Nonlinear Hall effect in topological Dirac semimetals in parallel magnetic field

This paper theoretically investigates the nonlinear Hall effect in two-dimensional topological Dirac semimetals under an in-plane magnetic field by solving the quantum kinetic equation for the Wigner distribution function and proposes experimental verification through anomalous Hall resistivity measurements in specific materials like SnTe, WTe$_2$, WSe$_2$, and Ce$_3$Bi$_4$Pd$_3$.

The Gist

Imagine you are driving a car on a very special, bumpy road. Usually, if you press the gas pedal (apply an electric field), the car moves forward. But in the world of quantum physics, specifically in materials called Topological Dirac Semimetals, things get weird. Sometimes, pressing the gas makes the car drift sideways, even if there's no steering wheel turned. This is the Hall Effect.

This paper explores a "super-powered" version of this drift, called the Nonlinear Hall Effect, and discovers a new way to control it using a magnetic field.

Here is the breakdown of their discovery using simple analogies:

1. The Setting: The Quantum Dance Floor

Think of the material (like a thin sheet of Tungsten Diselenide, or WSe₂) as a crowded dance floor. The dancers are electrons.

• The Geometry: The dance floor isn't flat; it has invisible "slopes" and "twists" built into the floor itself. In physics, this is called Berry Curvature.
• The Dipole: Because the floor is tilted in a specific way (due to strain in the material), the dancers naturally want to drift to one side when they move. This is the Berry Curvature Dipole. It's like a built-in bias that makes the crowd lean left or right.

2. The Old Trick: The Built-in Drift

Scientists already knew that if you push the dancers (apply an electric current), they would drift sideways because of that built-in tilt.

• The Catch: This drift is usually fixed. You can't easily change it without changing the material itself. It's like having a car with a broken steering wheel that always turns left.

3. The New Discovery: The Magnetic "Wind"

The authors of this paper asked: "What happens if we blow a strong wind (a magnetic field) parallel to the dance floor?"

They found that this "wind" does two amazing things:

1. It creates a new drift: The magnetic field pushes the dancers sideways in a way that depends on how strong the wind is.
2. It acts as a volume knob: By changing the direction of the magnetic wind, you can either boost the built-in drift or cancel it out completely.

The Analogy: Imagine you are walking on a moving walkway at an airport (the built-in drift).

• If you walk with the wind, you go super fast.
• If you walk against the wind, you might not move at all.
• The magnetic field is that wind. The researchers figured out exactly how to tune the wind to control your speed and direction perfectly.

4. The "Second Harmonic" (The Echo)

The title mentions "Second-Harmonic Response." In music, if you play a note, the second harmonic is an echo at double the frequency.

• In this experiment, they wiggle the electric field back and forth (like a wiggly dance move).
• The resulting sideways current doesn't just wiggle back and forth; it wiggles twice as fast.
• This "double-speed echo" is the signal they measure. It's a very sensitive way to detect these tiny quantum effects.

5. Why Does This Matter?

The paper suggests this isn't just theoretical math; it's something we can test in a lab.

• The Materials: They suggest testing this on materials like SnTe (a topological insulator), WTe₂, and Ce₃Bi₄Pd₃ (a material with heavy, slow-moving electrons).
• The Application: This gives us a new "knob" to control electricity. Instead of just turning current on or off, we can use magnetic fields to steer electron currents in complex ways without moving parts. This could lead to faster, more efficient electronic devices or new types of sensors.

Summary

Think of the electrons in these materials as cars on a curved track.

• Before: The track was curved, so the cars naturally drifted sideways. You couldn't stop it.
• Now: The authors found that by applying a magnetic field (like a strong gust of wind), you can steer those cars. You can make them drift harder, stop drifting, or even drift in the opposite direction.

They used complex math (Quantum Kinetic Equations) to prove this works and showed that the effect is strong enough to be measured in real experiments. It's a new tool for the "quantum toolbox" that lets us manipulate electricity with magnetic fields in a way we couldn't do before.

Technical Summary

1. Problem Statement

The paper addresses the theoretical understanding of the nonlinear Hall effect (NHE) in two-dimensional (2D) topological Dirac semimetals when subjected to an in-plane magnetic field.

• Context: The NHE is a second-order response where a transverse current ($j_y \propto E_x^2$) is generated in systems lacking inversion symmetry, even if time-reversal symmetry is preserved. The intrinsic mechanism is typically attributed to the Berry Curvature Dipole (BCD).
• Gap: While the BCD contribution is well-established, the specific role of an external in-plane magnetic field ($B_{\parallel}$) in modifying or inducing a nonlinear Hall response has not been fully explored microscopically. Specifically, the interplay between the geometric BCD contribution and field-induced contributions in realistic materials (like strained TMDs and Kondo lattices) requires a rigorous quantum kinetic treatment that goes beyond the semiclassical Boltzmann approximation.
• Goal: To derive a quantum kinetic theory for the second-harmonic current response in 2D Dirac systems under an in-plane magnetic field, distinguishing between geometric (BCD) and field-induced mechanisms, and identifying conditions where the field-induced term becomes significant.

2. Methodology

The authors employ a quantum kinetic equation (QKE) approach based on the Wigner distribution function (WDF), which offers advantages over the semiclassical Boltzmann equation by naturally incorporating interband transitions and disorder effects.

• Model Hamiltonian:
  • They utilize a two-band model of tilted massive Dirac fermions representing the low-energy physics of materials like strained Transition Metal Dichalcogenides (TMDs, e.g., WTe$_2$, WSe$_2$) and the surface states of topological crystalline insulators (SnTe).
  • The Hamiltonian includes:
    • Tilt parameter ($\alpha$) breaking mirror symmetry.
    • Mass term ($m$) opening a gap.
    • In-plane magnetic field coupling: Introduced via Zeeman coupling ($\gamma \mathbf{B} \cdot \boldsymbol{\sigma}$) and a crucial quadratic mass dispersion term ($\beta k^2$). The $\beta k^2$ term is essential for generating a non-zero field-induced response in the absence of tilt.
• Formalism:
  • The Wigner distribution function $w(\mathbf{k}, \mathbf{r}; \epsilon, t)$ is expanded perturbatively in powers of the electric field $\mathbf{E}$.
  • The QKE is solved to obtain the first-order ($w^{(1)}$) and second-order ($w^{(2)}$) corrections.
  • The current density is calculated via $j(t) = -e \int d^2k \, \text{Tr}[\hat{v} \hat{w}]$.
  • The analysis covers both AC (finite frequency $\omega$) and DC limits, accounting for disorder scattering via a relaxation time approximation ($\tau$).

3. Key Contributions

1. Derivation of the Second-Harmonic Response: The authors provide a complete microscopic derivation of the second-order current density in the presence of an in-plane magnetic field, separating contributions from diagonal and off-diagonal elements of the WDF.
2. Identification of Two Distinct Mechanisms:
   • Geometric (BCD) Contribution: Arises from the Berry curvature dipole, proportional to the tilt parameter $\alpha$. This term exists even without a magnetic field but is modified by it.
   • Field-Induced Contribution: A new term linear in the magnetic field ($B$) that arises specifically due to the quadratic dispersion of the mass term ($\beta k^2$). This contribution persists even when the tilt $\alpha = 0$ (where BCD vanishes).
3. Resonant Behavior: The theory predicts a sharp resonance in the nonlinear response at frequencies $\omega \approx \mu$ (chemical potential), corresponding to interband transitions between valence and conduction bands. This feature is captured by the QKE but missed by semiclassical Boltzmann treatments.
4. Analytical Expressions: Closed-form analytical expressions are derived for the nonlinear conductivity $\chi_{xxy}$ in both low-frequency ($\omega \tau \ll 1$) and high-frequency limits, explicitly showing the dependence on $B$, $\alpha$, $\beta$, and disorder $\tau$.

4. Key Results

• Nonlinear Current Components:
  • The total second-harmonic current is the sum of the BCD term and the magnetic-field-induced term.
  • BCD Term: $\mathbf{j}_{BCD} \propto (\mathbf{D} \cdot \mathbf{E}) (\mathbf{E} \times \hat{z})$. It scales with the tilt $\alpha$.
  • Field-Induced Term: In the low-frequency limit, the field-induced current component along the $y$-axis (for $E \parallel x$) scales as:
    $$j_y^{(2\omega)}(B) \propto \frac{e^3 E^2 \beta^2 \tau^2}{\mu^3} \gamma B_x$$
    This term is linear in $B_x$ and depends on the mass dispersion parameter $\beta$.
• Interplay and Tunability:
  • The magnetic field can either enhance or suppress the total nonlinear Hall signal depending on its orientation relative to the crystal axes and the sign of the BCD.
  • The ratio of the field-induced contribution to the BCD contribution is governed by a dimensionless parameter $\eta \sim (\gamma B/m) (\beta k_F^2/\mu) (\beta k_F/\alpha) \mu \tau$.
  • In clean systems (large $\mu \tau$) with small tilt $\alpha$, the field-induced term can become comparable to or even dominate the BCD term.
• Frequency Dependence:
  • The response exhibits a resonant peak at $\omega \approx \mu$.
  • At high frequencies ($\omega \gg \mu$), the response decays as $1/\omega^3$.
• Longitudinal vs. Transverse: The theory predicts a specific relationship between longitudinal and transverse second-harmonic currents: $j_x^{(2\omega)}(B) = (3B_y/B_x) j_y^{(2\omega)}(B)$.

5. Significance and Experimental Implications

• Material Candidates: The authors propose experimental tests in:
  • Strained TMDs: Monolayers of WTe$_2$ and WSe$_2$.
  • Topological Crystalline Insulators: Surface states of SnTe.
  • Kondo Lattices: Ce$_3$Bi$_4$Pd$_3$ (a Weyl-Kondo semimetal), where strong correlations and low temperatures ($T < 3$ K) make the effect potentially observable.
• Estimates: For WSe$_2$ and WTe$_2$, with magnetic fields in the range $1-10$ T, the ratio $\eta$ is estimated to be between $0.1$ and $1$, suggesting the effect is experimentally accessible.
• Theoretical Advancement: The work demonstrates that the quantum kinetic approach is superior to semiclassical methods for capturing resonant interband transitions and disorder effects in nonlinear transport. It highlights that mass dispersion (often neglected) is a critical factor in magnetic-field-induced nonlinear responses.
• Control Mechanism: The findings offer a practical route to tune the nonlinear Hall effect in Dirac materials simply by rotating the in-plane magnetic field, providing a new tool for investigating quantum geometry and topological properties.

In conclusion, this paper establishes a comprehensive theoretical framework showing that in-plane magnetic fields induce a significant, tunable nonlinear Hall response in 2D Dirac semimetals, distinct from and competitive with the intrinsic Berry curvature dipole mechanism.