Theory of frozen flux in a narrow uniform superconducting strip after cooling in a small magnetic field

This paper derives and solves a dynamic-balance equation to quantify how thermal activation and cooling rates determine the freezing temperature and resulting residual vortex density in narrow superconducting strips cooled through their transition temperature in a small magnetic field.

The Gist

Imagine a superconducting strip as a long, narrow hallway. Inside this hallway, tiny magnetic particles called "vortices" want to live. However, the walls of the hallway (the edges of the strip) and a special force called the "Meissner effect" create a bumpy energy landscape. Think of this landscape as a series of hills and valleys.

When the strip is hot, these vortices are energetic and jittery. They can easily jump over the hills (energy barriers) to enter the hallway or escape out of it. As the strip cools down, the vortices lose energy. Eventually, the hills become too high for them to climb, and they get stuck.

This paper, written by Alexei E. Koshelev, investigates exactly when and how these vortices get stuck (or "freeze") as the strip cools in a magnetic field. Here is the breakdown of the findings using everyday analogies:

1. The Setup: A Narrow Hallway

The study focuses on very thin, narrow strips of superconducting material. In these narrow strips, the physics is simpler than in wide ones. The "hills" that keep the vortices out are created by the geometry of the strip itself.

• The Minimum Expulsion Field ($H_e$): Imagine a magnetic field strength so weak that the "hills" are so high that no vortices can get in at all. This is the theoretical limit where the strip is perfectly clean.
• The Reality: In real experiments, scientists often see trapped vortices even when the magnetic field is stronger than this theoretical limit. The paper asks: Why?

2. The Race Against Time: Cooling Down

The key to the problem is cooling.

• The Equilibrium State: If you could cool the strip infinitely slowly, the vortices would have plenty of time to find the perfect balance. They would leave the hallway if the magnetic field was too strong, or stay if it was just right.
• The Freeze-Out: In the real world, we cool things down at a specific speed. As the temperature drops, the "hills" get steeper, and the vortices get slower. At a certain point, the vortices get so sluggish that they can't climb the hills fast enough to escape, even though the "perfect" balance says they should.
• The Freezing Temperature ($T_{fr}$): This is the specific moment (temperature) when the vortices stop running away and get trapped. The paper calculates exactly when this happens.

3. The "Freezing" Mechanism

The author describes a "dynamic balance." Think of it like a busy door in a hallway:

• Entering: Vortices try to jump in.
• Exiting: Vortices try to jump out.
• The Balance: At high temperatures, people (vortices) are running back and forth quickly. The number of people inside stays steady based on how crowded the hallway is outside.
• The Lock: As the temperature drops, the "exit door" becomes incredibly hard to open. The vortices inside can't get out. The "entry door" also becomes hard to open, but the ones already inside are now trapped.
• The Result: The number of trapped vortices stops changing and stays at a fixed number, even though the "ideal" number should be zero. This is the "frozen flux."

4. Key Findings in Plain English

• It Happens Very Close to the "Melting" Point: The vortices don't freeze when the strip is cold; they freeze just as the strip is starting to become superconducting (very close to the transition temperature).
• The "Logarithmic" Factor: The paper finds that the temperature at which freezing happens is slightly higher than the point where random thermal noise usually matters. It's a small difference, but mathematically significant (described as a "large logarithmic factor").
• Speed Matters: If you cool the strip slower, the vortices have more time to escape, so they freeze at a lower temperature, and fewer get trapped. If you cool it faster, they get trapped earlier and more of them stay.
• The Magnetic Field is a Switch: The amount of trapped flux depends heavily on the magnetic field strength.
  • Just above the minimum limit ($H_e$), the number of trapped vortices is tiny (almost zero).
  • As you increase the magnetic field slightly, the number of trapped vortices explodes (increases extremely fast).
  • Because of this sharp increase, scientists can define an "Effective Expulsion Field." This is the magnetic field strength where the trapped vortices become strong enough to be detected by instruments.

5. Why Real Experiments Differ from Theory

The paper explains a common puzzle: Experiments often show that strips need a much stronger magnetic field to be "clean" (free of vortices) than the simple math predicts.

• The Explanation: The math assumes a perfectly smooth, uniform hallway. Real strips have bumps, scratches, and impurities (inhomogeneities).
• The Effect: These imperfections can act like "traps" that hold vortices in place even when the magnetic field is low. This makes it look like the strip is trapping more flux than it should, pushing the "effective" expulsion field to higher values.

Summary

The paper provides a mathematical "recipe" to predict how many magnetic vortices will get stuck in a narrow superconducting strip when it cools down. It explains that the vortices get trapped not because the magnetic field is too strong, but because the strip cools down too fast for the vortices to escape the energy barriers. This "freezing" happens very close to the temperature where the material becomes superconducting, and the amount of trapped flux depends sharply on the cooling speed and the exact magnetic field strength.

Technical Summary

Technical Summary: Theory of Frozen Flux in a Narrow Uniform Superconducting Strip

Problem Statement
The paper addresses the issue of residual trapped magnetic flux in long, narrow, uniform superconducting strips cooled through their transition temperature ($T_c$) in a constant, perpendicular magnetic field ($H$). This phenomenon is critical for the performance of superconducting electronic devices, such as SQUIDs, transmon qubits, and single-photon detectors. While experimental imaging has established that trapped flux is absent below a certain "effective flux-expulsion field" ($H_{exp}$), this experimental value typically exceeds the theoretical minimum flux-expulsion field ($H_e$) by a factor of 3–4. A fundamental gap exists in the theoretical understanding of the dynamic process: specifically, what determines the frozen-flux density after cooling, and how does the system transition from equilibrium to a frozen state? Previous work lacked a quantitative analysis of the formation of this frozen state.

Methodology
The author employs a dynamic balance approach to model the evolution of vortex density ($n$) during cooling. The analysis focuses on the regime where the strip width $W$ is much smaller than the Pearl screening length ($\Lambda$), allowing for analytical treatment of the vortex energy profile.

1. Energy Landscape: The study utilizes the known vortex energy profile $E(X)$ in a narrow strip, which includes contributions from the core energy, the interaction with strip edges, and the Meissner screening current. This profile defines an energy barrier $U_{out}$ for vortex exit and $U_{in}$ for vortex entry.
2. Dynamic Balance Equation: The time evolution of the vortex density is governed by a rate equation describing thermally activated hopping over these energy barriers:
   $$\frac{dn}{dt} = -n \frac{1}{\tau} e^{-U_{out}/T} + n_\xi \frac{1}{\tau} e^{-U_{in}/T}$$
   where $\tau$ is the Kramers attempt time and $n_\xi$ is a pre-exponential factor related to the maximum possible linear density.
3. Cooling Protocol: The system is assumed to be cooled at a constant rate $R = -T_c^{-1} dT/dt$. The solution to the dynamic equation is analyzed to determine when the vortex density falls out of equilibrium with the rapidly changing temperature.
4. Freezing Condition: The "freezing" of the flux is defined not as an abrupt stop, but as the point where the relaxation time (exponentially dependent on the barrier) becomes too long to track the changing equilibrium density. The freezing time $t_{fr}$ and temperature $T_{fr}$ are derived by equating the activated escape rate to the cooling-induced rate of change of the entry Boltzmann factor.

Key Contributions and Results

• Derivation of Freezing Temperature ($T_{fr}$): The paper derives a quantitative expression for the reduced freezing temperature $\varepsilon_{fr} = 1 - T_{fr}/T_c$. The analysis shows that $\varepsilon_{fr}$ exceeds the fluctuation width of the transition ($\varepsilon_f$) by a large logarithmic factor. Crucially, $\varepsilon_{fr}$ increases rapidly as the applied field $H$ approaches the minimum flux-expulsion field $H_e$ (where the energy barrier vanishes) and increases logarithmically as the cooling rate decreases.
• Frozen Flux Density ($n_{fr}$): An analytical expression for the final frozen vortex density is obtained. The density is finite for any $H > H_e$ but is exponentially small near $H_e$. The density exhibits a very strong dependence on the magnetic field, increasing steeply as $H$ increases.
• Scaling Laws: Despite the complexity of material parameters (coherence length $\xi$, penetration depth $\lambda$, viscosity), the ratios $\varepsilon_{fr}/\varepsilon_f$ and $n_{fr}/n_\xi$ depend only on two reduced parameters: the dimensionless cooling rate ($R\tau_0/\varepsilon_f$) and the ratio of the strip width to the coherence length at the fluctuation width ($2w\sqrt{\varepsilon_f}/\xi_{GL}$).
• Effective Flux-Expulsion Field ($H_{eff}$): The paper proposes defining the effective flux-expulsion field as the field at which the calculated frozen flux density reaches a detectable level (e.g., one vortex per strip length). The theory predicts that $H_{eff}$ is always larger than the theoretical $H_e$.
• Material-Specific Estimates: The theory is applied to Niobium (Nb) and Niobium Titanium Nitride (NbTiN) strips.
  • For Nb strips, the calculated $H_{eff}$ is close to the crossover field $H_s$ (where single-vortex approximation breaks down) but remains significantly lower than experimental values ($H_{exp}$). The author suggests this discrepancy may arise from large-scale sample inhomogeneities rather than the ideal strip model.
  • For NbTiN strips, the calculated $H_{eff}$ is in reasonable agreement with experimental data, though slightly higher. The larger London penetration depth and smaller coherence length in NbTiN lead to a wider fluctuation regime and different scaling behavior compared to Nb.

Significance and Claims
The paper claims to provide the first definite quantitative prediction for the frozen-flux density in a narrow superconducting strip cooled in a fixed magnetic field. By solving the dynamic-balance equation, the author establishes that the final state is determined by the competition between thermal activation and the cooling rate, occurring at temperatures very close to $T_c$ but within the weak-fluctuation regime.

The primary significance lies in explaining the origin of the "effective flux-expulsion field." The paper argues that the experimentally observed $H_{exp}$ is not a fundamental thermodynamic limit but a kinetic threshold defined by the detection sensitivity and the specific cooling history. The model successfully captures the steep magnetic-field dependence of the trapped flux, suggesting that the large discrepancy between theoretical $H_e$ and experimental $H_{exp}$ in some materials (like Nb) may be attributed to sample inhomogeneities (pinning) rather than a failure of the narrow-strip theory itself. The work provides a framework for interpreting experimental flux-trapping data and defining effective expulsion fields based on material parameters and cooling rates.