The orbital-driven topological phase transition and planar Hall responses in ternary tellurides Weyl semi-metals

This study reveals that replacing Rh with Ir in ternary tellurides TaXTe$_4$ induces an orbital-driven topological phase transition from hybrid to type-II Weyl semimetals, which significantly enhances planar Hall responses through velocity-modulated effective mass anisotropy.

The Gist

The Big Picture: A Tale of Two Crystal Twins

Imagine you have two identical-looking crystal twins, TaRhTe₄ and TaIrTe₄. They are made of the same basic ingredients (Tantalum, Tellurium, and either Rhodium or Iridium) and they look the same under a microscope. However, inside these crystals, electrons are moving in very strange, "topological" ways.

The scientists in this paper discovered that by swapping just one ingredient (swapping Rhodium for Iridium), they can completely change the "personality" of how electricity flows through the crystal. It's like taking a car that drives on both highways and dirt roads and, by changing one engine part, turning it into a vehicle that only drives on dirt roads.

1. The "Traffic Jam" of Electrons (Weyl Semimetals)

To understand this, we need to talk about Weyl Semimetals.

• Normal Insulators: Think of these like a highway with a massive concrete wall (an energy gap) separating the "valence" lanes (where electrons live) from the "conduction" lanes (where they go to do work). No cars can cross.
• Weyl Semimetals: In these materials, the wall is gone. The lanes merge at specific points called Weyl Points. Imagine a traffic intersection where the road splits into two directions that touch at a single point.
• Type-I vs. Type-II:
  • Type-I: The intersection is perfectly balanced. Traffic flows smoothly in all directions.
  • Type-II: The intersection is tilted! It's like a ramp. Traffic is forced to flow mostly in one direction, creating a "cone" that leans over. This tilt creates a mix of "electron pockets" and "hole pockets" (like cars and empty parking spots) that can interact in wild ways.
  • Hybrid: A mix of both. Some intersections are balanced, others are tilted.

2. The Magic Switch: Orbital "Dancing"

The paper's main discovery is about what causes the switch from a balanced intersection to a tilted one.

Usually, scientists think the "Spin-Orbit Coupling" (SOC)—a fancy quantum effect where an electron's spin interacts with its movement—is the main driver. But here, the scientists found something more subtle: Orbital Character.

• The Analogy: Imagine the electrons are dancers. They have different "moves" (orbitals). Some dancers do a "flat spin" ($d_{xz}$), and others do a "vertical spin" ($d_{z^2}$).
• The Swap:
  • In TaRhTe₄ (with Rhodium), the dancers are doing a mix of flat and vertical spins. This creates a Hybrid state: some intersections are balanced, some are tilted.
  • In TaIrTe₄ (with Iridium), the Iridium atom is a bit heavier and changes the dance floor. Suddenly, the "vertical spin" dancers ($d_{z^2}$) take over the stage, and the "flat spin" dancers ($d_{xz}$) fade into the background.
• The Result: This change in the "dance routine" tilts the intersections so much that the balanced ones disappear. TaIrTe₄ becomes a pure Type-II Weyl Semimetal (all tilted ramps).

Key Takeaway: You don't need to break the crystal or change its shape to change its physics; you just need to change the "dance moves" of the electrons by swapping one atom.

3. The Planar Hall Effect: The "Slippery Slide"

The paper also looked at how these materials conduct electricity when you apply a magnetic field. This is called the Planar Hall Effect (PHE).

• The Analogy: Imagine sliding down a hill.
  • In a normal material, if you push a ball (electricity) and spin the ground (magnetic field), the ball goes straight.
  • In these Weyl crystals, because of the "tilted" intersections (Type-II), the ball gets pushed sideways, even though you didn't push it that way! It's like a slippery slide where the tilt of the slide forces you to drift to the side.
• The Discovery: The scientists found that the Hybrid material (TaRhTe₄) actually creates a stronger sideways drift (Planar Hall effect) than the pure Type-II material (TaIrTe₄).
• Why? It turns out that having a mix of balanced and tilted intersections creates a specific kind of "friction" or "mass" for the electrons that amplifies this sideways drift. It's like having a road with both smooth pavement and steep ramps; the combination creates a unique flow pattern that is very sensitive to magnetic fields.

4. The "Effective Mass" Connection

The researchers used a mathematical model (a tight-binding model) to explain why this happens. They looked at something called the off-diagonal effective mass.

• The Metaphor: Imagine the electrons are cars. "Effective mass" is how heavy the car feels. "Off-diagonal" means the car feels heavy when you try to turn it, not just when you drive straight.
• They found that the tilt of the Weyl points changes how "heavy" the electrons feel when they try to turn.
  • In the Hybrid phase, the "weight" of the electrons changes sign (from heavy to light) as they move between different points. This flipping of weight creates a massive boost in the sideways electrical current (the Planar Hall effect).

Summary: Why Does This Matter?

1. New Way to Engineer Materials: Previously, scientists thought you needed strong magnetic forces or heavy atoms to create these exotic states. This paper shows you can do it just by swapping atoms to change the orbital dance moves. It's a new recipe for building topological materials.
2. Better Sensors: Because the Planar Hall Effect is so sensitive to these changes, these materials could be used to make incredibly sensitive magnetic sensors or next-generation electronic switches.
3. Understanding the "Why": It solves a mystery about why these two twins behave differently, proving that the orbital character (the shape of the electron's wave) is the puppet master, not just the spin-orbit coupling.

In a nutshell: By swapping one atom in a crystal, the scientists changed the "dance style" of the electrons. This tilted the traffic intersections inside the material, which in turn made the electricity slide sideways much more dramatically. This gives us a new tool to design future electronics that are faster, more efficient, and smarter.

Technical Summary

1. Problem Statement

The study addresses the need to understand the mechanisms driving topological phase transitions in Weyl semimetals (WSMs) beyond the conventional reliance on strong spin-orbit coupling (SOC). Specifically, the authors investigate the ternary tellurides TaRhTe$_4$ and TaIrTe$_4$. While these materials are structurally similar and isoelectronic, they exhibit distinct topological phases. The central problem is to determine:

• How the substitution of Rh with Ir alters the Weyl point (WP) character (Type-I vs. Type-II).
• The role of orbital degrees of freedom (specifically $d$-orbitals) in driving these transitions.
• How these topological changes influence the Planar Hall Effect (PHE), a transport signature of chiral anomaly, and the underlying effective mass anisotropy.

2. Methodology

The authors employed a multi-faceted approach combining ab initio calculations with theoretical modeling:

• Density Functional Theory (DFT): Used to calculate the electronic band structures, Fermi surfaces, and spectral functions of TaRhTe$_4$ and TaIrTe$_4$ with and without SOC.
• Orbital Projection Analysis: To decompose the band contributions and identify the specific atomic orbitals (Ta-5$d$, Rh/Ir-$d$, Te-$p$) responsible for band crossings at the WPs.
• Planar Hall Conductivity Calculation: Utilized the semi-classical Boltzmann transport equation within the relaxation time approximation to compute the planar Hall conductivity ($\sigma_{xy}^\gamma$) as a function of chemical potential and magnetic field orientation.
• Tight-Binding (TB) Modeling: Constructed a model Hamiltonian for time-reversal symmetry (TRS) broken WSMs to simulate Type-I, Type-II, and Hybrid phases. This model was used to correlate the PHE response with velocity-modulated off-diagonal effective mass anisotropy ($q_{xz} = v_x v_z M_{xz}$).

3. Key Contributions and Results

A. Topological Phase Characterization

• TaRhTe$_4$: Identified as a Hybrid WSM. It hosts both Type-I and Type-II Weyl points simultaneously.
  • Without SOC: Hosts Type-II (V1) and Type-I (V2) WPs.
  • With SOC: Remains a Hybrid WSM. New Type-II WPs (W1) are generated below the Fermi level, while existing WPs shift but maintain their character (Type-II below, Type-I above).
• TaIrTe$_4$: Exhibits a SOC-driven phase transition.
  • Without SOC: A Type-I WSM (both WPs U1 and U2 are Type-I).
  • With SOC: Converts entirely into a Type-II WSM. The Type-I WPs above the Fermi level tilt significantly to become Type-II (W), while the Type-I WPs below the Fermi level are annihilated.

B. Orbital-Driven Mechanism

The study establishes that the phase transition is driven by orbital reorganization rather than just SOC strength:

• TaRhTe$_4$: The WPs are formed by a balanced hybridization of Rh-4$d_{xz}$ and Rh-4$d_{z^2}$ orbitals. This balance allows for the coexistence of Type-I and Type-II nodes.
• TaIrTe$_4$: Replacing Rh with Ir leads to a redistribution of orbital occupancy. The Ir-5$d_{z^2}$ orbital contribution near the Fermi level is significantly enhanced relative to the Ir-5$d_{xz}$ orbital.
• Consequence: This dominance of the $d_{z^2}$ orbital increases the tilting of the energy cones, converting the Type-I WPs into Type-II and annihilating the lower-energy Type-I nodes, effectively transforming the material from a Hybrid WSM to a pure Type-II WSM without changing the crystal symmetry.

C. Planar Hall Effect (PHE) Responses

• Enhancement in Hybrid Phases: The PHE is significantly enhanced in the Hybrid WSM (TaRhTe$_4$) compared to the Type-II WSM (TaIrTe$_4$).
• Chemical Potential Dependence: The PHE conductivity ($\sigma_{xy}^z$) increases by nearly an order of magnitude when the chemical potential is tuned to the energy of Type-II WPs compared to Type-I WPs.
• Correlation with Effective Mass: The authors found a strong correlation between the PHE magnitude and the velocity-modulated off-diagonal effective mass ($q_{xz}$).
  • In Type-II phases, $q_{xz}$ exhibits a specific sign profile around WPs leading to maximum PHE response.
  • In Type-I phases, the profile is different, yielding lower response.
  • In Hybrid phases, the sign of $q_{xz}$ changes between different WPs, resulting in an intermediate but distinct response profile.

4. Significance

• Beyond SOC: The work demonstrates that orbital degrees of freedom can be a primary driver for topological phase transitions, offering a new route for engineering topological materials that does not rely solely on heavy elements with strong SOC.
• Material Design: It provides a blueprint for tuning WSM phases (Type-I $\leftrightarrow$ Type-II $\leftrightarrow$ Hybrid) via isoelectronic substitution (Rh $\to$ Ir) by manipulating $d$-orbital contributions.
• Transport Probes: The study links the Planar Hall Effect directly to the nature of the Weyl points and the anisotropy of the effective mass. This suggests PHE can serve as a sensitive experimental probe to detect orbital-driven topological transitions in ternary tellurides.
• Theoretical Framework: The introduction of the velocity-modulated off-diagonal effective mass ($q_{xz}$) as a predictor for PHE magnitude bridges the gap between microscopic band structure details and macroscopic transport phenomena in topological semimetals.

In conclusion, the paper reveals that the interplay between specific $d$-orbital symmetries ($d_{xz}$ vs. $d_{z^2}$) dictates the topological nature of TaXTe$_4$ compounds, directly controlling their chiral anomaly-mediated transport properties.