Nonlinear response theory for orbital photocurrent in semiconductors

This paper establishes a general theoretical framework for calculating spin and orbital photocurrents in semiconductors, applying it to the Bernevig-Hughes-Zhang and Luttinger models to characterize optical responses across topological phase transitions and highlight distinct relaxation time dependencies in orbital conductivity.

The Gist

Imagine you have a crowd of people (electrons) in a giant, invisible dance hall (a semiconductor crystal). Usually, when you shine a light on them, they just jiggle in place or move randomly. But sometimes, if the light is just right, the whole crowd starts marching in a specific direction, creating a steady flow of energy. This is called a photocurrent, and it's the principle behind solar cells.

For a long time, scientists only cared about the flow of the people themselves (electric charge). But recently, they realized that these people also have "spins" (like tiny tops spinning) and "orbits" (like planets circling a sun). The paper you provided is a new instruction manual for predicting how these spins and orbits can also be made to flow in a straight line just by shining light on them.

Here is a breakdown of the paper's discoveries using simple analogies:

1. The New "Traffic Rules" for Light

The authors (Tanaka and Ishizuka) have written a new set of "traffic laws" (mathematical formulas) for how light pushes these spinning and orbiting particles.

• The Old Way: Scientists knew how to calculate the flow of electric charge (the people walking).
• The New Way: They figured out how to calculate the flow of spin (the spinning tops) and orbital motion (the circling planets).
• Why it matters: This is like discovering that not only can you move people from point A to B, but you can also make them spin in a specific direction or orbit in a specific pattern just by changing the color or angle of the light. This could lead to new types of computers that store information using "spin" or "orbit" instead of just electricity.

2. Two Ways to Make the Crowd Move

The paper explains that light can push these particles in two distinct ways, which the authors call the "Shift Current" and the "Injection Current."

• The Shift Current (The "Step"): Imagine the light hits the crowd, and everyone takes a sudden, synchronized step to the left. This happens instantly because of the shape of the dance floor (the crystal structure). This "step" doesn't depend on how slippery the floor is (how much friction/relaxation time there is). It's a pure geometric effect.
• The Injection Current (The "Run"): Imagine the light hits the crowd, and suddenly everyone starts running. How fast they keep running depends on how slippery the floor is. If the floor is very slippery (low friction), they run faster and longer. This part of the flow does depend on the "relaxation time" (how long they can keep moving before bumping into something).

The Big Surprise: The authors found that for orbital currents (the circling planets), the rules are a bit mixed up compared to normal electric currents.

• For linearly polarized light (light vibrating in one straight line), the orbital current behaves like the "Step" (it doesn't care about friction).
• For circularly polarized light (light spinning like a corkscrew), the orbital current behaves like the "Run" (it gets stronger if the floor is slippery).

3. Testing the Rules on Two Famous Dance Floors

To prove their new traffic laws work, the authors tested them on two famous theoretical models of crystals:

A. The BHZ Model (The "Topological Switch")

• The Setup: This model represents a material that can switch between being a "normal insulator" (a wall that stops traffic) and a "topological insulator" (a wall with a magic highway on the surface).
• The Discovery: When they shined light on this model, they saw that the direction of the orbital flow flipped depending on whether the material was in the "normal" or "topological" state.
• The Twist: They added a "Rashba term" (think of this as tilting the dance floor slightly). This broke the symmetry and allowed spin currents to flow, which usually can't happen in perfectly symmetrical rooms. It's like tilting a table so that marbles roll in a new direction they couldn't before.

B. The Luttinger Model (The "Complex Orchestra")

• The Setup: This model is more complex, representing materials like certain heavy metals. It's like a four-piece band instead of a soloist.
• The Discovery: They found that the way the material responds to light changes drastically depending on whether it's in a topological or normal state.
• The "Wine Bottle" Effect: In the "normal" state, the energy levels of the electrons look like the bottom of a wine bottle (wide at the bottom, narrow at the neck). This shape makes the light absorption look very different compared to the topological state.
• The Takeaway: By simply measuring how the material absorbs different colors of light, you can tell if it's in a topological state or not. It's like listening to a band play; if the sound changes from a smooth hum to a jagged noise, you know the band has switched instruments.

4. Why Should You Care?

This paper is a "user manual" for the future of Orbitronics.

• Current Tech: We use electricity (moving electrons) to power our phones and computers.
• Future Tech: We might use Spin (the direction electrons spin) or Orbit (how they circle) to carry information. This could make computers faster, smaller, and use less energy.
• The Paper's Role: Before we can build these devices, we need to know exactly how light will interact with these materials. This paper provides the mathematical tools to predict those interactions, helping scientists design better materials for next-generation electronics.

In a nutshell: The authors have created a universal translator that tells us how to use light to control the "spinning" and "orbiting" of electrons in crystals. They discovered that these movements follow unique rules that are different from regular electricity, and they showed how these rules can help us identify and build new, high-tech materials.

Technical Summary

1. Problem Statement

Recent theoretical advancements have identified analogs of the bulk photovoltaic effect (photocurrent) for spin and orbital degrees of freedom. While general formulas exist for spin photocurrents, a generic, quantitative framework for calculating orbital photocurrents in models relevant to real materials has been lacking. Specifically, there is a need to understand:

• How orbital currents behave in centrosymmetric materials (where traditional photocurrents are often forbidden).
• How these currents evolve across topological phase transitions.
• The distinct relaxation time dependencies of orbital currents compared to standard electrical photocurrents.

2. Methodology

The authors developed a general nonlinear response theory based on the density matrix formalism, extending the work of Sipe et al. regarding second-order optical responses.

• Hamiltonian and Operators: They considered a non-interacting electron system coupled to an electric field. They defined spin and orbital current operators ($\hat{J}$) involving the anticommutator of the velocity operator ($\hat{v}$) and the spin/orbital operator ($\hat{O}$).
• Perturbation Expansion: The density matrix $\rho$ was expanded to second order in the electric field ($E$). The authors solved the equation of motion for the density matrix to derive the second-order conductivity tensor $\sigma^{(2)}$.
• Decomposition: The second-order DC response was separated into two distinct contributions, analogous to electrical photocurrents:
  1. Shift Current ($\sigma^{(2A)}$): A geometric contribution independent of the relaxation time ($\tau$) in the clean limit. It arises from the shift vector in the wavefunction.
  2. Injection Current ($\sigma^{(2B)}$): A dynamic contribution proportional to the relaxation time ($\tau$), arising from the population imbalance of excited carriers.
• Symmetry Analysis: The authors analyzed the behavior of these currents under Time-Reversal Symmetry (TRS) and Spatial-Inversion Symmetry (IS), deriving selection rules for when spin or orbital currents can be non-zero.
• Model Applications: The theory was applied to two specific models:
  1. Bernevig-Hughes-Zhang (BHZ) Model: A 4-band model for HgTe/CdTe quantum wells, known for topological phase transitions.
  2. Luttinger Model: A 4-band model for materials like $\alpha$-Sn and Pr$_2$Ir$_2$O$_7$, describing $J=3/2$ hole systems with uniaxial strain.

3. Key Contributions

• Derivation of General Formulas: The paper provides a rigorous derivation of generic formulas for second-order orbital and spin photocurrents (Eqs. 16–18) expressed in the Lehmann representation. This allows for direct application to complex, multi-band models of real materials.
• Orbital Photocurrent in Centrosymmetric Materials: The theory demonstrates that orbital currents can exist in centrosymmetric materials (unlike electrical photocurrents), provided the orbital operator has specific symmetry properties.
• Distinct Relaxation Time Dependence: A crucial finding is that the relaxation time dependence of orbital currents differs from that of electrical photocurrents and varies depending on the polarization of light and the type of current (shift vs. injection).
• Topological Phase Transition Signatures: The study establishes that the frequency dependence of optical conductivity and the sign of nonlinear orbital responses serve as clear signatures of topological phase transitions.

4. Key Results

A. BHZ Model Results

• Centrosymmetric Orbital Current: For linearly polarized light, the orbital shift current exists even in the centrosymmetric BHZ model. The sign of the response coefficient flips between the trivial ($\lambda > 4.0$) and topological ($\lambda < 4.0$) phases.
• Circular Polarization: For circularly polarized light, an orbital injection current is generated. Its sign also reverses across the topological phase transition, driven by the change in the sign of the imaginary part of the product of Berry connections ($r^b_{mp}r^c_{pn}$) near the $\Gamma$ point.
• Rashba Interaction: Introducing a Rashba term (breaking inversion symmetry) allows for spin photocurrents. Interestingly, the shift and injection contributions are no longer purely real and imaginary parts of the conductivity; they mix. The spin current direction reverses with the sign of the Rashba term, while the orbital current does not.

B. Luttinger Model Results

• Optical Conductivity: The linear optical conductivity ($\sigma^{(1)}$) shows distinct behaviors between trivial and topological phases. In the topological phase, it resembles the joint density of states (jDOS). In the trivial phase, it exhibits a "wine-bottle" band structure effect, leading to a smooth monotonic increase with frequency, distinct from the jDOS.
• Nonlinear Orbital Current:
  • Relaxation Time Anomaly: Unlike standard electrical photocurrents, the real part of the orbital conductivity (for linearly polarized light) increases linearly with relaxation time ($\tau$), behaving like an injection current. Conversely, the imaginary part (related to circular polarization) converges to a finite value as $\tau \to \infty$, behaving like a shift current.
  • Sign Reversal: The orbital conductivity exhibits sign changes near specific frequencies (e.g., $\omega \approx 17$ meV). This is attributed to the interference between contributions from different valence-conduction band pairs, where the expectation values of the orbital angular momentum ($\langle J_x \rangle$) and group velocities have opposite signs for different bands.

5. Significance and Implications

• Characterization of Topological Materials: The distinct frequency dependence of optical conductivity and the sign reversal of nonlinear orbital currents provide robust experimental probes to distinguish between trivial and topological phases without requiring transport measurements.
• Orbitronics: The findings support the emerging field of "orbitronics," suggesting that orbital degrees of freedom can serve as new carriers for information, particularly in centrosymmetric materials where spin currents might be suppressed or difficult to generate.
• Quantitative Predictions: The derived formulas are applicable to complex real-material models, enabling quantitative predictions for future experiments involving spin and orbital photocurrents in topological insulators, Weyl semimetals, and other correlated systems.
• Fundamental Physics: The work clarifies the fundamental differences between shift and injection mechanisms in the context of orbital and spin degrees of freedom, particularly highlighting how symmetry breaking (e.g., Rashba terms) alters the interplay between these mechanisms.