Q015 · Barriers / resonance / current

Resonant Tunneling Transport

Move the barriers, tune the incoming energy, and count transmitted electrons. Build a spectrum before estimating current; a narrow resonance can hide between your measurements.

Recorded trials $\text{—}$

Transmission at probe $\text{—}$

01 / CHANGE THE BOUNDARY. ASK WHAT GETS THROUGH.

A second barrier can open a window.

Scan two barriers and save. Remove one and repeat. Then make the barriers thicker: does a coarse scan still find the narrow peak?

Enlarged conceptual chip · transport left to right · raised gates indicate rectangular potential regions · ideal energy-resolved counting

Incident energy

E/\mathrm{meV}

53

Ready. Each scan setting uses 200 independent incident particles.

TRY A CONTRAST

Left barrier height

V_0/\mathrm{meV}

180

Barrier thickness

a/\mathrm{nm}

1.5

Well width

L/\mathrm{nm}

6

Reservoirs, asymmetry & scan spacing

Right / left height

V_R/V_0

1

Reservoir bias

U/\mathrm{mV}

40

Reservoir temperature

T/\mathrm{K}

30

Scan spacing

\Delta E/\mathrm{meV}

2

Sampling seed

s_0

1729

Moving the probe preserves readings. Changing the device or preparation starts a fresh sample; save first. Nothing runs by itself.

One elastic channel with two spin states, uniform illustrative effective mass and rectangular barriers. Bias changes reservoir occupations only; the potential is rigid. Simulated single-particle counts and finite-window inferred current are separate from model predictions. No self-consistent device electrostatics or inelastic scattering.

02 / FIRST THE COUNTS. THEN THE CURVE.

Where does transmission return?

Model reference

$E/\mathrm{meV}$ · Blue: current records. Amber: last saved other sample. Dashed: optional model. Drag to inspect; the slider supports keyboard input.

AT THIS PROBE SETTING

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No recorded particles yet.

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95% pointwise Wilson interval. No simultaneous-band or optional-stopping guarantee.

LATEST 60 RAW OUTCOMES

Blue: transmitted. Gray: reflected or absorbed. No hidden trajectory is recorded.

03 / WHAT CAN THESE RECORDS TELL US?

A tall peak is not the whole current.

Declared reservoir weighting, independent of the counting run. Positive bias favors left-to-right flow. The barriers stay rigid.

INFERRED FROM SCAN RECORDS

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Independent model comparison

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MODEL INSPECTION · NO NEW RECORDS

The wave builds up between the barriers.

$x/\mathrm{nm}$ · Drag the right barrier; height is potential energy, not a physical wall. Dashed line: incident energy.

$|\psi|^2/|A_{\mathrm{in}}|^2$ · A stationary scattering state, not a normalized particle cloud or a trajectory. Vertical scale adapts and is labeled.

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04 / KEEP THE EVIDENCE

Save before changing the experiment.

Eight frozen samples. Restoring inspects the original records; acquiring again starts a new run. CSV contains individual outcomes, scan membership and original settings.

Physical model, units and sources

m^*=0.067m_e,\quad\mu_{L,R}=100\,\mathrm{meV}\pm eU/2

Illustrative uniform effective mass, equal lead velocities, one spatial channel and two spin states. The source, nanostructure and collector are enlarged conceptually; raised gates mark potential regions. Ideal single-energy transmission trials are not a simulation of a real single-electron detector. Reservoir bias does not tilt the potential. No mass mismatch, many-body charging, phonons or full device current outside the displayed window.

Olkhovsky, Recami & Zaichenko · double-barrier transmission, sections 2–3 · Beenakker & van Houten · Landauer transport

Start with the essentials

Focus question: Can two barriers transmit better than one?
One-sentence intuition: Interference can cancel reflection from two barriers, but the resonance width matters.

Core mathematical model

Stationary scattering

-\frac{\hbar^2}{2m^*}\psi^{\prime\prime}+V(x)\psi=E\psi

Wavefunction and derivative match at each interface. The effective mass is uniform.

Flux conservation

R+T=1

Identical left and right leads make transmission equal to the squared transmitted amplitude.

Finite-window Landauer current

I_{[1,301]}=\frac{2e}{h}\int_{1\,\mathrm{meV}}^{301\,\mathrm{meV}}T(E)[f_L(E)-f_R(E)]\,dE

Two spin channels. Scan counts estimate the same integral by a trapezoidal rule. Bias changes occupations, not the barrier profile.

Common difficulties

What was measured?

Typical misconceptionThe current readout is another independently simulated ammeter.

Better mental modelThe estimate uses recorded transmission counts and the declared reservoir occupations. The finer model integral is a separate reference.

Idealization

Typical misconceptionAgreement here certifies a real device.

Better mental modelOne elastic channel with two spin states, uniform illustrative effective mass and rectangular barriers. Bias changes reservoir occupations only; the potential is rigid. Simulated single-particle counts and finite-window inferred current are separate from model predictions. No self-consistent device electrostatics or inelastic scattering.

Run the experiment

01
Predict
Compare two barriers with one removed. Predict whether transmission can rise above the single-barrier result.
What to observe: Keep the reference curve hidden until after recording.
02
Scan and preserve
Scan energy and save. Widen the well, scan again and compare the positions of the peaks.
What to observe: Frozen samples retain the settings that produced them.
03
Expose the limitation
Use thick barriers and coarse scan spacing, then refine the scan. More counts at the same missed points cannot recover an unseen peak.
What to observe: Sampling uncertainty and energy-grid bias are distinct.