Unified theory of the photovoltaic Hall effect by field- and light-induced Berry curvatures

This paper presents a unified theoretical framework that coherently describes the photovoltaic Hall effect by demonstrating how bias electric fields and light jointly induce Berry curvatures and shift vectors to govern both field-induced circular photogalvanic effects and light-dressed anomalous Hall phenomena.

The Gist

Imagine you are trying to push a crowd of people (electrons) through a hallway (a material) using a gentle wind (an electric field). Usually, they just move straight ahead. But what if you could make them suddenly start dancing in a circle, creating a side current, just by shining a specific kind of flashlight on them?

This is the essence of the Photovoltaic Hall Effect, a phenomenon where light and electricity combine to push electrons sideways.

For a long time, scientists had two different "rulebooks" to explain how this happens, and they didn't quite fit together. One rulebook talked about the "shape" of the electron's path (geometry), and the other talked about the "momentum" of the electrons (how fast and in what direction they are moving). This paper, by Murotani, Fujimoto, and Matsunaga, is like writing a new, unified rulebook that explains both sides of the story at once.

Here is the breakdown of their discovery using simple analogies:

1. The Two Old Stories (The Problem)

• Story A (The "Floquet" Story): Imagine the light is like a DJ spinning a record. The music (light) changes the shape of the dance floor itself, creating a "Berry Curvature." This is like the floor suddenly tilting or curving, forcing the dancers to move in a circle even if they try to walk straight. This was thought to be the main cause of the sideways current.
• Story B (The "Momentum" Story): Recent experiments suggested that the light doesn't just change the floor; it pushes the dancers unevenly. Some get pushed harder to the left, others to the right. This "momentum asymmetry" (called the Field-Induced Circular Photogalvanic Effect) seemed to be the real driver, but scientists couldn't explain it using the same math as Story A.

2. The New Unified Theory (The Solution)

The authors realized these aren't two different stories; they are two sides of the same coin. They developed a single mathematical framework that shows how the electric field and the light work together to create three distinct mechanisms that all contribute to the sideways current:

Mechanism 1: The "Curved Floor" (Field-Induced Berry Curvature)

• The Analogy: Imagine the electric field is a strong wind blowing through the hallway. This wind actually warps the hallway itself, making the floor curve.
• The Result: When the light hits the electrons, they slide down this newly curved floor. Because the floor is curved (due to the "Berry Curvature"), they naturally drift sideways. The paper shows that the electric field creates this curvature, which wasn't fully understood before.

Mechanism 2: The "Slippery Step" (Field-Induced Energy Shift)

• The Analogy: Imagine the electrons are taking a step from one platform to another (jumping energy levels). Usually, the step is the same height. But the electric field acts like a ramp, tilting the platforms so the step is easier to take in one direction than the other.
• The Result: This is called the "Shift Vector." The electric field changes the "cost" of the jump depending on which way the electron is facing. This makes the electrons more likely to jump in one direction, creating a sideways flow.

Mechanism 3: The "Ghost Push" (Anomalous Velocity)

• The Analogy: This is the classic "spin" effect. Imagine the electrons are spinning tops. When they move through the material, the material's internal structure (topology) gives them a little nudge to the side, like a spinning top wobbling as it rolls.
• The Result: Even without the floor curving or the steps tilting, the electrons' own "spin" or "valley" nature causes them to drift sideways when pushed by the electric field.

3. The "Magic" of Gallium Arsenide (GaAs)

The team tested their theory on a common semiconductor called Gallium Arsenide (GaAs). They found something amazing:

• In GaAs, the "dance floor" (the valence band) has a special topological knot in it.
• Because of this knot, all three mechanisms (the curved floor, the slippery step, and the ghost push) go crazy at a specific energy level.
• The Result: When you shine light at just the right frequency, the sideways current explodes in strength. It's like finding the perfect resonance on a guitar string; the signal becomes incredibly loud. This explains why recent experiments saw such huge, sharp peaks in the current.

4. Why This Matters

• One Theory to Rule Them All: Before this, scientists had to use different math for "light-induced" effects and "electric-field-induced" effects. Now, they have one clean, geometric picture that covers everything.
• Better Electronics: Understanding how to control these sideways currents with light and electricity could lead to faster, more efficient electronic devices that don't rely on magnetic fields (which are bulky and hard to control).
• Engineering Geometry: The paper suggests we can "engineer" the geometry of materials (like bending the invisible floor) using light and electricity to control how electrons move.

Summary

Think of this paper as a master key. It unlocks the mystery of how light and electricity team up to push electrons sideways. It reveals that the electric field doesn't just push the electrons; it reshapes the world they live in (curving the floor, tilting the steps), and the light triggers the movement. By understanding this unified geometric dance, we can better design the next generation of ultra-fast, light-controlled computers.

Technical Summary

Here is a detailed technical summary of the paper "Unified theory of the photovoltaic Hall effect by field- and light-induced Berry curvatures" by Murotani, Fujimoto, and Matsunaga.

1. Problem Statement

The photovoltaic Hall effect (PHE) refers to the generation of a photocurrent perpendicular to an applied bias electric field ($E_0$) when a material is illuminated by circularly polarized light. While this phenomenon has been studied extensively, existing theoretical frameworks suffer from fragmentation:

• Floquet Engineering: Describes light-induced Berry curvature arising from light-dressed states (Floquet states), often associated with topological phase transitions and the light-induced Anomalous Hall Effect (AHE).
• Field-Induced Circular Photogalvanic Effect (CPGE): Recent studies suggest that momentum asymmetry of photoexcited carriers, driven by the breaking of inversion symmetry ($I$) via the bias field, contributes significantly to the Hall current.
• The Gap: These two mechanisms have historically been described using different theoretical techniques and frameworks. There was no unified theory capable of describing both the light-induced AHE (Floquet states) and the field-induced CPGE on an equal footing, hindering a coherent understanding of the geometric and topological aspects of light-matter interactions in the third-order nonlinear regime.

2. Methodology

The authors developed a general unified theory based on density matrix calculations in the length gauge.

• Framework: They treat the system under a static bias electric field ($E_0$) and an optical field ($E_1$) simultaneously.
• Perturbation Theory: The theory is formulated in the third-order nonlinear regime. The static field $E_0$ is treated as a perturbation that modifies the band structure, transition dipole moments, and transition energies.
• Geometric Quantities: The derivation explicitly tracks corrections to:
  • The Berry connection ($\xi_{nm}$).
  • The Berry curvature ($\Omega_{nm}$).
  • The Shift vector ($R_{nm}$), which describes the spatial shift of electron clouds during interband transitions.
• Models: The theory is applied to two specific systems to validate the results:
  1. A Massive Dirac electron system (representing lead halide perovskites).
  2. Gallium Arsenide (GaAs) using an eight-band Kane model to capture the complex valence band structure.
• Handling Degeneracy: The authors utilize Singular Value Decomposition (SVD) to handle degenerate bands, allowing them to resolve independent excitation paths and define shift vectors and energy shifts for each path.

3. Key Contributions

The paper unifies three distinct physical mechanisms contributing to the photovoltaic Hall effect, all derived within a single geometric framework:

A. Field-Induced Berry Curvature ($J_{inj, dip}$)

• Mechanism: The bias field $E_0$ modifies the interband Berry connection, inducing a correction to the band-resolved Berry curvature ($\Delta \Omega_{nm} \propto E_0$).
• Result: This field-induced curvature creates a momentum asymmetry in the photoexcited carriers, generating a transverse current even in centrosymmetric materials.
• Significance: This is the first explicit theoretical link establishing that the field-induced CPGE is governed by a field-induced Berry curvature.

B. Field-Induced Energy Shift ($J_{inj, ene}$)

• Mechanism: The bias field alters the transition energy ($\hbar \Delta \omega_{nm}$) between bands. This shift is coupled to the shift vector ($R_{nm}$), a geometric quantity measuring the spatial displacement of the electron wavefunction during the transition.
• Result: The energy shift depends on the light polarization. The difference in transition energies for left- and right-circularly polarized light leads to an asymmetric population of carriers, generating a Hall current.
• Significance: This identifies a second geometric contribution to the CPGE, distinct from the Berry curvature mechanism but equally driven by the bias field.

C. Anomalous Velocity of Photocarriers ($J_{inj, ano}$)

• Mechanism: This is the conventional mechanism where photoexcited carriers possess an intrinsic anomalous velocity ($v_{anom} \propto E_0 \times \Omega_n$) due to the Berry curvature of the bands they occupy.
• Result: This term dominates in simple scenarios but has distinct dynamical properties compared to the other two terms (it depends on the carrier density history rather than instantaneous field intensity).

D. Unification with Light-Dressed States (Floquet)

• The authors extend the theory to metallic systems to describe the light-induced AHE arising from light-dressed (Floquet) states.
• They demonstrate that the light-induced Berry curvature ($\delta \Omega_n$) in Floquet theory is equivalent to the light-induced correction to the Berry connection ($\delta \xi_{nn}$) derived in their density matrix approach.
• Crucially, they reveal a direct link between the light-induced AHE and optical rectification, showing both are governed by the light-induced Berry connection.

4. Key Results

• Unified Tensor Formulation: The authors express the total photocurrent as a sum of three tensor components ($K_{i}$ for $i=$ dip, ene, ano), providing a clear geometric picture of the third-order nonlinear response.
• Resonant Enhancement in GaAs:
  • In GaAs, the valence band possesses a topological character (monopole charges $Q=\pm 3$ at the $\Gamma$ point).
  • The theory predicts that the field-induced Berry curvature ($\Delta \Omega_{12}$) and the shift vector ($R_{12}$) diverge as $1/k$ near the zone center due to the mixing of heavy-hole (HH) and light-hole (LH) bands.
  • This divergence leads to a resonant enhancement of the transverse photocurrent at the bandgap energy, explaining recent experimental observations of sharp peaks in the photon energy dependence of the PHE.
• Dynamic Response Distinction:
  • The field-induced terms ($J_{dip}$ and $J_{ene}$) scale linearly with the instantaneous bias field $E_0(t)$ and light intensity $I(t)$.
  • The anomalous velocity term ($J_{ano}$) scales with $E_0(t)$ and the integrated carrier density (history of light intensity).
  • This qualitative difference allows experimentalists to distinguish between the mechanisms using time-resolved terahertz spectroscopy.
• Comparison with Previous Work: The paper clarifies that previous theories (e.g., Fregoso, Ahn et al.) mixed these distinct geometric contributions. The unified theory separates them, showing that the "Hermitian curvature" in Riemannian geometry approaches is a mixture of the anomalous velocity and energy shift terms.

5. Significance

• Theoretical Unification: The paper resolves the fragmentation in the field by providing a single, physically transparent framework that encompasses both Floquet engineering (light-induced) and field-induced CPGE.
• Geometric Insight: It establishes that the photovoltaic Hall effect is fundamentally a geometric phenomenon driven by the interplay of Berry curvature and shift vectors, modulated by external fields.
• Experimental Guidance: The theory provides specific signatures (resonant peaks in GaAs, distinct temporal responses) to guide future experiments in distinguishing between different Hall mechanisms.
• Berry Curvature Engineering: It offers a new pathway for "Berry curvature engineering" not just via light (Floquet) or strain, but via the synergistic application of static electric fields and light, potentially enabling new types of ultrafast optoelectronic devices and spintronic applications.
• Foundation for Future Work: The formalism is robust enough to be extended to correlated systems and provides a solid foundation for interpreting complex nonlinear optical responses in topological materials.