Reaching Quantum Critical Point by Adding Non-magnetic Disorder in Single Crystals of Superconductor $(\text{Ca}_x\text{Sr}_{1-x})_3\text{Rh}_4\text{Sn}_{13}$

This study demonstrates that controlled non-magnetic disorder induced by electron irradiation can suppress the charge-density wave order in $(\text{Ca}_x\text{Sr}_{1-x})_3\text{Rh}_4\text{Sn}_{13}$ single crystals, driving the system from a Fermi liquid to a non-Fermi liquid regime and refining the location of its non-magnetic quantum critical point to the composition range of $x=0.75$–$0.85$.

The Gist

Imagine a bustling city where two different groups of people are trying to organize a parade. One group wants to march in a perfect, rigid formation (this represents the Charge-Density Wave, or CDW). The other group wants to dance freely and spontaneously (this represents Superconductivity, where electricity flows with zero resistance).

Usually, these two groups fight for space. If the rigid marchers take over the streets, the dancers can't move. If the dancers take over, the marchers can't form their lines.

In the material studied in this paper, a special alloy called $(Ca_xSr_{1-x})_3Rh_4Sn_{13}$, scientists found a very rare spot where these two groups are trying to coexist right under a "dome" of superconductivity. They suspected there was a hidden "tipping point" (called a Quantum Critical Point or QCP) where the rigid marchers would disappear completely, allowing the dancers to take over the whole city.

The Problem: How to Find the Tipping Point?

Usually, to find this tipping point, scientists try to change the "recipe" of the city (adding more Calcium or less Strontium) or squeeze the city with pressure. But changing the recipe is messy. It's like trying to fix a car engine by swapping out the entire chassis; you change the shape of the engine and the fuel lines at the same time, making it hard to know which change caused the result.

The Solution: The "Controlled Chaos" Experiment

The researchers in this paper came up with a clever, new way to test the city. Instead of changing the recipe, they introduced controlled chaos.

Think of the city as a smooth, icy skating rink.

• The Clean Rink: When the ice is perfect, the rigid marchers (CDW) can glide in a perfect, synchronized line.
• The Chaos: The scientists shot high-speed electrons at the material. Imagine this as throwing thousands of tiny, invisible pebbles onto the ice. These pebbles create tiny bumps and holes (disorder) but don't change the chemical makeup of the ice itself.

What Happened?

As they added more and more "pebbles" (disorder):

1. The Marchers Stumbled: The rigid marchers (the CDW) couldn't keep their perfect formation anymore. The bumps in the ice broke their synchronization. The temperature at which they could march dropped lower and lower.
2. The Tipping Point Arrived: At a specific amount of chaos (a specific dose of electron radiation), the marchers completely lost their ability to form a long line. They vanished. This was the Quantum Critical Point.
3. The Dancers Took Over: Once the marchers were gone, the material behaved like a "non-Fermi liquid." In simple terms, the electrons stopped acting like individual billiard balls bouncing around (normal metal behavior) and started acting like a strange, collective fluid. The electrical resistance became perfectly proportional to the temperature (a straight line), which is a hallmark of this critical state.

The "Overshoot"

Here is the most interesting part: The scientists kept adding chaos even after the marchers disappeared.

• Too much chaos: Eventually, the city became too messy. The dancers (superconductivity) started to stumble too. The material went from being a "strange fluid" back to acting like a normal metal again, just with a lot of bumps.
• The Lesson: They proved that you can use "disorder" (the pebbles) as a precise dial to tune a material. You can turn the dial to find the exact moment of quantum criticality, and even turn it past that point to see what happens on the other side.

Why Does This Matter?

This is a big deal for physics because:

• New Tool: It proves that "disorder" isn't just a nuisance; it's a powerful tool. Scientists can now use it to tune materials to their most interesting states without changing their chemical recipe.
• Superconductivity: Understanding these critical points might help us figure out how to make superconductors work at higher temperatures (maybe even room temperature), which would revolutionize power grids, maglev trains, and computers.
• Universal Rule: The paper suggests that this "Imry-Ma" rule (that disorder breaks long-range order) applies to many different materials, not just this one.

In a nutshell: The scientists took a material where two quantum states were fighting, threw in some carefully measured "pebbles" of disorder, and successfully pushed the material to a magical tipping point where the rules of physics change, proving that chaos can be used as a precise control knob for the quantum world.

Technical Summary

1. Problem Statement

The study addresses the challenge of tuning quantum phase transitions (QPTs) to reach a Quantum Critical Point (QCP) without altering the chemical composition or applying external pressure.

• Context: In the Remeika series superconductor $(Ca_xSr_{1-x})_3Rh_4Sn_{13}$, a competition exists between a Charge-Density Wave (CDW) structural transition and superconductivity. Previous studies suggested a QCP exists at a critical calcium concentration of $x_c \approx 0.9$ (at ambient pressure), where the CDW transition temperature ($T_{CDW}$) drops to zero under the superconducting dome.
• Limitation of Traditional Methods: Tuning via chemical doping (changing $x$) or pressure simultaneously alters the electronic band structure, Fermi surface topology, and introduces substitutional disorder. This makes it difficult to isolate the specific effect of disorder on the suppression of long-range order versus changes in the electronic structure.
• Objective: To demonstrate that controlled, non-magnetic point-like disorder can serve as an independent tuning parameter to suppress the CDW order, drive the system to a QCP, and potentially push it beyond the critical point into a Fermi-liquid regime, all while keeping the chemical composition constant.

2. Methodology

The researchers employed a controlled disorder approach using electron irradiation on high-quality single crystals.

• Sample Preparation: Single crystals of $(Ca_xSr_{1-x})_3Rh_4Sn_{13}$ with varying compositions ($x = 0, 0.5, 0.75, 0.8, 1.0$) were grown using tin flux.
• Disorder Induction: Samples were subjected to 2.5 MeV electron irradiation at the SIRIUS accelerator facility (France).
  • Mechanism: High-energy electrons create Frenkel pairs (vacancy-interstitial defects) without changing the chemical potential (no "charge doping").
  • Control: Irradiation was performed at low temperatures (~22 K) to prevent defect recombination, followed by warming to room temperature to stabilize a metastable population of point defects.
  • Dosimetry: The accumulated dose was measured in Coulombs per square centimeter ($C/cm^2$), with samples irradiated in incremental steps (up to ~5.15 $C/cm^2$).
• Measurements:
  • Electrical Resistivity ($\rho(T)$): Measured in a four-probe configuration to track the evolution from Fermi-liquid ($T^2$) to non-Fermi-liquid ($T^n$, where $n \le 1$) behavior.
  • Hall Effect: Measured to confirm that irradiation did not alter carrier concentration (ruling out charge doping effects).
  • Data Analysis: The resistivity data was fitted to a power law $\Delta\rho = A + BT^n$ to extract the exponent $n$. The derivative $d\rho/dT$ was analyzed to distinguish between linear and quadratic temperature dependencies.

3. Key Contributions

1. Disorder as a Tuning Parameter: The paper provides the first experimental demonstration that controlled non-magnetic disorder can tune a system with coexisting CDW and superconductivity to a QCP and beyond, validating theoretical predictions (e.g., Imry-Ma argument) that disorder destabilizes long-range order.
2. Refinement of the QCP Location: By starting with a composition ($x=0.75$) below the previously suggested $x_c=0.9$ and driving the CDW to zero via disorder, the authors refine the QCP location to the interval $x \in [0.75, 0.85]$.
3. Decoupling Scattering from Band Structure: By using irradiation on a fixed composition, the study isolates the effect of scattering on the CDW order, proving that disorder alone can suppress the CDW transition temperature to absolute zero.

4. Key Results

• Suppression of CDW Order:

  • In the $x=0.75$ sample, increasing the irradiation dose progressively suppressed the CDW transition temperature ($T_{CDW}$).
  • At a critical dose of 4.1 C/cm², the long-range CDW order was fully suppressed ($T_{CDW} \to 0$).
  • The transition broadened as disorder increased, indicating the destruction of long-range coherence while short-range correlations persisted.

• Evolution of Resistivity (Fermi Liquid to Non-Fermi Liquid):

  • Pristine State: The $x=0.75$ sample exhibited a mix of linear and quadratic terms in resistivity.
  • At the QCP: As the CDW was suppressed, the resistivity evolved into a perfectly linear temperature dependence ($\rho \propto T$) at the critical dose. This is the hallmark signature of a quantum critical point.
  • Beyond the QCP: Further increasing the disorder (higher doses) caused the resistivity to revert to a quadratic dependence ($\rho \propto T^2$), indicating the system had been "overshot" the QCP and entered a clean Fermi-liquid regime where CDW fluctuations were completely eliminated.

• Superconducting Transition ($T_c$):

  • In stoichiometric $Sr_3Rh_4Sn_{13}$, disorder initially enhanced $T_c$, suggesting an intrinsic interplay where suppressing the CDW gap increases the density of states available for pairing.
  • In alloyed compositions ($x=0.75, 0.8$), $T_c$ was suppressed by disorder, consistent with unconventional pairing mechanisms sensitive to scattering.

• Hall Effect Verification:

  • Hall coefficient measurements showed no change in carrier concentration before and after irradiation, confirming that the observed effects were due to point-like disorder (scattering) and not chemical doping.

5. Significance

• Novel Tuning Mechanism: The study establishes controlled disorder as a viable, non-thermal tuning parameter comparable to pressure or composition. This is crucial for systems where chemical doping introduces complex, hard-to-deconvolute changes in the electronic structure.
• Validation of Theory: The results strongly support the theoretical framework (Imry-Ma argument) that random fields (disorder) destabilize long-range order in dimensions $D \le 4$, driving the system toward a continuous quantum phase transition.
• Implications for Unconventional Superconductivity: By demonstrating that disorder can tune a system to a QCP where superconductivity is enhanced (in specific cases) or modified, the work opens new avenues for exploring the "glue" of unconventional superconductivity. It suggests that the interplay between CDW fluctuations and superconductivity can be manipulated without changing the material's stoichiometry.
• Future Applications: This methodology is applicable to a wide range of correlated electron systems, including high-temperature superconductors, heavy-fermion materials, and topological materials, offering a new tool to map phase diagrams and explore quantum critical phenomena.