Magnetoconductance signatures of subband structure in semiconductor nanowires

Gregory W. Holloway, Daryoush Shiri, Chris M. Haapamaki, Kyle Willick, Grant Watson, Ray R. LaPierre, and Jonathan Baugh

Phys. Rev. B 91, 045422 – Published 16 January 2015

Abstract

The radial confining potential in a semiconductor nanowire plays a key role in determining its quantum transport properties. Previous reports have shown that an axial magnetic field induces flux-periodic conductance oscillations when the electronic states are confined to a shell. This effect is due to the coupling of orbital angular momentum to the magnetic flux. Here, we perform calculations of the energy level structure, and consequently the conductance, for more general cases ranging from a flat potential to strong surface band bending. The transverse states are not confined to a shell, but are distributed across the nanowire. It is found that, in general, the subband energy spectrum is aperiodic as a function of both gate voltage and magnetic field. In principle, this allows for precise identification of the occupied subbands from the magnetoconductance patterns of quasiballistic devices. The aperiodicity becomes more apparent as the potential flattens. A quantitative method is introduced for matching features in the conductance data to the subband structure resulting from a particular radial potential, where a functional form for the potential is used that depends on two free parameters. Finally, a short-channel InAs nanowire field-effect transistor device is measured at low temperature in search of conductance features that reveal the subband structure. Features are identified and shown to be consistent with three specific subbands. The experiment is analyzed in the context of the weak localization regime; however, we find that the subband effects predicted for ballistic transport should remain visible when backscattering dominates over interband scattering, as is expected for this device.

Received 14 May 2014
Revised 17 November 2014

DOI:https://doi.org/10.1103/PhysRevB.91.045422

Authors & Affiliations

Gregory W. Holloway^1,2,3, Daryoush Shiri¹, Chris M. Haapamaki^1,4,5, Kyle Willick^1,2,3, Grant Watson¹, Ray R. LaPierre⁵, and Jonathan Baugh^1,3,4,*

¹Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
²Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
³Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
⁴Department of Chemistry, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
⁵Department of Engineering Physics, Centre for Emerging Device Technologies, McMaster University, Hamilton, Ontario, Canada L8S 4L7

^*Corresponding author: baugh@iqc.ca

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Vol. 91, Iss. 4 — 15 January 2015

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Figure 1
(a) The cylindrical nanowire geometry is shown with an axial magnetic field $B_{z}$ . (b) Schematic of the nanowire FET used to measure magnetoconductance. The two-terminal conductance is measured between the source and drain contacts as a function of $B_{z}$ and gate voltage. (c),(d) Radial wave functions $R_{0, 0} (r)$ and $R_{0, 4} (r)$ , normalized by $\int_{0}^{\infty} \int_{0}^{2 π} {| e^{i l θ} R_{n, l} (r) |}^{2} r d r d θ = 1$ , calculated for a cylindrically symmetric radial potential $V (r)$ defined in the main text with $b = 2.75$ and (c) $A = 0.25$ eV, (d) $A = 0$ eV. $R_{n, l} (r)$ is characterized by the radial and angular quantum numbers $n$ and $l$ , respectively. The effective mass used is for InAs. Strong band bending results in a wave function proximate to the nanowire surface for all states. (e) Real part of the transverse electron wave function for the two states shown in (d).
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Figure 2
(a),(d) Calculated energy levels $E_{k} (B_{z})$ in a nanowire with radius $r_{0} = 38$ nm, for radial potentials $V (r)$ with $b = 2.75$ and $A = 0.25$ eV (a) and $A = 0$ eV (d). The radial quantum number is distinguished by color, where black denotes $n = 0$ , blue denotes $n = 1$ , and red denotes $n = 2$ . In (a), the curvatures of $E_{k} (B_{z})$ in the $n = 1$ manifold, appearing above 0.19 eV, are smaller than those of the radial ground state manifold because the radial expectation value $r_{eff}$ is closer to the nanowire center for $n > 0$ . In the lower part of (d), the successive subband minima move upwards in energy. This is due to an effective increase in confinement as the quantum number $| l |$ increases, since the wave function becomes more narrowly peaked. (b),(e) Ballistic magnetoconductance calculated from the energies in (a) and (d) using the Landauer equation. The conductance increases (decreases) stepwise by $2 e^{2} / h$ when a new transverse mode is populated (emptied). The vertical axis is to be identified with the chemical potential in the nanowire, modulated by gate voltage. (c),(f) Fast-Fourier transform (FFT) of the conductance in (b) and (e). The color scale is labeled $Δ G$ since the FFT peak intensity reflects the amplitude of magnetoconductance oscillations at a particular frequency. The mean of each conductance trace is subtracted prior to the FFT in order to avoid low frequency artifacts.
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Figure 3
(a) Derivative with respect to gate voltage of the experimental conductance of the InAs nanowire FET, where values below 0.43 $\frac{2 e^{2}}{h V}$ have been removed for clarity. White curves are least squares fits to parabolas consistent with transverse subbands. Red lines are linear fits to the parabolas from 0 to 2 T, used for extracting the zero-field slope of each curve. $s_{1} - s_{3}$ indicate the slopes and $d_{12}, d_{23}$ indicate the vertex separations in gate voltage. (b) Raw experimental magnetoconductance for this device, the source of the data shown in (a). (c) Ratios $d_{i, i + 1} / s_{j}$ calculated across the parameter ranges $0 \leq A \leq 0.2$ eV and $2 \leq b \leq 9$ . Colors indicate different sets of $l$ values: purple, $l = (- 1, - 2, - 3)$ ; blue, $l = (- 2, - 3, - 4)$ ; green, $l = (- 3, - 4, - 5)$ ; red, $l = (- 4, - 5, - 6)$ . These are plotted for three different radial excitation manifolds, $n = 0, 1, 2$ . Experimental values (black diamonds) from the fits to the data in (a) show best overall agreement with the $n = 1, l = (- 1, - 2, - 3)$ states. The $↑ A$ symbol indicates the direction of increasing $A$ values, i.e., stronger band bending. (d) Simulated magnetoconductance for $A = 0.11$ eV and $b = 2.75$ , over a range of chemical potential consistent with the experimental data, as described in the text. The subband transmissions are of order $L_{0} / L$ (see text), where $L_{0} = 20$ nm is a characteristic length on the order of the mean free path, and is a free parameter for matching the simulated conductance to the experimental values. Note that the conductance scales in (b) and (d) are the same.
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