Thermal Hall conductivity of electron-doped cuprates: Electrons and phonons

This study demonstrates that in electron-doped cuprates, the thermal Hall conductivity arises from competing contributions of electrons and phonons, where the latter's negative signal persists across doping levels and suggests a fundamental link to antiferromagnetic correlations within the pseudogap phase.

The Gist

Imagine a cup of hot coffee. Usually, heat just flows straight from the hot center to the cold rim. But in certain special materials called cuprates (which are famous for being high-temperature superconductors), things get weird when you put them in a strong magnetic field. Instead of flowing straight, the heat starts to drift sideways, like a river current being pushed by the wind. This sideways flow of heat is called the Thermal Hall Effect.

For a long time, scientists knew that in these materials, electrons (the tiny charged particles that carry electricity) could do this. But recently, they discovered something surprising: even the phonons (which are just vibrations of the crystal lattice, like sound waves traveling through a solid) can also carry heat sideways and create this effect.

This paper is like a detective story trying to figure out who is doing the driving in a very clean, high-quality crystal of an electron-doped cuprate (a material called NCCO).

The Cast of Characters

1. The Electrons: Think of these as race cars. They are fast, charged, and when you put them in a magnetic field, they naturally curve to the right. They generate a "positive" sideways heat flow.
2. The Phonons: Think of these as bicycles or pedestrians. They are neutral (no electric charge) and represent the vibration of the material itself. Surprisingly, in these cuprates, they curve to the left, creating a "negative" sideways heat flow.
3. The Impurities: These are like potholes or speed bumps in the road. In a dirty crystal, there are lots of them. In a clean crystal, the road is smooth.

The Mystery: Who Wins the Race?

The researchers looked at two samples of the same material, but with slightly different "cleanliness":

• Sample A (The "Dirty" One): This sample had more impurities (potholes). Here, the phonons (bicycles) were the main drivers. Because they curve left, the total heat flow was negative. It looked like the phonons were in total control.
• Sample B (The "Clean" One): This sample was much smoother. Here, the electrons (race cars) could zoom around without hitting as many potholes. Because they curve right, they started to dominate. In fact, in this clean sample, the electrons and phonons were fighting each other with almost equal strength, but in opposite directions. The electrons won, so the total heat flow was positive.

The Big Reveal: In the cleanest crystals, the electrons and phonons are equally strong, but they pull in opposite directions. It's like a tug-of-war where one team is pulling left and the other is pulling right. In the dirty samples, the electron team is too weak to pull hard, so the phonon team wins easily.

Why Does This Matter? (The "Why" Behind the Curve)

The scientists asked a big question: Why do the phonons (bicycles) curve at all? They don't have an electric charge, so the magnetic field shouldn't push them.

• The Old Theory (The "Charged Impurity" Idea): Some scientists thought phonons curved because they were bouncing off charged impurities (like a ball hitting a magnetized wall).
• The New Evidence: The researchers found that this "charged impurity" theory is wrong. Why? Because in the "metallic" state (where the material conducts electricity well), the electrons act like a shield that cancels out any local electric charges. If the phonons were curving because of charged impurities, the effect should disappear in the clean, metallic samples. But it didn't! The phonons kept curving left, even when the "charged" theory said they shouldn't.

The Real Culprit: The "Spin Texture"

So, what is making the phonons curve? The paper suggests it's the magnetic order of the material.

Imagine the atoms in the crystal are like a crowd of people holding hands. In these cuprates, even when they are conducting electricity, there is a hidden "spin texture"—a subtle, organized pattern of magnetic spins (like everyone in the crowd subtly turning their heads in a specific direction).

The researchers propose that the phonons (vibrations) are scattering off this invisible magnetic pattern. It's like a bicycle rider trying to ride through a crowd that is subtly swaying in a specific rhythm; the rider gets pushed sideways by the rhythm of the crowd, not by hitting a specific person.

The Takeaway

1. Two Forces at Play: In these materials, heat is carried by both electrons (positive effect) and phonons (negative effect). In very clean samples, they cancel each other out partially, revealing that they are both powerful.
2. The Phonon Mystery: The fact that phonons curve even when the material is a good metal proves that the cause isn't simple charged impurities.
3. The Magnetic Connection: The most likely cause is the material's internal magnetic structure (antiferromagnetic correlations). This suggests that the "pseudogap" phase (a mysterious state in these superconductors) is deeply connected to these magnetic patterns.

In short: The paper shows that in these special crystals, heat is a tug-of-war between electrons and vibrations. The vibrations are being pushed sideways by the material's hidden magnetic heartbeat, a discovery that helps us understand the secret life of high-temperature superconductors.

Technical Summary

1. Problem Statement

The thermal Hall effect ($\kappa_{xy}$) in cuprate superconductors has been a subject of intense study, particularly the discovery of a large, negative $\kappa_{xy}$ in hole-doped cuprates persisting into the Mott insulating phase. This signal is attributed to phonons (lattice vibrations) acquiring a "handedness" (chirality) in a magnetic field, a phenomenon whose microscopic mechanism remains unknown.

While previous studies established that phonons dominate the negative $\kappa_{xy}$ in electron-doped cuprates (NCCO) at low doping (insulating/antiferromagnetic regime), the behavior at high doping (metallic regime) was unclear. In a metal, mobile electrons are expected to generate a positive thermal Hall signal via the Lorentz force (the thermal analog of the electrical Hall effect). The central problem addressed by this study is to disentangle the contributions of electrons and phonons to the total thermal Hall conductivity in high-doping electron-doped cuprates and to determine if the phonon mechanism changes between insulating and metallic states.

2. Methodology

The researchers investigated single crystals of the electron-doped cuprate $\text{Nd}_{2-x}\text{Ce}_x\text{CuO}_4$ (NCCO) with two nominal doping levels:

• $x = 0.16$ (Sample 1 & 2): High-quality crystals, with Sample 1 being exceptionally clean.
• $x = 0.17$: A slightly "dirtier" crystal.

Experimental Techniques:

• Thermal Transport Measurements: Conducted in magnetic fields up to 15 T (sufficient to suppress superconductivity) and temperatures ranging from 50 mK to 100 K.
  • Longitudinal thermal conductivity ($\kappa_{xx}$) and thermal Hall conductivity ($\kappa_{xy}$) were measured using a steady-state method with RuO$_2$ thermometers (low T) and Type-E thermocouples (high T).
  • The Wiedemann-Franz (WF) law ($\kappa_0/T = L_0/\rho_0$) was utilized to extract residual resistivity ($\rho_0$) and quantify the electronic contribution to thermal transport, as heat transport is not contaminated by $c$-axis currents.
• Quantum Oscillations (TDO): Tunnel Diode Oscillator measurements were performed in pulsed magnetic fields up to 85 T at 1.8 K to detect Shubnikov-de Haas oscillations. This served as a probe for Fermi surface reconstruction and sample purity (mean free path).
• Data Analysis: The total $\kappa_{xy}$ was decomposed into electronic ($\kappa^e_{xy}$) and phononic ($\kappa^{ph}_{xy}$) components by comparing samples of different purity and utilizing the known behavior of electrons at very low temperatures (where phonon contribution vanishes).

3. Key Contributions

• Separation of Channels: The study successfully isolates and quantifies two competing contributions to the thermal Hall effect in the metallic state of NCCO: a positive electronic contribution and a negative phononic contribution.
• Sample Quality Dependence: It demonstrates that the sign of the total $\kappa_{xy}$ depends on sample purity. In the cleanest sample ($x=0.16$), the electronic signal dominates (positive $\kappa_{xy}$), whereas in the dirtier sample ($x=0.17$), the phonon signal dominates (negative $\kappa_{xy}$).
• Mechanism Constraint: The authors provide strong evidence that the phonon thermal Hall effect is independent of the electronic state (insulator vs. metal). Since the negative phonon signal persists in the metallic regime where charged impurities should be screened, mechanisms relying on skew scattering off charged defects (like oxygen vacancies) are ruled out.

4. Key Results

A. Thermal Hall Conductivity ($\kappa_{xy}$)

• $x = 0.17$ Sample: Exhibits a negative $\kappa_{xy}$ across all temperatures (3 K to 100 K). This is attributed entirely to phonons, consistent with lower-doping behavior.
• $x = 0.16$ (Clean) Sample: Exhibits a positive $\kappa_{xy}$ at all temperatures.
  • At $T < 4$ K, $\kappa_{xy}/T$ is constant, confirming the signal is purely electronic (fermionic).
  • The residual linear term $\kappa^e_{xy}/T$ is 10 times larger in the $x=0.16$ sample than in $x=0.17$, confirming that the electronic contribution scales with sample cleanliness (mean free path).
• Decomposition: By subtracting the phonon-dominated signal of the $x=0.17$ sample from the $x=0.16$ data, the authors estimate the electronic contribution. They find that in the cleanest samples, the positive electronic and negative phononic contributions are of comparable magnitude.

B. Electronic Transport and Fermi Surface

• Residual Resistivity: The $x=0.16$ sample has a residual resistivity $\rho_0 \approx 2.7 \, \mu\Omega\text{cm}$, roughly half that of the $x=0.17$ sample ($6.1 \, \mu\Omega\text{cm}$), confirming higher purity.
• Hall Coefficient: The extracted Hall coefficient ($R_H \approx 0.36 \, \text{mm}^3/\text{C}$) is roughly half the value expected for a large, unreconstructed Fermi surface. This indicates Fermi surface reconstruction.
• Quantum Oscillations:
  • Oscillations were observed only in the $x=0.16$ sample, not in $x=0.17$, confirming the former has a longer electronic mean free path.
  • Two frequencies were detected: a low frequency ($F_1 \approx 300$ T) corresponding to small hole pockets, and a high frequency ($F_2 \approx 11.3$ kT) corresponding to the large unreconstructed Fermi surface area (via magnetic breakdown).
  • The presence of these oscillations confirms that short-range antiferromagnetic correlations persist even at high doping ($x=0.16$), leading to Fermi surface reconstruction below the critical doping $x^* \approx 0.175$.

C. Phonon Behavior

• The negative phononic $\kappa_{xy}$ in the metallic state ($x=0.16, 0.17$) has the same magnitude and temperature dependence as in the insulating state ($x < 0.15$).
• This persistence across the insulator-metal transition rules out mechanisms based on scattering off charged impurities, as mobile electrons in the metal would screen these charges, suppressing such an effect.

5. Significance and Implications

• Mechanism of Phonon Chirality: The results strongly suggest that the mechanism generating the phonon thermal Hall effect is linked to antiferromagnetic correlations (spin texture) rather than charged impurities. The signal persists as long as these correlations are significant (up to $x^*$).
• Unified Picture for Cuprates: The study proposes that the same mechanism (phonon scattering by spin textures or defects embedded in magnetic order) likely explains the phonon thermal Hall effect in hole-doped cuprates within their pseudogap phase (below $p^*$).
• Pseudogap Connection: If the phonon $\kappa_{xy}$ is a signature of antiferromagnetic correlations, its presence below the critical doping $p^*$ (and absence above it) in hole-doped systems provides evidence that the pseudogap phase is characterized by significant short-range spin correlations.
• Material Quality: The work highlights the critical importance of sample quality in thermal transport studies of cuprates, where the competition between electronic and phononic signals can invert the observed sign of the Hall effect.

In conclusion, this paper resolves the nature of thermal Hall transport in high-doping electron-doped cuprates, revealing a delicate balance between electrons and phonons, and provides compelling evidence that the enigmatic phonon thermal Hall effect is driven by magnetic correlations intrinsic to the cuprate lattice.