Abstract
Quantum ballistic transport in electron waveguides (EWGs)^{1,2} is based on coherent quantum states arising from the onedimensional (1D) confinement in nanometrescale constrictions. Semiconductor EWGs have received considerable renewed interest for quantum logic devices^{3,4,5,6,7} and theoretical concepts^{8,9,10,11} in the context of solidstate quantum information processing^{12}. Implementation in realworld quantum circuits requires the unambiguous experimental distinction between all involved energy levels. However, such knowledge of EWGs investigated for wavefunction hybridization^{13,14,15} is solely based on estimates. Here, we present coupled EWGs that allow singlemode control and manipulation of mode coupling at temperatures as high as that of liquidhelium (4.2 K) and above. We demonstrate highresolution energy spectroscopy of each EWG subband ladder and the 1D coupled states involved. The results verify the power of advanced nanolithography and its ability to open the door to the scalable semiconductor quantum circuits envisaged today.
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Main
Quantum 1D conductors formed from twodimensional (2D) electron gases (2DEGs) are important in nanoscopic and mesoscopic semiconductor devices when studying the physics of coherent electron flow^{1,2}. Spatial 1D constrictions are formed of the order of the Fermi wavelength λ_{F}∼35 nm and much below the mean free path length of l∼10 μm as achieved in highelectronmobility (∼1×10^{6} cm^{2} V^{−1} s^{−1}) 2DEGs in AlGaAs/GaAs heterostructures. Various techniques can be used, such as metaldeposited split gates^{16,17}, etching^{18,19,20} or local anodic oxidation^{21}. Conductance quantization^{16,17} in linear and nonlinear transport is reported for etched single EWGs for temperatures up to 30 K (ref. 19), which requires large 1D subband separations (>10 meV). So far, coupled EWGs have been restricted to 1D subband spacings of a few millielectronvolts, hampering the direct highresolution energy spectroscopy of each single EWG and their coupled modes^{13,14,15}. Only estimates of splitting energies are known^{13,22}. Our objective here is to demonstrate direct highresolution energy spectroscopy applicable to various coupled EWGs. The complete knowledge of modecoupled 1D energy spectra is provided fulfilling a prerequisite for quantum engineering of proposed complex EWG devices^{3,4,5,6,7,8,9,10,11}. Nanolithography with an atomic force microscope (AFM, Digital Instruments, Nanoscope III) enables us to produce EWGs showing 1D subband spacings above 10 meV allowing singlemode operation and control of mode coupling at liquidhelium temperature (4.2 K) and above.
Spatially separated coupled EWGs can be realized using a tunnel barrier, either from two vertically stacked 2DEGs or from a single 2DEG with a lateral tunnel barrier. Spatially coincident coupled EWGs can also be realized, by either injecting electrons from different modes of the 2D reservoirs or by laterally merging the electron flow from two EWGs into one junction. In each case, mode coupling occurs as wavefunction hybridization^{13} Ψ_{+/−}〉=a_{1}Ψ_{s1,n1}〉±a_{2}Ψ_{s2,n2}〉, which results in splitting ΔE of two degenerate 1D energy levels E_{n}^{s} (see Fig. 1d). Here, s distinguishes between the EWGs and the quantum number n denotes the 1D transverse mode index. The symmetry of the confining potential V (x,y,z) defines the mode spectra^{23} including all experimentally observed level crossings and anticrossings^{13,14,15,22,24}. In our study, the mixing between n1 and n2 subbands of the wires labelled s1 and s2 is determined by the matrix element 〈Ψ_{s1,n1}V (z)Ψ_{s2,n2}〉, where V (z) is the conductionband profile of the quantum well under consideration as detailed in ref. 13. Mode coupling is expected in aligned wires of equal width for transverse modes of n1=n2 and in aligned wires of different width if n1 and n2 are either both odd or even. In finite magnetic fields the interaction can be strongly modified (see below).
We fabricated spatially coincident and spatially separated coupled 1D electron systems. The devices are short EWGs in threeterminal AlGaAs/GaAs modulationdoped fieldeffect transistor structures (Fig. 1a). Ohmic source and drain contacts to the 2DEG reservoirs allow us to pass a source–drain current through the 1D channel. A Schottkycontact top gate covers the 2DEG reservoirs and the 1D channel. The barriers forming the 1D constriction are established by nanogrooves wetetched into the semiconductor surface beneath which the 2D electron gas is locally depleted (Fig. 1b). Spatially coincident EWGs are formed from the ground and first excited states of a 2DEG in a 30nmwide GaAs quantum well^{22} (see Supplementary Information, Fig. S1). Spatially separated tunnelcoupled EWGs, on the other hand, are based on two vertically stacked 2D electron systems hosted in a 30nmwide GaAs quantum well with 1 nm of AlGaAs barrier in its middle (see Supplementary Information, Fig. S1). Details of the heterostructures and the measurements are given in the Methods section.
Coupled EWGs show deviations from the normal conductance increase^{22,24,25} quantized by 2e^{2}/h (e is the electron charge and h Planck’s constant, see Fig. 1c and Supplementary Information, Fig. S1), revealing that two 1D subband edges are in close proximity. Double conductance steps of 4e^{2}/h are formed if 1D levels become degenerate. The contributing modes can unambiguously be identified by shifting the subladders relative to each other either by purely electrostatic means^{24} or by magnetotransport spectroscopy^{22}. Mode coupling has been observed in terms of level anticrossings in the transconductance^{13,14,15,22}.
Energy spectroscopy of mode coupling is demonstrated for spatially coincident EWGs first. The two 1D electron systems arising from the occupation of the ground and first excited states of 2D reservoirs (see Supplementary Information, Fig. S1) are depicted as the first (s=1) and second (s=2) vertical mode, respectively (Fig. 1d). The 1D electron wavefunctions can be denoted as Ψ_{s,n}(x,y,z)=exp(i k_{x}x)ψ_{s,n}(y,z) with directions x for propagating modes, y and z for lateral and vertical confinement, respectively. For decoupled lateral and vertical components, the envelope wavefunction ψ_{s,n}(y,z)=ϕ_{n}(y)Φ_{s}(z), where ϕ_{n}(y) denotes the nth lateral mode and Φ_{s}(z) the vertical wavefunction of the quantum well. Inplane magnetic fields B energetically shift the two 1D subbladders, E_{n}^{1} and E_{n}^{2}, relative to each other^{22}. In transverse magnetic fields, see Fig. 2a, 1D subbands for s=2 experience a stronger diamagnetic shift^{26} proportial to B^{2} than for s=1 on behalf of larger spatial extensions of the vertical wavefunction^{22} (see Supplementary Information, Fig. S2). Mode coupling between wires of slightly different lateral confinement occurs for 1D levels of equal parity in the transverse mode index^{13,22} n (see Supplementary Information, Fig. S4). In longitudinal magnetic fields, see Fig. 2b, lateral and vertical components of the confining potential are coupled leading to Fock–Darwinlike spectra^{14,22} and complex mode mixing^{13}.
A relative shift of 1D subladders can be performed purely electrostatically by cooling the sample under gate bias^{22}. Here, the confining potential is varied by persistent recharging of the doping layer^{27} (see Supplementary Information, Fig. S3). This enables the observation of coincidences of different pairs of modes in the magnetotransport spectra^{22,24} because of a shifted subladder onset.
Recording the quantized conductance under d.c. source–drain bias allows direct imaging of mode coupling and spectroscopy of the corresponding energy splittings as demonstrated in Fig. 2c and d. The transconductance maxima (in black) reflect coincidences of the chemical potential μ_{S,D} of the source or drain reservoir with 1D subband edges E_{n}^{s}. For a single EWG, extensions of the rhombic patterns of transconductance maxima on the drain bias voltage scale are a measure of energy separations ΔE_{n,n+1}^{s} between subsequent 1Dsubband edges^{28}. For coupled EWGs, two superimposed sequences of rhombic pattern depict the two subladders. Furthermore, e patterns can appear as a consequence of mode coupling. Comparing Fig. 2d with the corresponding magnetotransport spectrum in Fig. 2b also leads to an unambiguous identification of the onset of the second 1D subladder for each applied magnetic field. Level anticrossings evident in the mode spectrum, such as (1,5) and (2,2) at B=3.7 T encircled in Fig. 2b, are directly reflected in the bias spectroscopy (Fig. 2d, encircled). Here, the energy splitting ΔE=2 meV of the coupled modes (1,5) and (2,2) depicts an intermediate coupling strength when compared with the 1D subband spacing ΔE_{1,2}^{1}=10 meV of the first vertical mode. At zero magnetic field, energy spectroscopy helps to identify anticrossings: unphysical increases in subsequent subband spacings are observed in the independent singlewire subladder if uncoupled modes are assumed (an example is given in Supplementary Information, Fig. S4). Furthermore, bias spectroscopy enables us to untangle different mode contributions as visible by the relative shift of the 1D subladders obtained by cooling under different gate bias voltages in Fig. 3a,b. Here, anticrossings such as (1,4) and (2,2) (Fig. 3a) disappear and a superimposed image of the two independent, undisturbed 1D subladders becomes visible (Fig. 3b). Detailed evaluation of bias spectroscopy allows the verification of assumptions made about each EWG confining potential (an example is given in Supplementary Information, Fig. S5).
Onedimensional subband spacings of more than 10 meV allow us to operate EWGs in direct contact with a liquidhelium bath, which is favourable for any longterm measurement. Figure 3a,b demonstrates highresolution energy spectroscopy at 4.2 K. Figure 3a corresponds to the 2 K data in Fig. 2c. The anticrossing can be resolved up to 5.2 K, see Fig. 3d.
Tunnelcoupled EWGs are of much interest for application in quantum device circuits ^{3,4,6,7,8,9,10,11}. In our vertically stacked EWGs of symmetric quantum wells, the 1D sublevels of the bottom EWG (b,n) vary more strongly with applied back gate voltage V_{bg} than levels from the top EWG (t,n) owing to electrostatic screening. Therefore, a purely electrostatic shift of 1D subladders is possible. Degenerate 1Dsubband edges of unequal transverse mode index n lead to level crossing with varied backgate voltage^{13} (see Supplementary Information, Fig. S6). Mode coupling occurs for 1D levels from the bottom (s=b) and top (s=t) EWG with equal transverse mode indices^{13} n. This special situation is fulfilled for EWGs of nearly equal threshold as depicted by the greyscale plot of 4.2 K transconductance measurements versus top and back gate voltage in Fig. 4a for different cooling biases. Bias spectroscopy at V_{bg}=−210 V (Fig. 4c) and 210 V (Fig. 4e) enables us to determine energy separations between the first and second 1D subband of the top and bottom EWGs to 12.3 and 10.8 meV, respectively. Furthermore, at zero back gate voltage (Fig. 4d) for which mode coupling occurs, bias spectroscopy reveals splitting energies of 5.4, 3.9 and 2.9 meV for n=1, 2 and 3, respectively. Mode coupling is seen for temperatures above 10 K. In Fig. 4b, the transconductance peaks of the split levels are unambiguously distinguishable.
On increasing temperature T the conductance plateaux in single EWGs acquire a finite slope^{1,2}, , as the Fermi–Dirac distribution f(E−E_{F})=(1+exp(E−E_{F}/k_{B}T))^{−1} becomes smeared (E_{F} is the Fermi energy and k_{B} is the Boltzmann constant). The observed energy resolution ΔE corresponds to the width of the thermal broadening df/dE_{F} of ∼4k_{B}T. As our measurements of coupled EWGs show, subband spacings of 10 meV are also resolved up to 20 K and level splittings of 2.5 meV are accessible up to 5 K. The thermal dephasing time τ_{th}=πħ/k_{B}T (where ħ is the reduced Planck constant) decreases from 6 ps at 4.2 K to 2.4 ps at 10 K and is the dominating timescale in our heterostructures above 4 K. In comparison, the scattering time τ_{sc}=m^{*}μ/e is 32 ps for the two subband 2DEGs and 9 ps for the vertically stacked 2DEGs, where m^{*} is the effective mass and μ the electron mobility. Phasecoherence times τ_{ph} of about 60 ps ensure the socalled coherent quantumwire states in single EWGs^{1,2,29}. Superposition states formed by coupling require coherence in the transverse modes. They lead to splitting of otherwise degenerate 1D subband edges and are experimentally detectable by means of bias spectroscopy.
As we have demonstrated, level spectroscopy and identification of single and coupled quantumwire states can be established by allelectrical means. This paves the way for quantum engineering of coupled EWG devices. Further, we have shown that the manipulation and the control of coupled 1D modes is feasible at temperatures of liquid helium and above. This opens the field of research to standard lowtemperature laboratory equipment. Widely accessible nanopatterning techniques may be used to prepare coupled EWGs: local probe techniques, electronbeam lithography and largescale imprint lithography. Possible applications of multiple EWGs are bidirectional couplers^{3,4}, quantum waveguide inverters^{5} and, for an allelectrical control of coherent quantumwire superposition states, quantum networks capable of processing quantum information^{8,9,10,11,12}.
Methods
Two series of EWGs were fabricated from GaAs/AlGaAs heterostructures with different quantum wells. Spatially coincident EWGs were formed from a 30nmwide GaAs square quantum well situated 60 nm below the sample and Si deltadoped from both sides. The asgrown carrier mobility was determined in the dark as 8.5×10^{5} cm^{2} V^{−1} s^{−1} at a density of 4.2×10^{11} cm^{−2} at 4.2 K. Spatially separated, tunnelcoupled EWGs were made from two 14.5nmwide GaAs layers separated by a 1nmthick Al_{0.32}Ga_{0.68}As barrier. The upper interface of the top quantum well lies 60 nm below the heterostructure surface. A Si deltadoped supply layer is situated on each side of the double quantum well. A sheet electron density of 4.3×10^{11} cm^{−2} and a mobility of 2.4×10^{5} cm^{2} V^{−1} s^{−1} were measured in the dark at 4.2 K for the structure without a top gate.
Onedimensional constrictions were defined introducing local lateral barriers by means of etched nanogrooves. Dynamic ploughing of a 7nmthick resist with an AFM and subsequent wetchemical etching of a line pattern as shown in Fig. 1c lead to the complete depletion of electrons underneath in each heterostructure^{20}. Twoterminal differential conductance and transconductance measurements were performed by means of a standard lockin technique. The source–drain excitation voltage was 0.3 mV r.m.s. at 433 Hz. Series resistances were determined from the deviation of the conductance from the fundamental values. In addition, for transconductance measurements, the topgate voltage was modulated with a 3 mV r.m.s. voltage. The error for energylevel separations determined by source–drain d.c. bias spectroscopy amounts to 0.5 meV.
The conduction band edges and charge distribution of the quantum wells were obtained with the numerical simulation program AQUILA (ref. 30). The quantumwell width and barrier thickness were taken from the growth parameters as described above; the Fermi energy was assumed to be pinned by surface states at the middle of the bandgap. The heterostructure doping was chosen such that the 2D subband electron densities match the measurement results (see above) and are determined with donor concentrations of N_{d,top}=2.1×10^{12} cm^{−2} and N_{d,bottom}=2.9×10^{11} cm^{−2} for the top and bottom deltadoping layers, respectively, and N_{A}=10^{15} cm^{−3} for the acceptor concentration of GaAs. The latter results from ptype carbon doping unintentionally introduced during growth.
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Acknowledgements
Part of this work was supported by the Bundesministerium für Bildung und Forschung under grant no. 01BM920. G.A. gratefully acknowledges the financial support of the foundation Isolde Dietrich. S.F.F. is grateful to Y. Milev, A. Rüdinger, L. Murokh, C. van der Wal and R. Akis for valuable discussions.
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Fischer, S., Apetrii, G., Kunze, U. et al. Energy spectroscopy of controlled coupled quantumwire states. Nature Phys 2, 91–96 (2006). https://doi.org/10.1038/nphys205
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DOI: https://doi.org/10.1038/nphys205
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