The theory of planar ballistic SNS junctions at $T=0$

This paper presents an exact analytical theory for planar ballistic SNS junctions at zero temperature that incorporates phase gradients in superconducting leads, resolving charge conservation issues and revealing a distinct current-phase relation for short junctions that aligns with recent numerical calculations and experimental observations on InAs nanowires.

The Gist

Imagine a superhighway for electrons, but with a twist. In a special type of electrical junction called an SNS junction, you have two superhighways (Superconductors, or "S") separated by a short, ordinary road (a Normal metal, or "N"). Electrons can zip through this setup without any resistance, creating a "supercurrent."

For over 50 years, physicists have had a specific rulebook for how this traffic flows. However, this new paper by Edouard B. Sonin argues that the old rulebook was missing a crucial piece of the puzzle, especially when the "N" road is very short.

Here is the breakdown of the paper's discovery using simple analogies:

1. The Old View: The "Static" Highway

The traditional theory treated the superconducting highways like two separate, static pools of water.

• The Assumption: It assumed the "phase" (a property of the electron waves that drives the current) was perfectly flat and constant in the superconducting parts, only changing abruptly in the middle section.
• The Problem: This created a "leak" in the laws of physics. Specifically, it violated the Law of Charge Conservation. In the old model, current would flow in the middle section but seemingly disappear or appear out of nowhere in the superconducting leads. It was like a car driving onto a bridge and vanishing before reaching the other side.
• The Fix (Previous): Physicists thought, "Well, maybe there are tiny, invisible ripples in the water at the edges that fix this, but they are so small we can ignore them."

2. The New View: The "Moving" Highway

Sonin says: "No, those ripples aren't just tiny; they are essential, and they change the whole picture."

• The Insight: He applied a concept called Galilean Invariance. Think of this like being on a moving train. If you walk forward on a train, your speed relative to the ground is your walking speed plus the train's speed.
• The Discovery: In these junctions, the superconducting leads aren't static pools; they are moving trains. The "phase" (the rhythm of the electron waves) actually has a constant slope or gradient across the leads, just like the train is moving.
• The Result: When you account for this "train motion," the current flows smoothly everywhere. The "leak" disappears. The total current is a sum of two things:
  1. The Condensate Current: The "train" moving the whole crowd of electrons together.
  2. The Vacuum Current: The individual "cars" (electrons) trying to move against the flow.
     In the old theory, people thought the total current was just the "cars." The new theory shows it's the "train" plus the "cars," and they balance each other perfectly to obey the laws of physics.

3. The Short vs. Long Junction

The paper focuses heavily on what happens when the "N" road is very short (short junctions).

• Long Junctions: If the road is very long, the old and new theories happen to agree on the final result (a jagged "saw-tooth" pattern of current). This is why the mistake went unnoticed for so long.
• Short Junctions: If the road is very short (or disappears entirely, making it just a uniform superconductor), the two theories give completely different answers.
  • Old Theory: Predicts the current peaks at a specific angle (phase) that makes the curve look "forward-skewed" (leaning to the right).
  • New Theory: Predicts the current peaks earlier, making the curve "backward-skewed" (leaning to the left).

4. Why It Matters (According to the Paper)

The author points out that this isn't just a math correction; it fixes a fundamental error in how we view charge conservation in these ideal models.

• Real-world Confirmation: The paper notes that recent experiments on tiny nanowires (InAs nanowires) and new computer simulations have already observed this "backward-skewed" shape.
• The "Bridge" Analogy: The old theory was like describing a bridge between two massive continents as if the continents were flat and still. The new theory realizes the continents are actually moving, and you have to account for that motion to understand how the traffic flows across the bridge.

Summary

In simple terms, this paper says: "We've been modeling these super-conducting bridges as if the ends are frozen in place. They aren't. They are moving. Once we account for that motion, the math finally works correctly, and it explains why recent experiments see a different shape of current flow than the old textbooks predicted."

The paper does not claim this will lead to new medical devices or immediate technology changes; it is a fundamental correction to the physics of how electrons move in these specific, idealized quantum bridges.

Technical Summary

Technical Summary: The Theory of Planar Ballistic SNS Junctions at T = 0

Problem Statement
This work addresses a long-standing inconsistency in the theoretical description of planar ballistic Superconductor-Normal-Superconductor (SNS) junctions at zero temperature ($T=0$). The widely accepted theory, based on the "steplike pairing potential" model (where the superconducting gap is constant in leads and zero in the normal layer), assumes constant phases in the superconducting leads. This assumption leads to a violation of the charge conservation law, as the calculated current flows exclusively within the normal layer while the superconducting leads carry no current. While it was previously argued that this violation could be ignored by assuming infinitesimal phase gradients in the leads, this paper demonstrates that for planar SNS junctions (which are not weak links at $T=0$), these gradients significantly affect the current in the normal layer. The core problem is to derive a current-phase relation (CPR) that satisfies charge conservation within the steplike pairing potential model without resorting to the more complex self-consistent field method.

Methodology
The author employs the Bogolyubov–de Gennes (BdG) equations under the assumption that the Fermi energy is much larger than the superconducting gap ($\epsilon_F \gg \Delta_0$), allowing for the neglect of normal scattering and the reduction of second-order differential equations to first-order equations.

The central methodological tool is Galilean invariance. The paper establishes that the ballistic SNS junction possesses Galilean invariance despite broken translational invariance. The analysis proceeds by:

1. Defining a state with a constant phase gradient $\nabla\phi$ in the superconducting leads, characterized by a superfluid phase $\theta_s = L\nabla\phi$ and a vacuum phase $\theta_0$.
2. Demonstrating that a Galilean transformation of the ground state (zero phase gradient) yields a state where the total current $J$ is the sum of a "condensate current" ($J_s = en v_s$) flowing uniformly through all layers and a "vacuum current" ($J_v$) confined to the normal layer.
3. Enforcing the charge conservation law by requiring that the total current deep in all layers be equal. This necessitates that the vacuum current in the normal layer be compensated by an "excitation current" ($J_q$) arising from the occupation of Andreev bound states.
4. Deriving the CPR analytically by applying the Landau criterion: the transition to a non-zero vacuum current occurs when the energy of the lowest Andreev level reaches zero in the laboratory frame.

Key Contributions and Results

• Resolution of Charge Conservation: The paper resolves the charge conservation paradox within the steplike pairing potential model. It proves that a condensate current can flow through the entire junction (including leads) without violating charge conservation, provided the phase gradients in the leads are accounted for. This refutes the notion that solving the self-consistency equation is strictly necessary to restore charge conservation in this model.
• Analytical CPR for Arbitrary $L$: An exact analytical CPR is derived for any normal layer thickness $L$ at $T=0$.
  • Short Junctions ($L \to 0$): In the limit where the normal layer vanishes, the junction becomes a uniform superconductor. The derived CPR is $J = J_{cr} \cos(\theta/2)$, where $J_{cr}$ is the depairing current. This results in a backward-skewed CPR (maximum current at $\theta < \pi/2$). This contradicts the previous theory (Kulik-Omel'yanchouk), which predicted a forward-skewed $J \propto \sin(\theta/2)$ by ignoring phase gradients.
  • Long Junctions ($L \to \infty$): The theory recovers the standard saw-tooth CPR ($J \propto \theta$) in the limit of long junctions, consistent with previous results, but attributes it to the correct physical mechanism involving phase gradients.
• Dimensionality: The analysis is extended from 1D single-channel junctions to 2D and 3D systems by integrating over transverse wave vectors. The results show that while the critical current density changes with dimensionality, the functional form of the CPR in the short-junction limit ($L=0$) remains identical to the 1D case ($J/J_{cr} = \cos(\theta/2)$).
• Phase Slip Branch: For phases exceeding a critical value $\theta_{cr}$, the CPR enters a "phase slip branch" where the vacuum current is compensated by the excitation current. The transition point $\theta_{cr}$ depends on the ratio of the junction length to the coherence length.

Significance and Claims
The paper claims that the inclusion of phase gradients in superconducting leads is essential for a physically consistent description of planar ballistic SNS junctions at $T=0$. The primary significance lies in:

1. Correcting the Short-Junction Limit: The work demonstrates that the previously accepted CPR for short junctions (Eq. 28 in the text) is invalid for planar SNS junctions because it violates charge conservation. The correct limit yields a backward-skewed CPR, consistent with recent numerical calculations (Krekels et al.) and experimental observations in short InAs nanowire junctions (Spanton et al.), though the paper notes its specific applicability to planar geometries.
2. Validity of the Steplike Model: It establishes that the steplike pairing potential model, often criticized for charge conservation issues, can yield exact and physically consistent results if Galilean invariance and phase gradients are properly treated, without needing the full self-consistent field calculation.
3. Physical Insight: The analysis clarifies the distinct roles of condensate and vacuum currents, showing that the condensate current dominates in short junctions below the critical phase, a feature previously obscured by the assumption of constant phases in the leads.

The author concludes that while the steplike model is idealized, understanding the supercurrent flow in this ideal problem is crucial for the physical intuition of more complex, realistic systems.