Quantum transport reveals spin glass correlations in a 2D network of TbPc$_{2}$ single-molecule magnets grafted on graphene

This study demonstrates that a 2D network of TbPc$_2$ single-molecule magnets grafted on graphene induces long-range, 2D Ising spin-glass magnetic correlations in the graphene substrate, as evidenced by the analysis of low-temperature magnetoresistance, 1/f noise, and universal conductance fluctuations.

The Gist

Imagine a piece of graphene as a perfectly smooth, ultra-thin sheet of graphene (a single layer of carbon atoms). It's like a super-fast highway for electrons, the tiny particles that carry electricity. Normally, this highway is very orderly.

Now, imagine sprinkling a layer of TbPc2 molecules onto this highway. These aren't just any molecules; they are "single-molecule magnets." Think of them as tiny, individual compass needles that are stuck to the road. Each one has a strong magnetic personality, pointing either "up" or "down."

The scientists in this paper wanted to see what happens when these magnetic compass needles interact with the electrons zooming by on the graphene highway. They discovered something surprising and fascinating: the electrons and the magnets start to "dance" together in a chaotic, frozen way that resembles a spin glass.

Here is a breakdown of their findings using simple analogies:

1. The "Traffic Jam" of Magnetism (Spin Glass)

In a normal magnet (like a fridge magnet), all the tiny compass needles line up perfectly in the same direction. In a "spin glass," it's a mess. The compass needles want to point in different directions because they are arguing with their neighbors, but they can't decide on a single order. They get "stuck" in a frozen, disordered state.

Usually, this happens in 3D metal alloys (like mixing copper and manganese). But here, the scientists found this chaotic, frozen state happening in a 2D layer (just the flat graphene sheet). It's like watching a crowd of people on a flat dance floor who can't agree on a dance move, so they just freeze in random poses.

2. The "Static" on the Radio (1/f Noise)

How did they know this was happening? They didn't just look at the magnets; they listened to the electricity.

When you turn on an old radio, you hear a hiss or static. In electronics, this is called noise. The scientists found that the electricity flowing through their graphene device had a very specific type of "hiss" called 1/f noise.

• The Analogy: Imagine a crowd of people whispering. If they all whisper at once, it's a steady hum. But if the crowd is made of groups of people who start whispering, stop, start again, and change their volume at random intervals, you get a "hiss" that sounds like static.
• The Discovery: This "hiss" in their device wasn't random electrical noise. It was the sound of the magnetic compass needles (the TbPc2 molecules) slowly flipping and changing their minds. The fact that the noise followed a "1/f" pattern meant these magnets were interacting over long distances, creating a complex, glassy network.

3. The "Magic" Magnetic Field

The scientists played a trick on the system: they applied a magnetic field (like holding a giant magnet near the device).

• What happened: When they turned on the magnetic field, the "hiss" (the noise) got quieter and eventually disappeared.
• The Analogy: Imagine that chaotic crowd of people arguing again. If you suddenly shout a loud command (the magnetic field), everyone stops arguing and points in the same direction. The chaos (noise) vanishes because the magnets are now forced to align.
• The Significance: This proved that the noise was indeed coming from the magnetic molecules, not just the graphene itself. The magnetic field "tamed" the wild, glassy behavior.

4. The "Ghost" in the Machine (Irreversibility)

In a normal, predictable system, if you run a test forward and then backward, you get the same result.

• The Finding: In this graphene device, the results were different depending on whether they were increasing or decreasing the magnetic field. The system had "memory."
• The Analogy: It's like walking through a field of tall grass. If you walk forward, you flatten the grass. If you try to walk backward, the grass is still flattened; you can't retrace your exact path. The system remembers where it has been. This "memory" is a classic sign of a spin glass.

Why Does This Matter?

This experiment is a big deal for a few reasons:

1. New State of Matter: They found a way to create a "spin glass" state in a flat, 2D world, which is theoretically very difficult to do.
2. Quantum Computing: Understanding how these tiny magnets interact with electrons is a stepping stone toward building better quantum computers. These molecules act like tiny, controllable magnets that talk to electrons.
3. A New Tool: They showed that by listening to the "static" (noise) in a material, you can learn about the magnetic secrets hidden inside it, even if you can't see the magnets directly.

In Summary:
The scientists took a super-fast electron highway (graphene), parked a layer of tiny, stubborn magnetic compass needles on it, and listened to the electricity. They heard a chaotic "hiss" that proved the magnets were stuck in a frozen, disordered argument (a spin glass). When they applied a magnetic field, they forced the magnets to agree, and the noise stopped. It's a beautiful demonstration of how tiny magnetic atoms can change the behavior of electricity in a whole new way.

Technical Summary

1. Problem Statement

The research addresses the challenge of inducing and detecting long-range magnetic correlations in two-dimensional (2D) systems. While graphene is highly sensitive to adsorbed atoms, creating a hybrid system with stable magnetic order is difficult due to the delicate balance required: the interaction must be strong enough to mediate exchange between the molecule and graphene's charge carriers, yet weak enough to preserve the molecule's magnetic moment (avoiding Kondo screening or quenching).

Specifically, the authors investigate whether a monolayer of Terbium Phthalocyanine (TbPc2) single-molecule magnets (SMMs) grafted onto graphene can induce a 2D spin-glass phase. Unlike 3D spin glasses which have a finite transition temperature, 2D Ising spin glasses are theoretically predicted not to order at finite temperatures. The study aims to detect if magnetic correlations and glassy dynamics (characterized by slow relaxation and $1/f$ noise) can still emerge in this 2D limit through transport measurements.

2. Methodology

The study employs low-temperature quantum transport measurements on hybrid devices:

• Sample Fabrication:
  • Substrates: Two types of graphene field-effect transistors (FETs) were used: (1) Graphene on a SiO$_2$/Si substrate, and (2) Graphene on a monolayer Tungsten Disulfide (WS$_2$) flake (to induce strong spin-orbit interactions).
  • Functionalization: Devices were coated with TbPc2 molecules via drop-casting from solutions of varying concentrations ($10^{-5}$ M and $10^{-6}$ M). A control sample coated with Iron Porphyrin (FeTPP) was also prepared to isolate magnetic effects.
  • Characterization: Gate-voltage sweeps confirmed significant hole doping and charge transfer from graphene to TbPc2 molecules.
• Measurement Conditions:
  • Experiments were conducted in a dilution refrigerator at temperatures as low as 50 mK.
  • A 3D superconducting vector magnet allowed for both perpendicular (out-of-plane) and parallel (in-plane) magnetic fields up to several Tesla.
• Techniques:
  • Universal Conductance Fluctuations (UCFs): Analyzed for reproducibility and symmetry to detect phase coherence and time-reversal symmetry breaking.
  • Resistance Noise Spectroscopy: Measured low-frequency ($1/f$) resistance noise as a function of magnetic field ($B$), temperature ($T$), and gate voltage ($V_g$).
  • Weak Localization: Fitted magnetoconductance data to extract the phase coherence length ($L_\phi$).

3. Key Contributions

• Detection of 2D Spin-Glass Signatures: The paper provides the first experimental evidence of spin-glass-like correlations in a purely 2D network of Ising spins mediated by graphene charge carriers, despite the theoretical absence of a finite-temperature phase transition in 2D Ising systems.
• Noise as a Probe: It establishes that low-temperature resistance noise (specifically $1/f$ noise) is a more sensitive probe of magnetic phase transitions in mesoscopic 2D systems than standard DC magnetization or resistance measurements.
• Exchange Interaction Mechanism: It demonstrates that the coupling between the TbPc2 molecular spins and graphene carriers is strong enough to create a disordered magnetic network, distinct from the weak coupling observed in Fe-porphyrin systems.

4. Key Results

A. Mesoscopic Signatures and Symmetry Breaking

• Non-Reproducible Fluctuations: Unlike the FeTPP control sample (which showed reproducible, even-in-field UCFs), the TbPc2-coated samples exhibited time-dependent, non-reproducible conductance fluctuations.
• Time-Reversal Symmetry Breaking: The average resistance $\langle R(B) \rangle$ displayed an odd component with respect to the magnetic field (i.e., $\langle R(B) \rangle \neq \langle R(-B) \rangle$). This indicates broken time-reversal symmetry, a hallmark of frozen magnetic correlations in spin glasses.
• Reduced Phase Coherence: The phase coherence length ($L_\phi$) was significantly reduced in TbPc2 samples compared to FeTPP samples, attributed to spin-flip scattering by the magnetic moments.

B. $1/f$ Noise Characteristics

• Field Dependence: The resistance noise amplitude was maximal at zero field and decreased sharply as the magnetic field increased beyond 0.1–0.2 T. This suppression corresponds to the energy scale of the exchange interaction between neighboring molecules ($\Delta J_{Tb-Tb} \approx 0.1$ meV).
• Temperature Dependence: A sharp increase in noise power was observed below $T_g \approx 0.4$ K (for the WS$_2$/Graphene sample). This "freezing" temperature is consistent with the exchange energy scale, marking a crossover from fast to slow relaxation regimes.
• Gate Voltage Dependence: The noise was strongest near the Dirac point (low carrier density) and showed an asymmetry between electron and hole doping, suggesting the exchange coupling is mediated by the delocalized spins on the phthalocyanine ligands.

C. The "Freezing" Line

The authors defined a characteristic line $T_g(B)$ in the temperature-magnetic field plane. Below this line, the system exhibits slow relaxation dynamics (glassy behavior) detectable as excess $1/f$ noise. Above this line, the noise is suppressed, indicating a paramagnetic-like state. This behavior mimics the de Almeida-Thouless line found in 3D spin glasses, despite the system being 2D.

D. Control Experiments

• FeTPP Control: Samples coated with FeTPP showed no $1/f$ noise enhancement, no time-dependence, and no odd-field components, confirming that the effects are specific to the strong magnetic moment and coupling of TbPc2.
• In-Plane Field: Applying an in-plane magnetic field up to 1 T did not suppress the noise, consistent with the strong Ising-like uniaxial anisotropy of the TbPc2 moments (which are pinned perpendicular to the graphene plane).

5. Significance

• Fundamental Physics: The work challenges the strict theoretical limit that 2D Ising spin glasses cannot order at finite temperatures. It suggests that while a true thermodynamic phase transition may not occur, a dynamic crossover (freezing) into a glassy state with long relaxation times is observable in mesoscopic 2D systems.
• Platform for Quantum Simulation: The TbPc2/graphene system serves as a versatile platform for studying 2D magnetic phase transitions, disorder, and quantum interference effects.
• Future Applications: The ability to tune magnetic correlations via gate voltage and substrate engineering (e.g., using WS$_2$) opens pathways for developing molecular spintronic devices and exploring quantum spin-glass states using transverse fields (by using molecules with lower anisotropy).

In summary, the paper demonstrates that graphene functionalized with TbPc2 molecules acts as a highly sensitive detector for magnetic correlations, revealing a 2D spin-glass-like state characterized by $1/f$ noise, time-reversal symmetry breaking, and a field-dependent freezing transition.