What holes in superconductors reveal about superconductivity

This paper argues that the inability of a type I superconducting body with interior holes to reach thermodynamic equilibrium during a phase transition in a magnetic field reveals a fundamental flaw in the conventional BCS theory's explanation of the Meissner effect, suggesting instead that the phenomenon requires physical elements central to the alternative theory of hole superconductivity.

The Gist

The Big Question: Can a Superconductor "Clean" Itself?

Imagine you have a block of metal (a superconductor) with a small, empty hole drilled right through the middle. You place this block in a magnetic field and then cool it down until it becomes a superconductor.

The Standard View (The "Dream"):
According to the conventional theory of superconductivity (called BCS theory), the metal should instantly become a perfect "magnetic shield." It should push all the magnetic field lines out of the block, including the ones trapped inside that little hole. The system is supposed to be smart enough to find the most efficient, lowest-energy state, just like water freezing into a solid block of ice, even if there's a pebble inside the water.

The Author's View (The "Reality Check"):
J. E. Hirsch argues that this is impossible. He claims that if there is a hole inside the metal, the magnetic field cannot be pushed out of that hole. The metal will get stuck in a "half-finished" state where the field remains trapped inside the hole, and a small ring of the metal around the hole stays "normal" (not superconducting) to let the field lines escape.

The paper argues that the conventional theory fails to explain how the metal pushes the field out, and when you look closely at the physics of that "pushing," a hole makes it impossible.

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The Analogy: The "Orbit Expansion" Mechanism

To understand why the author thinks the field gets stuck, we need to look at his alternative theory: Hole Superconductivity.

1. The Electron as a Swinging Ball
Imagine electrons in a normal metal are like tiny balls swinging on very short, tight strings (microscopic orbits). They are jittery and chaotic.

2. The Magic of Superconductivity
When the metal becomes a superconductor, the author says these electrons don't just "pair up"; they expand their orbits. They stretch their strings out to become much larger loops (mesoscopic size).

• The Catch: To stretch that string out, the electron has to move outward (radially) from the center of its orbit.

3. The Magnetic "Push"
Here is the crucial part: The author claims that the magnetic field itself acts like a hand that pushes the electron sideways as it moves outward.

• As the electron moves out, the magnetic field pushes it sideways (azimuthally).
• This sideways push creates the electric current that generates the magnetic shield (the Meissner effect).
• The Metaphor: Think of a child on a swing. If you push the child outward (away from the pivot) while they are swinging, they start spinning faster. The "outward push" is required to create the "sideways spin" that blocks the magnetic field.

Why the Hole is a Problem

Now, let's look at the hole in the metal.

• Inside the metal: Electrons can move outward, get pushed sideways by the magnetic field, and create the current that expels the field.
• Inside the hole: There is no metal. There are no electrons.
• The Result: You cannot have an electron moving outward inside an empty hole. If there is no outward movement, there is no sideways push. If there is no sideways push, there is no current. If there is no current, the magnetic field cannot be expelled.

The "Traffic Jam" Analogy:
Imagine the magnetic field is a crowd of people trying to leave a stadium (the metal).

• In a solid stadium, the crowd can push through the exits (the electrons moving outward) to get out.
• But if there is a giant, empty pit in the middle of the stadium (the hole), the people inside the pit have nowhere to go. They can't push outward because there is no floor to push against. They are trapped.
• The author argues that the magnetic field lines in the hole are like those people. They are stuck because the "mechanism" to push them out (electron expansion) cannot happen in empty space.

The Thermodynamic Paradox

The paper points out a strange contradiction in the standard theory:

1. Thermodynamics says: Systems always want to reach the lowest energy state. A state with no magnetic field inside is lower energy than a state with a trapped field. So, the system should find a way to get the field out.
2. The Author's Logic: The paper argues that the process of getting the field out requires specific physical steps (electrons moving outward). If those steps are physically impossible (because of a hole), the system gets stuck in a "metastable" state. It's like a ball rolling down a hill but getting stuck in a small dip; it wants to go lower, but it can't get over the bump.

The author claims that standard theory ignores the "how" (the dynamic process) and just assumes the system magically finds the bottom. But if you look at the "how," the hole blocks the path.

The "Meissner Pressure" vs. "Maxwell Pressure"

The author uses a pressure analogy to explain why the field stays in the hole:

• Maxwell Pressure: The magnetic field inside the hole pushes outward, trying to expand. It's like air in a balloon.
• Meissner Pressure: The superconductor needs to generate an "outward pressure" to push the field back. This pressure comes from the electrons expanding their orbits.
• The Conflict: Inside the hole, there is no material to generate this "Meissner pressure." There is no one to push back against the balloon. Therefore, the magnetic field stays trapped.

What the Paper Proposes as a Test

The author suggests a simple experiment to prove his point:

1. Take a type-I superconductor (like pure tin or indium).
2. Drill a tiny hole in the middle.
3. Cool it down while it is in a magnetic field.
4. The Prediction:
   • If Standard Theory is right: The metal will eventually find a way to push the field out of the hole, even if it takes a long time or requires super-cooling. The field will disappear completely.
   • If the Author is right: The field will remain trapped in the hole forever. The metal will never reach the "perfect" state because the mechanism to expel the field is broken by the hole.

Summary

The paper argues that the conventional theory of superconductivity is incomplete because it doesn't explain the mechanics of how magnetic fields are expelled. The author proposes that expulsion requires electrons to physically move outward, which creates a sideways current.

Because a hole is empty space, electrons cannot move outward inside it. Therefore, the magnetic field inside a hole cannot be expelled. The system gets "stuck" with the field trapped, proving that the process of becoming a superconductor is not just about reaching a lower energy state, but about following specific physical rules that a hole breaks.

Technical Summary

Technical Summary: "What holes in superconductors reveal about superconductivity"

Problem Statement
The paper addresses a fundamental gap in the conventional theory of superconductivity (BCS): the lack of a dynamic explanation for the Meissner effect, specifically how a system transitions from a normal state to a superconducting state in the presence of a magnetic field. While thermodynamics dictates that a type I superconductor should reach a state of minimum free energy by completely expelling magnetic flux, the conventional theory does not describe the physical mechanisms required to achieve this state dynamically. The author argues that the presence of internal holes (cavities) in a superconducting body creates a scenario where the system cannot reach this thermodynamic equilibrium state, a fact that challenges the conventional view that the system will inevitably "find its way" to the lowest energy state.

Methodology
The author employs a combination of thermodynamic analysis, electrodynamics (Poynting's theorem), and the specific dynamical framework of the "theory of hole superconductivity."

1. Thermodynamic Analysis: The paper analyzes a type I superconductor with a volume $V$ containing a small hole of volume $V_h$. It calculates the energy balance required to expel magnetic flux from the entire volume versus a scenario where flux remains trapped in the hole. The analysis considers the energy supplied by the power source maintaining the magnetic field, the change in magnetic field energy, and the condensation energy released by the material.
2. Electrodynamic Analysis: Using Poynting's theorem, the paper examines the flow of electromagnetic energy at the boundary of a hole during flux expulsion. It investigates whether the electric field can perform the necessary work on charges to expel the field in the absence of material (inside the hole).
3. Dynamical Modeling: The paper contrasts the conventional view (where the transition is driven by potential energy lowering) with the theory of hole superconductivity (where the transition is driven by kinetic energy lowering and involves specific radial charge flows and orbit expansion).

Key Contributions and Results

• Thermodynamic Impossibility of Complete Expulsion: The analysis demonstrates that for a system cooled to a temperature infinitesimally below the critical temperature $T_c$ in a magnetic field $H = H_c(T)$, complete expulsion of magnetic flux from a hole is energetically impossible without violating energy conservation. The energy available from the condensation of the material volume $(V - V_h)$ is insufficient to supply the work required to return energy to the power source and account for the magnetic energy reduction in the hole volume $V_h$. A "missing energy" term proportional to $V_h$ exists.
• Electrodynamic Impossibility: Application of Poynting's theorem to the surface of a hole reveals that magnetic field expulsion requires the electric field to perform negative work on charges within the volume. Since a hole contains no charges, this condition cannot be satisfied. Consequently, the magnetic field cannot be expelled from the interior of a hole unless a current flows within the hole itself, which is impossible in a vacuum.
• The Role of Radial Charge Flow: Within the theory of hole superconductivity, the expulsion of magnetic fields is driven by the radial expansion of electronic orbits from microscopic to mesoscopic scales. This expansion generates a radial flow of negative charge (or holes flowing inward), which, via the Lorentz force, imparts the necessary azimuthal momentum to create the screening currents. Since a hole contains no electrons to expand their orbits, this mechanism is blocked.
• Trapped Flux and Metastability: The paper concludes that a type I superconductor with an internal hole, when cooled in a magnetic field, will not reach the global thermodynamic minimum (complete flux expulsion). Instead, it will become trapped in a metastable state where the magnetic field remains in the hole, and a normal phase persists in the material surrounding the hole to allow field lines to escape. This state persists regardless of how slowly the cooling occurs or how small the hole is (down to atomic dimensions, provided the material is type I).
• Contrast with Conventional Theory: The conventional BCS theory, which treats the transition as a potential energy lowering without specifying the dynamical mechanism for current generation, predicts that the system should eventually reach the state of complete flux expulsion. The paper argues this prediction fails to account for the macroscopic constraints imposed by the hole.

Significance and Claims
The paper claims that the behavior of superconductors with internal holes serves as a critical test to distinguish between the conventional BCS theory and the alternative theory of hole superconductivity.

• Challenge to BCS: The inability of the system to reach the thermodynamic equilibrium state in the presence of a hole suggests that the Hamiltonians used in BCS theory are missing essential physical elements required to describe the Meissner effect. Specifically, BCS theory does not account for the necessity of radial charge flow and the kinetic energy-driven nature of the transition.
• Validation of Hole Superconductivity: The results support the central tenet of the theory of hole superconductivity: that magnetic field expulsion is a local, dynamic process driven by the expansion of electron orbits and radial charge flow. Because this process cannot occur in a void, the field remains trapped.
• Experimental Implication: The paper proposes that controlled experiments on type I superconductors with internal holes could validate or invalidate these theories. If experiments show that the system can expel the field from the hole, the conventional view holds. If the system consistently fails to expel the field and remains in the higher-energy metastable state, it supports the theory of hole superconductivity and the necessity of the proposed dynamical mechanisms.

The author emphasizes that the presence of a hole reveals that the Meissner effect is not merely a thermodynamic drive toward lower potential energy, but a specific dynamical process involving momentum transfer and kinetic energy changes that are absent in standard theoretical descriptions.