SUPERCONDUCTING INTERFACE / ELECTRON–HOLE CONVERSION

Andreev Reflection

What comes back when an electron hits a superconductor with an energy inside its gap? Mostly a hole. Inside the gap, the superconductor has no states a lone electron can enter. The electron gets in only by taking a partner electron along as a Cooper pair, and the missing partner travels back as a hole. A barrier at the interface blocks this conversion, so the electron bounces back as an ordinary electron. Above the gap, the wave passes into the superconductor.

Send an electron wavepacket toward a superconductor and watch the wave come back as a hole. Raise a barrier to suppress that conversion, then give the packet enough energy to cross the gap and pass into the superconductor.

The model is a one-dimensional tight-binding Bogoliubov–de Gennes model with a fixed s-wave pair potential. The electron and hole ribbons are two components of the wave on the same wire, not two separate paths. It uses a finite wavepacket and zero-temperature mean field theory. The condensate is not solved self-consistently, and collisions and calibrated conductance measurements are not included.

One interface, two ways to come back

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. Ribbon height is linear in probability with a fixed gain. 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 make 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 BY REGION

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.

$k\ \longrightarrow\quad E/J$

Sample records from the model

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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Gold is the record; cyan is the model.

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: What comes back when an electron hits a superconductor with an energy inside its gap?
One-sentence intuition: Pairing mixes the electron and hole parts of the wave, so an electron arriving with an energy inside the gap can come back 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

At an energy inside the bulk gap, no quasiparticle mode can propagate deep in the superconductor.

Common difficulties

What the model can and cannot claim

Typical misconceptionA hole is a positron emitted by the interface.

Better mental modelA hole stands for an electron state left empty compared with the reference condensate. Andreev conversion exchanges charge with that condensate.

The ribbons are not particle paths

Typical misconceptionThe raised ribbons are paths followed by individual particles.

Better mental modelThe ribbon height shows a probability or spectral density, as labeled. Both colors are plotted against the same lattice-site coordinate.

Run the experiment

01
Make a prediction
Compare the three presets, then change one continuous control.
What to observe: The barrier suppresses conversion into a hole, and enough energy lets the packet pass into the superconductor.
02
Find the cause
Scrub from the incoming packet to the separated outgoing packets.
What to observe: At early times the electron weight on the left still includes the incoming packet, so it is not yet a reflection probability.
03
Collect data
Collect repeated records, inspect the histogram and interval, then export the complete CSV.
What to observe: The model curve is shown separately as a reference; the gold histogram is built only from recorded events.