Orbital Tuning of Tunnel Coupling in InAs/InP Nanowire Quantum Dots

We report results on the control of barrier transparency in InAs/InP nanowire quantum dots via the electrostatic control of the device electron states. Recent works demonstrated that barrier transparency in this class of devices displays a general trend just depending on the total orbital energy of the trapped electrons. We show that a qualitatively different regime is observed at relatively low filling numbers, where tunneling rates are rather controlled by the axial configuration of the electron orbital. Transmission rates versus filling are further modified by acting on the radial configuration of the orbitals by means of electrostatic gating, and the barrier transparency for the various orbitals is found to evolve as expected from numerical simulations. The possibility to exploit this mechanism to achieve a controlled continuous tuning of the tunneling rate of an individual Coulomb blockade resonance is discussed.

(b) the common-mode gate lever arm, α cm ; (c) the common-mode capacitance C cm and the total capacitance C Σ of the quantum dot. The parameters were calculated starting from a t of the slopes of the Coulomb diamond's edges of the stability diagram. The ∆U values for odd diamonds (blue dots in the graph) correspond to the charging energies (E c ), while the energy dierence between even (grey dots in the graph) and odd diamonds can be used to estimate the level spacing (∆ε). The charging energy in the QD of D#1 is about 10 meV, while the level spacing reaches the values up to 10-15 meV. The common-mode lever arm used for the energy conversion in Figure 5 of the main paper was obtained by a linear t of the data reported above (around N=14). The tunneling rate Γ for each peak was extracted by tting the zero bias conductance peaks ( Figure 2.c in the main text) using the typical line-shape for a non-degenerate delta-like where α k is the gate lever arm, V g is the gate voltage and V N g is peak gate voltage at lling N.
Corrections can be expected considering the spin degeneracy, but these eects were beyond the scope of the current paper. The results of the described tting procedure are reported in the following

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In Figure S2 we report the stability diagram and extracted parameters for D#2 and D#3.
In the plot, the back gate electrode was used to control the electron population inside the QD and an abrupt threshold to larger tunneling was observed. Devices D#2 and D#3 displayed a similar tunability using lateral gates but a full investigation of the spectral dependence could not be performed due to the presence of too many charge rearrangements over the large voltage swings necessary to perform the study. The threshold for the observation of large coupling orbitals was observed to be N = 22 and N ≈ 24 for devices D#2 and D#3, respectively. Note the electron lling in D#3 cannot be reliably identied due to the presence of a non-negligible charge rearrangement. The main target of the simulations was to estimate the expected spacing between radial and axial excitations in the studied QD, i.e. the number of orbitals that need to be lled before larger tunneling resonances can be experimentally observed. To this end, we made a set of simplied assumptions and used a single particle model to predict the expected energy spacings. The nominal geometry of the studied QDs is reported in Figure S3; the InAs island is a hexagonal box with an axial thickness τ = 19.5 nm in the z direction and a corner-to-corner diameter d = 48 nm.

S.2 Numerical models
Calculations were performed using the PDE solver Comsol Multiphysics and the following further approximations were made: (i) the connement potential is innite and barrier penetration is neglected; (ii) band bending at the NW surface and the non-parabolicity eects are neglected. In these approximations, the wave function can be factored as the product ψ(x, y, z) = A(z)B(x, y), where the axial component has a particularly simple expression A(z) = sin(kz) and k z = πn a /τ where n a is an integer. This leads to eigenvalues of the form where n r is the quantum number of the radial problem and the transvers electric eld acts only on ε nr . While the adopted approximations are clearly strong, the resulting estimates on the lling numbers were found to be reliable in the limit of low occupation numbers.
Importantly, it has to be noted that in such a limit the transmission across the barrier only depends to the z part of the eigenvalue problem, and thus only on n a . In particular, larger transmission amplitudes can be expected for larger k z values and thus for larger n a quantum numbers. This is in full agreement with the experimental observations reported in the main text in Figure 2c.    In Figure S6 we report a sequence of diamond scans (dierential conductance) as a function of the common-mode gate voltage V cm , for a discrete set of imbalance values ∆V covering the anomalous crossing occurring at N=14 for ∆V ≈ −9V . Few excited state lines are visible in measurement but they appear to be connect with the partial or total lling of the available orbitals. For instance, in the sequence starting from imbalance -7V we highlighted an excitation line for the tunneling of the 14 th electron. The excitation energy is found to decrease until the size of the N=14 diamond is minimized at imbalance -8.12 V. This voltage marks the beginning of the gating region where the even-odd lling scheme is violated. This excitation line is obviously related to a partial lling of both the crossing orbitals and the process becomes the lowest energy one after imbalance -8.12 V. In the case  4 No obvious evidence for such a further resonance could be highlighted.

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S.5 Device structure and measurement set-up The structure of the studied devices is shown in Figure S7 along with a sketch of the measurement setup. Dierent gaps between the gates (GG) have been studied in dierent devices.
The device analyzed in the main text is reported in the SEM picture and had a gate gap of about 250 nm. Figure S7: Device structure, SEM picture of the device studied in the main text, and sketch of the measurement setup.