Photogalvanic Effects in Surface States of Topological Insulators under Perpendicular Magnetic Fields

This paper theoretically investigates the tunable nonlinear magneto-optical shift conductivity in the surface states of Bi$_2$Se$_3$ under perpendicular magnetic fields, demonstrating that the effect arises from discrete Landau level transitions and can be effectively controlled by adjusting the chemical potential and magnetic field strength.

The Gist

Imagine a Topological Insulator (like the material Bi₂Se₃ discussed in this paper) as a very special kind of "smart" material. Think of it like a chocolate-covered ice cream bar: the inside (the bulk) is an insulator, meaning electricity can't flow through it at all (like the frozen ice cream). However, the outside (the surface) is a conductor, meaning electricity flows freely there (like the chocolate shell).

The scientists in this paper are studying what happens when you shine a special light on that "chocolate shell" while also putting the whole thing in a strong magnetic field.

Here is the breakdown of their discovery in simple terms:

1. The "Traffic Jam" of Electrons (Landau Levels)

Normally, electrons on the surface of this material can move around freely, like cars on a highway. But when you apply a strong magnetic field perpendicular to the surface, it's like putting up a series of concrete barriers on the highway.

Suddenly, the electrons can't just drive anywhere. They are forced into specific, narrow lanes called Landau Levels. Think of these as parking spots. An electron can only sit in spot #1, #2, or #3, but it cannot sit in the space between the spots. This turns a smooth highway into a set of discrete, isolated parking spots.

2. The "Shift" Current (The Main Character)

The paper focuses on a phenomenon called the Shift Current.

• Normal Current: Usually, when light hits a material, it kicks electrons, and they run in a straight line, creating a current.
• Shift Current: This is different. Imagine an electron is a person standing on a trampoline. When a photon (a particle of light) hits them, they don't just jump up; they also slide sideways to a new spot on the trampoline before jumping. This "sideways slide" creates a current even without a battery or voltage pushing them.

The scientists wanted to know: If we force electrons into those "parking spots" (Landau Levels) with a magnetic field, how does this sideways slide change?

3. The Rules of the Game (Selection Rules)

In this new "parking lot" world, electrons can't just jump from any spot to any other spot. There are strict rules (Selection Rules).

• An electron can jump from spot #1 to #2, or #1 to #3, but maybe not #1 to #4 directly.
• The paper maps out exactly which "jumps" are allowed. It's like a game of chess where the magnetic field changes the rules of how the pieces can move.

4. The Magic of Tuning (Chemical Potential)

One of the coolest findings is how easy it is to tune this effect.

• The Chemical Potential is basically the "filling level" of the parking lot. Are the bottom spots full? Are the top spots empty?
• By changing this "filling level" (like adding more electrons or removing them), the scientists can turn the current on or off or change its strength.
• Analogy: Imagine a water slide. If the water level (chemical potential) is low, the slide is dry and no one goes down. If you raise the water level just right, the slide becomes super fast. The scientists found they could dial the "water level" to get exactly the amount of electricity they wanted.

5. The "Pure" Light Problem

The paper also discovered a funny quirk: If you shine pure circularly polarized light (light that spins perfectly like a corkscrew) on this material, nothing happens. No current is generated.

• Why? Because the symmetry of the material and the spinning light cancel each other out.
• The Fix: You need light that is a mix of spinning and straight (elliptical or linear polarization) to get the electrons to slide. It's like trying to push a swing; if you push exactly when the swing is at the top of its arc, it won't move. You need to push at the right angle.

6. Why Does This Matter?

The researchers found that this "Shift Current" is:

• Super Tunable: You can control it easily with magnets and chemical adjustments.
• Strong: It produces a significant electrical current.
• Robust: It doesn't care much about how "dirty" or imperfect the material is (damping).

The Big Picture:
This research suggests that we could build super-efficient solar cells or light detectors using these materials. Because we can tune the current so precisely with magnets and chemical tweaks, we might be able to create devices that harvest energy from light much better than current technology allows, potentially breaking the limits of what we thought was possible for solar energy.

In a nutshell: The paper shows how to turn a special material into a highly controllable "light-to-electricity" machine by using magnets to organize electrons into neat rows and then nudging them with the right kind of light.

Technical Summary

1. Problem Statement

Topological insulators (TIs), such as $\text{Bi}_2\text{Se}_3$, possess insulating bulk states and conducting surface states with unique spin-momentum locking. While nonlinear optical phenomena like the shift current (a second-order bulk photovoltaic effect) have been observed on TI surfaces due to broken inversion symmetry, most existing studies focus on:

• In-plane magnetic fields (which do not quantize electronic states).
• Strain or chemical potential tuning without magnetic quantization.
• Second-harmonic generation (SHG), which is a non-resonant process.

There is a gap in understanding how perpendicular quantizing magnetic fields affect the photogalvanic shift current. Specifically, it is unclear how discrete Landau levels (LLs) influence the selection rules, spectral lineshapes, and tunability of the shift current, particularly given that shift current requires real photon absorption (unlike SHG).

2. Methodology

The authors employ a theoretical framework combining perturbation theory with the Landau level formalism for the surface states of $\text{Bi}_2\text{Se}_3$.

• Hamiltonian: The system is modeled using the Hamiltonian for $\text{Bi}_2\text{Se}_3$ surface states under a perpendicular magnetic field ($B \parallel \hat{z}$). This includes:
  • Linear Dirac term ($v_F$).
  • Hexagonal warping term ($\lambda$).
  • Zeeman coupling ($\Delta$).
• Electronic States: The electronic states are quantized into Landau levels ($\Psi_s$) with index $s$. The authors utilize previously derived Berry connections ($\xi_{s_1 s_2}$) between these levels.
• Perturbative Approach: The shift conductivity tensor $\sigma^{(2)}_{\alpha\beta\gamma}(-\omega, \omega)$ is derived microscopically up to the second order of the electric field. The expression involves a summation over Landau level transitions ($s_1, s_2, s$) and includes relaxation parameters ($\Gamma_1, \Gamma_2$) to account for damping.
• Symmetry Analysis: The system possesses $C_3$ point group symmetry under the magnetic field. The authors analyze the tensor components to determine which polarization configurations generate non-zero currents.
• Parameters: Calculations assume $T=0$ K, with specific values for Fermi velocity ($v_F$), warping strength ($\lambda$), and Landé g-factor ($g$).

3. Key Contributions

• Derivation of Microscopic Expression: The paper provides an explicit analytical expression for the shift conductivity tensor in the presence of a quantizing magnetic field, decomposing it into interband and intraband contributions.
• Identification of Selection Rules: The study establishes specific selection rules for Landau level transitions contributing to the shift current: $|s_1| - |s_2| = \pm 1, \pm 2$. This differs from the selection rules found in Second-Harmonic Generation (SHG) for the same system.
• Symmetry Constraints on Polarization: It is rigorously proven that due to $C_3$ symmetry, pure circularly polarized light generates zero shift current. Non-zero currents require elliptical or linear polarization.
• Distinction from SHG: The authors highlight fundamental differences between shift current and SHG in Landau levels, particularly regarding energy conservation (Dirac delta vs. Lorentzian lineshapes in the clean limit) and the nature of the spectral functions.

4. Key Results

A. Spectral Characteristics and Selection Rules

• Discrete Peaks: In the clean limit (zero damping), the shift conductivity consists of discrete peaks at specific photon energies corresponding to Landau level transitions ($\hbar\omega = \varepsilon_{s_1} - \varepsilon_{s_2}$).
• Lineshapes: With finite damping, the peaks transform into Lorentzian lineshapes.
  • The real part ($\sigma_r$) is dominated by $L_1$ functions (centered at transition energy).
  • The imaginary part ($\sigma_i$) is dominated by $L_2$ functions (peaked at $\pm \Gamma$ from the transition energy).
• Degenerate Transitions: Transitions with the same energy (e.g., $|s_1| - |s_2| = 1$ and $-2$) can interfere. For $\sigma_r$, degenerate transitions often cancel out due to opposite signs, leading to suppressed peaks. For $\sigma_i$, they do not cancel, leading to increasing peak values at higher energies.

B. Dependence on Chemical Potential ($\mu$)

• Pauli Blocking: The chemical potential acts as a switch for transitions.
  • Low $\mu$ (near Dirac point): Only interband transitions contribute.
  • High $\mu$ (doped): Intraband transitions (within the same band) become active, appearing at lower photon energies (Drude-like behavior).
• Tunability: By tuning $\mu$, one can selectively activate or deactivate specific transition channels, creating a "gap" in the spectrum where no shift current exists between the intraband and interband regimes.

C. Dependence on Magnetic Field ($B$)

• Scaling: Peak positions scale with the cyclotron energy $\hbar\omega_c \propto \sqrt{B}$. As $B$ increases, peaks shift to higher energies and the spacing between them increases.
• Low Field Limit: As $B \to 0$, the discrete peaks merge into a smooth curve.
  • The real part ($\sigma_r$) vanishes linearly with $B$ (proportional to $B$).
  • The imaginary part ($\sigma_i$) remains non-zero and exhibits a resonant peak at $\hbar\omega \approx 2|\mu|$.

D. Magnitude and Practicality

• The calculated effective bulk shift conductivity reaches $\sim 326 \, \mu\text{A/V}^2$ at $B=5$ T and $\hbar\omega \approx 297$ meV.
• This magnitude is comparable to or exceeds that of known nonlinear materials like GeSe, and can be further enhanced by higher doping or stronger magnetic fields.

5. Significance

• Fundamental Physics: The work elucidates the distinct physical mechanisms of shift currents in quantized systems compared to conventional crystals or SHG processes, specifically highlighting the role of discrete Landau levels and selection rules.
• Device Applications: The results demonstrate that topological insulators under perpendicular magnetic fields offer a highly tunable platform for nonlinear magneto-optical devices.
  • The ability to switch between interband and intraband responses via chemical potential allows for broadband photodetection.
  • The magnetic field dependence offers a new degree of freedom for controlling photocurrent direction and magnitude.
• Material Potential: The predicted conductivity magnitude suggests that $\text{Bi}_2\text{Se}_3$ surface states are viable candidates for next-generation photodetectors and solar energy conversion devices capable of surpassing the Shockley-Queisser limit.

In conclusion, this paper provides a comprehensive theoretical foundation for utilizing the shift current in topological insulators under quantizing magnetic fields, revealing a highly controllable and robust nonlinear optical response.