Q015 · Barriers / resonance / current

Resonant Tunneling Transport

Can two barriers transmit better than one? At the right energies, yes. Waves bouncing between the two barriers interfere so that their reflections cancel, and nearly all of the wave gets through. These resonances can be so narrow that a coarse energy scan steps right over them.

Move the barriers, tune the energy of the incoming electrons and count how many get through. Measure the transmission spectrum before estimating the current, because a narrow resonance can fall between your measurement points.

Recorded trials $\text{—}$

Transmission at probe $\text{—}$

01 / Change the boundary and see what gets through

A second barrier can open a window.

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

Enlarged conceptual chip. Transport runs left to right, and raised gates mark rectangular potential regions. Counting is ideal and energy-resolved.

Incident energy

E/\mathrm{meV}

53

Ready. Each scan setting uses 200 independent incident particles.

Try a comparison

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.

Electrons scatter elastically in one channel with two spin states. The effective mass is a uniform illustrative value, and the barriers are rectangular. The bias only changes how the reservoirs are filled; the potential stays fixed. The simulated single-electron counts, and the current estimated from them over a finite energy window, are kept separate from the model predictions. Self-consistent electrostatics of the device and inelastic scattering are not included.

02 / Counts first, 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. It is not a simultaneous band and carries no guarantee under optional stopping.

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 does not mean a large current.

This reservoir weighting is set separately from the counting run. A positive bias favors flow from left to right. The barriers keep their shape.

Inferred from the scan records

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Independent model comparison

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Model only; nothing new is recorded

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.

$|\psi|^2/|A_{\mathrm{in}}|^2$ · This is a stationary scattering state, not a normalized particle cloud or a trajectory. The vertical scale adjusts automatically and is labeled.

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04 / Save 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 the reflections from the two barriers, but the width of that resonance matters just as much.

Core mathematical model

Stationary scattering

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

The wavefunction and its derivative are continuous at each interface, and the effective mass is the same everywhere.

Flux conservation

R+T=1

Because the left and right leads are identical, the transmission equals the squared magnitude of the transmitted amplitude.

Landauer current over a finite energy window

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

Both spin channels contribute. The scan counts estimate the same integral with the trapezoidal rule. The bias changes the occupations, not the shape of the barriers.

Common difficulties

What was measured?

Typical misconceptionThe current readout comes from a separately simulated ammeter.

Better mental modelThe current is estimated from the recorded transmission counts and the given reservoir occupations. The model integral on a finer grid is shown separately as a reference.

An idealized model

Typical misconceptionAgreement here certifies a real device.

Better mental modelElectrons scatter elastically in one channel with two spin states. The effective mass is a uniform illustrative value, and the barriers are rectangular. The bias only changes how the reservoirs are filled; the potential stays fixed. The simulated single-electron counts, and the current estimated from them over a finite energy window, are kept separate from the model predictions. Self-consistent electrostatics of the device and inelastic scattering are not included.

Run the experiment

01
Predict
Compare two barriers with one barrier removed. Predict whether two barriers can transmit more than one.
What to observe: Keep the reference curve hidden until you have recorded data.
02
Scan and save
Scan energy and save. Widen the well, scan again and compare the positions of the peaks.
What to observe: A saved sample keeps the settings that produced it.
03
Find the limits
Use thick barriers and a coarse scan, then refine the scan. More counts at points that miss the peak cannot recover it.
What to observe: Counting noise and the bias from a coarse energy grid are two separate problems.