First principles study of thermoelectric properties of $\text{Nb}_2\text{Co}_2\text{InSb}$ and $\text{Nb}_2\text{Co}_2\text{GaSb}$ double half-Heuslers

This study demonstrates that the double half-Heusler compounds $\text{Nb}_2\text{Co}_2\text{InSb}$ and $\text{Nb}_2\text{Co}_2\text{GaSb}$ exhibit significantly reduced lattice thermal conductivity compared to the parent NbCoSn system due to mass disorder, positioning them as promising candidates for high-temperature thermoelectric applications.

The Gist

Imagine you are trying to build a better thermoelectric generator. Think of this device as a magical bridge that turns waste heat (like the heat from a car engine or a factory chimney) directly into electricity. To make this bridge efficient, you need a material that acts like a "one-way street" for electricity but a "dead-end alley" for heat.

This paper is about finding the perfect material for that job. The researchers are looking at a family of materials called Half-Heuslers.

The Problem: The "Highway" for Heat

Think of the original material, NbCoSn, as a well-built road.

• The Good News: It's great for electricity. Electrons (the cars) can zoom down this road very easily.
• The Bad News: It's too good at letting heat travel. Heat travels as vibrations in the material's atoms (called phonons). In NbCoSn, these vibrations move like a high-speed train on a smooth track. Because the heat escapes so fast, the temperature difference needed to generate electricity disappears, and the device becomes inefficient.

The researchers wanted to keep the "electric highway" open but put up speed bumps and detours for the "heat train."

The Solution: The "Double Half-Heusler" Shuffle

The researchers decided to play a game of atomic musical chairs.

1. The Original Setup: Imagine a dance floor where three types of dancers (Niobium, Cobalt, and Tin) are arranged in a perfect, repeating pattern.
2. The New Idea: They replaced the Tin dancers with a mix of Indium/Gallium and Antimony dancers.
   • This created a "Double Half-Heusler" (a fancy name for a four-atom dance floor).
   • Crucially, they mixed these new dancers in different ways:
     • Ordered: The dancers stand in a strict, perfect grid (like a military formation).
     • Disordered (SQS): The dancers are shuffled randomly, like a crowd at a concert.

The Analogy: The "Mass Disorder" Speed Bump

Why does shuffling the dancers help?

• Mass Disorder: Imagine the Tin dancer was replaced by a mix of a very light dancer (Gallium) and a very heavy dancer (Indium/Antimony).
• The Effect: When the "heat train" (phonons) tries to move through the material, it hits these random heavy and light dancers. It's like driving down a road where the pavement suddenly changes from smooth asphalt to gravel, then to mud, then to ice. The heat vibrations get scattered, confused, and slowed down.
• The Result: The heat can't escape as easily, so the material stays hot on one side and cool on the other, generating much more electricity.

The Findings: What Worked Best?

The researchers tested four different "dance formations" for two new materials: Nb2Co2InSb and Nb2Co2GaSb.

1. The Heat Trap:

   • The original material (NbCoSn) let heat flow at about 13–18 units.
   • The new materials were incredible. They reduced heat flow to about 5–7 units.
   • Analogy: They turned a superhighway for heat into a bumpy, winding country road. The heat flow was cut by more than half, sometimes even by five times!

2. The Electricity Flow:

   • They needed to make sure the "electric cars" could still drive fast.
   • Surprisingly, the ordered formations (the strict military grid) were actually better at conducting electricity than the messy, random ones. The electrons preferred the organized dance floor.

3. The Winner (The "Figure of Merit"):

   • The ultimate score is called zT. A higher score means a better energy converter.
   • The old material (NbCoSn) had a score of roughly 0.3. It was okay, but not great.
   • The new materials scored between 1.7 and 2.6.
   • Analogy: If the old material was a bicycle, the new materials are Formula 1 race cars. They are nearly 8 to 10 times more efficient at turning heat into electricity.

The Conclusion

This paper shows that by simply rearranging atoms and introducing a little bit of "atomic chaos" (mixing heavy and light atoms), scientists can create materials that are perfect for harvesting waste heat.

• Nb2Co2GaSb (with a specific ordered structure) and Nb2Co2InSb (with a specific disordered structure) are the new champions.
• They are so efficient that they could be used to build both the "positive" and "negative" sides of a thermoelectric device, making them the perfect building blocks for future green energy technology.

In short: The researchers found a way to build a material that says "Yes" to electricity and "No" to heat, turning waste into a powerful new energy source.

Technical Summary

1. Problem Statement

Half-Heusler (hH) alloys are promising thermoelectric materials due to their tunable electronic structures, mechanical robustness, and thermal stability. However, their widespread application is hindered by inherently high lattice thermal conductivity ($k_L$), which limits the thermoelectric figure of merit ($zT$).

• Case Study: The parent compound NbCoSn (a ternary half-Heusler) has a high power factor but suffers from a very high $k_L$ (experimentally 13.25 W/mK, theoretically ~18 W/mK). Consequently, its $zT$ is extremely low (0.05 at room temperature).
• Objective: The study aims to develop derivative quaternary double half-Heusler (DHH) structures by replacing Sn atoms in NbCoSn with combinations of In/Ga and Sb. The goal is to maintain favorable electronic properties while drastically reducing $k_L$ through mass disorder scattering at lattice sites, thereby achieving a significantly higher $zT$.

2. Methodology

The research employed First-Principles Density Functional Theory (DFT) combined with semi-classical transport theories.

• Structural Models: Four structural phases were investigated for both Nb$_2$Co$_2$InSb and Nb$_2$Co$_2$GaSb:
  1. Ordered Structure 1 (OS1): Generated by substituting Sn with In/Ga and Sb in a conventional unit cell.
  2. Ordered Structure 2 (OS2): Taken from the Open Quantum Materials Database (OQMD), predicted as the most stable for Nb$_2$Co$_2$InSb.
  3. Special Quasirandom Structures (SQS): Two models (SQS1 and SQS2) representing disordered solid solutions to simulate high-temperature synthesis conditions.
• Computational Tools:
  • Electronic Structure: Quantum ESPRESSO (PBE-GGA, ultrasoft pseudopotentials).
  • Transport Properties: BoltzTraP2 code (constant relaxation time approximation) combined with Deformation Potential Theory to calculate carrier relaxation times ($\tau$) and mobilities ($\mu$).
  • Lattice Thermal Conductivity: Calculated using the Debye-Callaway model. Crucially, the model included scattering mechanisms from Normal (N) and Umklapp (U) processes, as well as mass and strain field fluctuation scattering arising from the random distribution of atoms in the DHH structures.
  • Stability: Phonon dispersion calculations (DFPT) were used to assess dynamical stability.

3. Key Contributions

• Systematic Comparison: A comprehensive comparison of ordered vs. disordered (SQS) phases for two specific DHH systems derived from NbCoSn.
• Stability Analysis: Determination of the most stable configurations: OS2 is the ground state for Nb$_2$Co$_2$InSb, while SQS1 is the most stable for Nb$_2$Co$_2$GaSb.
• Mechanism Elucidation: Demonstrated that the introduction of mass disorder in DHHs creates significant point-defect scattering, reducing $k_L$ to less than half (approx. 1/5th) of the parent ternary compound.
• High-Performance Prediction: Identified specific structural phases that achieve high $zT$ values (>2.0) at elevated temperatures, making them viable for both n-type and p-type thermoelectric legs.

4. Key Results

A. Structural and Electronic Properties

• Stability: All structures were dynamically stable except for the SQS2 phase of Nb$_2$Co$_2$GaSb, which exhibited imaginary phonon modes.
• Band Gaps: Disorder generally increased the band gap. For Nb$_2$Co$_2$InSb, gaps ranged from 0.21 eV (OS1) to 0.74 eV (SQS2).
• Carrier Mobility: Ordered structures (OS1, OS2) exhibited significantly higher carrier mobility compared to disordered SQS structures.
  • Nb$_2$Co$_2$GaSb (OS1): Holes showed exceptional mobility of 1746.7 cm$^2$/Vs.
  • Nb$_2$Co$_2$InSb: Holes also had lower effective masses than electrons in ordered phases.

B. Electronic Transport (Power Factor)

• Power Factor ($PF = S^2\sigma$): Ordered structures generally outperformed disordered ones due to higher mobility.
  • Nb$_2$Co$_2$GaSb (OS1): Achieved a peak p-type power factor of 34.85 mW/mK$^2$ at 1000 K.
  • Nb$_2$Co$_2$InSb (OS1): Achieved a peak p-type power factor of 45.40 mW/mK$^2$.
• Seebeck Coefficient: Peak values ranged between 277–421 $\mu$V/K for n-type and 151–415 $\mu$V/K for p-type, depending on the structure and carrier concentration.

C. Lattice Thermal Conductivity ($k_L$)

• Drastic Reduction: The $k_L$ of the DHH compounds was found to be approximately one-fifth that of the parent NbCoSn.
  • NbCoSn: ~13–18 W/mK.
  • Nb$_2$Co$_2$InSb (300 K): Ranged from 5.40 to 6.78 W/mK (with mass/strain scattering).
  • Nb$_2$Co$_2$GaSb (300 K): Ranged from 4.70 to 5.66 W/mK (with mass/strain scattering).
• Scattering Mechanism: The reduction is attributed to strong phonon scattering by mass and strain field fluctuations caused by the random occupation of the 4c Wyckoff position by In/Ga and Sb atoms.

D. Figure of Merit ($zT$)

The combination of high power factors and low $k_L$ resulted in exceptional $zT$ values at 1200 K:

• Nb$_2$Co$_2$InSb:
  • Best phase: SQS2 (disordered).
  • Peak $zT$: 1.73 (n-type) and 2.34 (p-type).
• Nb$_2$Co$_2$GaSb:
  • Best phase: OS2 (ordered).
  • Peak $zT$: 2.61 (n-type) and 2.31 (p-type).
• Comparison: These values are ~7–8 times higher than the parent NbCoSn ($zT \approx 0.32$ n-type, $0.12$ p-type at 1200 K).

5. Significance

This study provides a robust theoretical framework for designing high-efficiency thermoelectric materials based on half-Heusler alloys.

1. Strategy Validation: It confirms that converting ternary half-Heuslers into quaternary double half-Heuslers via aliovalent substitution is an effective strategy to decouple electronic transport from thermal transport.
2. Material Selection: It identifies Nb$_2$Co$_2$GaSb (OS2) and Nb$_2$Co$_2$InSb (SQS2) as top-tier candidates for high-temperature thermoelectric applications, offering $zT > 2.0$.
3. Device Viability: The high $zT$ for both electron and hole carriers suggests these materials can serve as both n-type and p-type legs in thermoelectric modules, a critical requirement for practical device fabrication.
4. Thermal Stability: The high melting points and thermal stability of the parent NbCoSn system are retained, ensuring these materials can operate in high-temperature waste heat recovery applications.