes and are termed diodic quantum materials  [10],  [35],  [36],  [37]. By virtue of their junction-free nature, diodic quantum materials transcend the constraints associated with semiconductor junctions, functioning as high-efficiency rectifiers capable of operating at terahertz frequencies, zero bias voltage, and ultralow input power. Thus, the quest for quantum materials offering a new platform for quantum phenomena and technology is of paramount importance.

Our study demonstrates that the AFM topological material EuSn₂As₂, with even-numbered layer slab geometry, exhibits significant non-linear Hall conductivity $\sigma^{yxx}$, stemming from Type-II surface Dirac band dispersion and anti-crossing band gaps of band-edge-states. Moreover, appropriate P doping can substantially enhance the value of $\sigma^{yxx}$. Given the direct relationship between device application efficiency and non-linear Hall voltage, the large non-linear Hall conductivity observed in the EuSn₂As₂ thin-film holds promise for improving device operating efficiency. An exciting future breakthrough would be to demonstrate room-temperature wireless rectification based on the INHE in an AFM material.

Beyond the A-type AFM configuration explored in this work, recent studies propose that NLHE can also manifest in more intricate magnetic configurations, such as altermagnetic materials  [38] and non-collinear antiferromagnetism  [39]. In addition to intrinsic effects arising from the band structure, disorder plays a crucial role in inducing NLHE in PT-symmetric antiferromagnetic materials  [40],  [41],  [42],  [43],  [44]. Intriguingly, this extrinsic effect can surpass intrinsic contributions in magnitude, emphasizing the need for further theoretical investigations to clarify its impact on real materials. Exploring the interplay between these intrinsic and extrinsic mechanisms, as well as their implications for potential spintronics applications, remains an open and exciting avenue for future research.

T.-R.C. was supported by National Science and Technology Council (NSTC) in Taiwan (Program No. MOST111-2628-M-006-003-MY3 and NSTC113-2124-M-006-009-MY3), National Cheng Kung University (NCKU), Taiwan, and National Center for Theoretical Sciences, Taiwan. This research was supported, in part, by the Higher Education Sprout Project, Ministry of Education to the Headquarters of University Advancement at NCKU. T.-R.C. thanks the National Center for High-performance Computing (NCHC) of National Applied Research Laboratories (NARLabs) in Taiwan for providing computational and storage resources. H.L. acknowledges the support by Academia Sinica in Taiwan under grant number AS-iMATE-113-15. L.F. was supported by the Air Force Office of Scientific Research under award num ber FA2386-24-1-4043. The Global Seed Funds provided by MIT International Science and Technology Initiatives (MISTI) under the program of MIT Greater China Fund for Innovation is acknowledged.