Electron tunneling — tunneling through a double barrier

1. Four interfaces, one product of matrices#

A double barrier has four interfaces. The transfer-matrix method of Lecture 6.1 handles it without any change: one 2×2 matrix per interface, multiplied in order, and \(T = 1/|M_{11}|^2\) when the contacts on both sides are the same material. The lecture draws the same structure as a cascade of four scattering matrices; either way no particles are lost (\(R + T = 1\)), the incident wave from the left is normalized to one, and nothing is incident from the right. The closed form is a homework problem. Here the product is written out numerically and compared with open-nemo, which solves the discretized Schrödinger equation with open boundaries.

The first example is the lecture's shallow double barrier: two 2 nm barriers of 0.2 eV around a 5 nm GaAs well (effective mass \(m = 0.067\,m_0\) everywhere, \(m_0\) the free-electron mass). The contacts are 10 nm of GaAs on each side.

Both methods give the same curve; the small difference is open-nemo's discretization error on the 0.02 nm grid. Two features have no counterpart in the single barrier of Lecture 6.2:

• Resonant tunneling. At 0.072 eV, well below the 0.2 eV barrier top, the transmission is one and the reflection drops to \(2\cdot10^{-9}\) on this energy grid (a deep dip on the logarithmic scale). A single 2 nm barrier of the same height never reaches one below its top. Classically the electron could not cross even one barrier.
• A quasi-bound state above the barriers. At 0.319 eV a second, much broader peak reaches one. The state is only weakly bound: the well "holds" it only by reflection at the barrier edges.

2. Taller and thicker barriers sharpen the resonance#

The lecture varies the barriers in two ways: the height, from 0.2 to 0.5 eV at fixed width 2 nm (slide 7), and the thickness, from 2 nm to about 3.5 nm at a height of 0.4 eV (slide 8, read off the PCPBT figure). For each structure nemo.resonances finds the quasi-bound states as complex eigenvalues \(E_0 - i\Gamma/2\) of the open-system Hamiltonian; \(E_0\) is the resonance energy and \(\Gamma\) the full width at half maximum (FWHM) of the transmission peak. The same quantities are also read directly from the computed \(T(E)\).

For the lowest resonance the peaks of \(T(E)\) and the complex eigenvalues agree: the widths to within 1.5 %, the energies to within about 3 meV (the transmission maximum sits above \(E_0\) when \(\Gamma\) is large, because the escape rate grows with energy). Raising the barriers from 0.2 to 0.5 eV narrows the lowest resonance from 22 meV to 6.4 meV and pulls a second resonance, at 0.405 eV, below the barrier top: the two strong resonances of slide 7. Thickening the 0.4 eV barriers from 2 to 3.5 nm narrows the lowest resonance by almost a factor of ten, from 9.8 meV to 1.1 meV. The resonance energy moves much less than the width.

The second resonances are broad (23 to 63 meV in \(T\)) and lie close to the barrier top. For them the transmission peak is the robust quantity. The complex eigenvalue that nemo.resonances returns for the 2 nm, 0.4 eV structure depends on the length of the contact layers included in the device: 0.311 eV with 10 nm contacts, 0.362 eV with 20 nm, and no state at all with 40 nm, while the transmission peak stays at 0.379 eV. Even the narrow lowest state changes its width by about 7 % (9.6 to 10.3 meV) between these contact lengths.

3. Resonance width and lifetime#

Klimeck describes the double barrier as a resonator: an electron placed in the well bounces between the barriers and eventually leaks out. If \(\tau\) is its lifetime, the energy of the state is uncertain by \(\Gamma = \hbar/\tau\), and the transmission near the resonance is a Lorentzian,

where \(\Gamma_L\) and \(\Gamma_R\) are the escape rates (times \(\hbar\)) through the left and right barrier. The table above converts the widths: 30 fs for the shallow structure, 580 fs for the thick barriers.

A semiclassical estimate makes the dependence on the barriers explicit. The electron moves in the well with velocity \(v = \hbar k/m\); it hits one barrier every \(2W_\mathrm{eff}/v\) seconds and escapes with the single-barrier probability \(T_1(E)\) of Lecture 6.2 each time:

\(W\) is the well width and \(W_\mathrm{eff}\) adds the decay length \(1/\gamma\) of the wave function into each barrier (Section 5). The estimate uses the resonance energy \(E_0\) from nemo.resonances.

The escape-rate estimate is within about 20 % for the shallow 0.2 eV structure and within 8 % for the others (1 % for the thick barriers), with no adjustable parameter. It explains the trends of slides 7 and 8: the width is proportional to the single-barrier transmission, which falls exponentially with the barrier thickness and with \(\sqrt{V_0 - E}\) (Lecture 6.2). A sharper resonance means a longer residence time in the well.

4. The well as a particle in a box#

If the barriers confine the electron well, the resonances should be close to the levels of a particle in a box (Lecture slide 10) of width \(W\),

A better closed model is the finite square well of depth \(V_0\): the same well, but with barriers that extend to infinity, so that the wave function decays into them without leaking out. Its levels solve \(\tan(kW/2) = \gamma/k\) (even states) and \(-\cot(kW/2) = \gamma/k\) (odd states). The lecture compares two structures: thick, tall barriers (10 nm, 1 eV, 20 nm well; slide 11) and thinner, lower ones (8 nm, 0.25 eV, 10 nm well; slide 12).

For the thick, tall barriers the lowest resonances lie 14 % below the particle-in-a-box levels, the discrepancy grows towards the barrier top, and the widths are tiny: \(5\cdot10^{-15}\) eV for the ground state, a lifetime of the order of a tenth of a second. For the thinner, lower barriers even the ground state lies at 0.6 of the box energy, as the lecture states: "the system doesn't resemble very well anymore the particle in a box problem".

The finite well, by contrast, reproduces every resonance to within 0.03 meV in both structures. What the infinite box misses is not the openness itself (the widths are far smaller than the level spacings) but the finite barrier height: the wave function leaks into the barriers, which makes the box effectively wider and lowers the levels. Openness matters for the width, and for the energy only when the width becomes comparable to the spacing.

5. Penetration into the barriers and the effective well width#

The lecture's slide 13 plots \(|\psi|^2\) of the two resonances of the 8 nm, 0.25 eV structure, computed with tight binding in the tool. nemo.resonances returns the same densities, normalized over the device. The fraction of the density inside the barriers measures the penetration, and an effective width \(W_\mathrm{eff} = \pi n\,\hbar/\sqrt{2mE_0}\) makes the box formula reproduce each resonance exactly.

The ground state places 3 % of its density inside the barriers and the first excited state, which sits closer to the barrier top, 15 %. Measured by its energy, the ground state behaves like a particle in a box 13.1 nm wide instead of 10 nm; the excited state behaves like one 13.4 nm wide. The simple estimate \(W + 2/\gamma\), one decay length added on each side, gives 13.2 nm and 14.3 nm: good for the deep state, too large for the shallow one. Klimeck's remark that one can "play some number games to make them match" applies: an effective width can be fitted level by level, but it is a different width for each level, which is why the finite-well or the open-system calculation is the reliable route.

6. Exploration: an asymmetric double barrier#

In the symmetric structure the peak transmission is exactly one. If the two barriers differ, the Lorentzian of Section 3 gives at \(E = E_0\)

using the escape-rate estimate \(\Gamma_{L,R} \propto T_1^{L,R}\). The left barrier is kept at 2 nm and the right one is thickened, at 0.4 eV around a 5 nm well.

With a thicker right barrier the peak falls below one, to 0.60 at 3 nm and 0.19 at 4 nm, and the estimate from the two single-barrier transmissions follows the computed peaks to within 0.03. The resonance also narrows, from 9.8 to 5.3 meV, but only to about half: the total escape rate becomes dominated by the 2 nm barrier alone, which keeps its share \(\Gamma_L\). Unit transmission requires matched barriers: the reflected waves from the two sides cancel only when the electron leaks out of the well equally fast in both directions.

Summary#

• A double barrier is four interfaces; the transfer matrix (or scattering matrix) handles it in the same way as one barrier, and open-nemo's open-boundary transmission agrees with it.
• It transmits with probability one at discrete energies below the barrier top (resonant tunneling), which neither a single barrier nor classical physics allows. Above the top it has broad, weakly bound states.
• Each resonance is a quasi-bound state \(E_0 - i\Gamma/2\) with a finite lifetime \(\tau = \hbar/\Gamma\). Taller or thicker barriers lengthen the lifetime and sharpen the peak, in proportion to the single-barrier transmission: \(\Gamma \approx (\hbar v/2W_\mathrm{eff})(T_1^L + T_1^R)\).
• The particle in a box overestimates the resonance energies because the wave function penetrates into the barriers; the finite well reproduces them as long as the width is small compared with the level spacing.
• Unequal barriers lower the peak transmission to \(4\Gamma_L\Gamma_R/(\Gamma_L + \Gamma_R)^2\).

Exercises#

1. For the 2 nm | 5 nm | 2 nm structure, find the barrier height at which the lowest resonance width equals 1 meV, first with the escape-rate estimate and then with nemo.resonances.
2. Fit the Lorentzian of Section 3 to the computed \(T(E)\) of the 3.5 nm, 0.4 eV structure. How far from \(E_0\) does the Lorentzian stay within 10 % of \(T\)?
3. Make the barriers of the 8 nm, 0.25 eV structure out of a material with mass \(0.1\,m_0\). Predict the direction of the change of \(E_0\) and \(\Gamma\) before computing them.
4. Repeat Section 6 with equal widths but unequal heights (0.3 and 0.5 eV). Which barrier controls the width of the resonance?

Further reading#

• Lecture 6.2: the single barrier and Lecture 6.4: N barriers and the formation of bands.
• The resonant-tunneling diode tutorial: the resonant-tunneling diode built from the same double barrier.
• Benchmark: piecewise-constant barriers against nanoHUB's PCPBT.
• The tool of the lecture: PCPBT on nanoHUB.