SUPERCONDUCTING INTERFACE / ELECTRON–HOLE CONVERSION

Andreev Reflection

Launch an electron wavepacket into a superconducting interface. Watch the returning wave switch character, raise a barrier to stop the conversion, then cross the gap to transmit into the superconductor.

A one-dimensional tight-binding Bogoliubov–de Gennes model with a prescribed s-wave pair potential. Electron and hole ribbons are components on the same wire, not two paths. Finite wavepacket, zero-temperature mean field; no self-consistent condensate, collisions or calibrated conductance measurement.

ONE INTERFACE. TWO KINDS OF RETURN.

HOLE WEIGHT IN THE NORMAL REGION

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GOLD PLANE · NS INTERFACE

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CYAN · ELECTRONCORAL · HOLEHEIGHT · PROBABILITY PER SITE

The two ribbons are Nambu components on ONE wire, not two trajectories. Fixed linear probability height. The shaded right half has superconducting pairing.

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01 / THE COMPLETE COLLISION

Follow the color that comes back.

Position left to right; time 0 at the top and 75 at the bottom. Cyan is electron weight, coral is hole weight. Color uses a fixed square-root gain; the gold line is the interface.

DARK 0 → BRIGHT 0.04 per site

02 / A LOCAL LOOK

Two components. One normalized state.

Cyan: electron probability per site. Coral: hole probability per site. Both components use the same position axis. Gold marks the interface at site 161.

$j\ \longrightarrow\quad |u_j|^2,\ |v_j|^2$

03 / PROBABILITY BUDGET

Where did the excitation go?

Cyan: electrons in the normal region. Coral: holes in the normal region. Purple: total weight inside the superconductor. Early left-electron weight includes the incoming packet; only separated late packets describe reflection.

$\tau=Jt/\hbar\ \longrightarrow\quad P$

04 / INDEPENDENT BULK REFERENCE

The gap decides what can propagate.

Analytic bands of the infinite uniform superconductor. The gold line is the incident packet’s central energy, which has a finite spread.

$k\ \longrightarrow\quad E/J$

TURN THE MODEL INTO RECORDS

Count what came back.

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Ideal component-and-position Born readouts from independent preparations at the selected time. These are simulated quasiparticle snapshots, not a calibrated detector or a count of Cooper pairs. Fixed reproducible seed.

Recorded normal-hole fraction

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95% Wilson interval · shot noise

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Model probability

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Norm drift

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Records in gold. Model in cyan.

Three exhaustive outcomes: normal electron, normal hole, or either component inside the superconductor. The bars are built only from saved records.

Latest 20 records · export every trial

Trial	Site	Component

Model, units and sources

320 sites with open ends. The normal region ends at site 160; pairing begins at 161. A one-site barrier sits at 160. The initial electron is centered at site 81 with position standard deviation 12 sites.

Position increases to the right in lattice-spacing units. Hopping sets the energy unit; time is measured in reduced Planck constant divided by hopping. Ribbons are a diagnostic representation, with schematic spacing between components.

H=\begin{pmatrix}h&\Delta_s\\\Delta_s&-h\end{pmatrix},\quad h_{j,j+1}=-J,\quad h_{160,160}=V

Symmetric split evolution uses exact onsite and disjoint-bond rotations, with time step 0.0125. No renormalization, absorbing boundary or probability clipping. The displayed interval ends before significant boundary returns; the numerical boundary weight is monitored. The pair potential is prescribed, not solved self-consistently.

$P_{\mathrm{edge}}=0$ · Probability in the outer eight sites on either end.

Sources: Blonder, Tinkham and Klapwijk’s interface theory and the BdG formulation in the Kwant superconductivity tutorial. The lattice wavepacket is not the continuum BTK approximation.

Start with the essentials

Focus question: Send an electron. Get a hole back.
One-sentence intuition: Pairing mixes electron and hole amplitudes. A subgap excitation can return as a hole.

Core mathematical model

Electron and hole amplitudes

i\hbar\partial_t\begin{pmatrix}u\\v\end{pmatrix}=H_{\mathrm{BdG}}\begin{pmatrix}u\\v\end{pmatrix}

These are two components of a quasiparticle, not independent physical paths.

Uniform superconducting spectrum

E(k)^2=4J^2\cos^2 k+\Delta_s^2

An energy inside the bulk gap has no propagating quasiparticle mode deep in the superconductor.

Common difficulties

Respect the scope

Typical misconceptionA hole is a positron emitted by the interface.

Better mental modelA hole describes a missing electronic occupation relative to the reference condensate. Andreev conversion exchanges charge with that condensate.

A diagnostic is not a photograph

Typical misconceptionThe raised ribbons are paths followed by individual particles.

Better mental modelRibbon height encodes a declared probability or spectral density. The two colors share the same physical site coordinate.

Run the experiment

01
Make a prediction
Compare the three presets, then change one continuous control.
What to observe: The barrier suppresses hole conversion; sufficient energy permits transmission.
02
Inspect the cause
Scrub from the incoming packet to the separated outgoing packets.
What to observe: Early-time left-electron weight includes the incoming packet; it is not yet a reflection probability.
03
Acquire evidence
Collect repeated records, inspect the histogram and interval, then export the complete CSV.
What to observe: The model curve is a separate reference. Only recorded events build the gold histogram.