Tunneling Between Parallel Quantum Wires

Abstract

Recent developments in fabrication techniques and experimental design have allowed the observation of tunneling conductance between parallel quantum wires of very high quality, in GaAs heterostructures, using the cleaved edge overgrowth technique [1]. To a first approximation momentum is conserved during the tunneling process, except for a momentum boostħQ = Bed/cproduced by an applied magnetic fieldBperpendicular to the plane containing the two wires, wheredis the separation between the wires. The differential tunnel conductanceG = dI/dVis proportional to the spectral density for creating a hole in one wire and and electron in the other, with total momentumQand energyE =eV, where V is the voltage difference between the two wires.

For two infinite wires, with non-interacting electrons, a gray-scale plot ofGwould show positive and negative S-function peaks along curves in the V —Bplane which are directly related to the one-electron dispersion curves in the two wires. If electron-electron interactions are included according to the LuttingerLiquid theory for a pair of interacting one-dimensional wires, the 6-functions are replaced, at small but finite V, by more complicated structures singularities along lines of different slopes that reflect the existence of separate velocities for spin and charge propagation, instead of the simple Fermi-velocity, in each of the two wires. Detailed predictions for the form ofGhave previously been given, for a pair of infinite wires, by Carpentier, Peca and Balents [2], and by Zülicke and Governale [3]. Experimental results in an appropriate regime were found to be consistent with the theoretical predictions, giving an indication of the separation of spin and charge velocities, but the experimental resolution did not permit an accurate measurement of the these velocities.

In recent work, together with Yaroslav Tserkovnyak, Ophir Auslaender, and Amir Yacoby, we analyzed in greater detail the experimental results in the range of small voltages and small magnetic fields, which in the sample studied were controlled by tunneling between one-dimensional bands in the two wires whose Fermi-momenta differed by only a few percent. [4 5] For non-interacting electrons, one can then neglect the difference in the Fermi-velocitiesvFbetween the two wires, and for infinite wires one would then expect to observe the 8-function contributions toGalong two lines, with slopes ±1/vF in theE—Qplane. However, the measurements show striking additional structure, which reflects the finite length of the region of tunneling, determined by the lengthLof the shorter of the two wires. In particular, in the gray-scale plot ofGthere are a series of light and dark bands parallel to the lines expected for an infinite sample, reflecting oscillations inGwith an approximate periodSBin the magnetic-field direction that varies inversely with the lengthLfor different samples. The period corresponds roughly toSQ =2π/L, as might be expected from a diffraction pattern. Unlike a conventional diffraction pattern, however, the observed patterns were one-sided, appearing only on the positiveVside of the infinite-length lines (where positive V favors electron-tunneling from the longer of the two wires into the shorter wire). We have been able to explain this effect by considering that the confinement at the ends of the shorter wire is not sharp, but is gradual, due to the relatively large distance of the wire from the confining electrostatic gates. Thus the electron wave functions in the shorter wire are not sine waves but are WKB wave functions containing about 100 nodes, that are spaced further apart near the ends than in the center of the line segment. By making a reasonable assumption for the shape of the confining potential, we have been able to get good agreement for both the varying spacing between maxima of the oscillatory pattern and for the fall off of intensity with distance from the beginning of the pattern.

Another feature of the observed diffraction pattern is a series of bands parallel to the B-axis, where the modulation ofGis greatly reduced. We can explain this a Moire pattern from superposition of two sets of parallel lines with different slopes, corresponding to two different velocities, which we identify with the spin velocity and the charge velocity of the wires. (Only the antisymmetric charge mode, with charge fluctuations of opposite sign in the two wires should be excited in the tunneling experiment). The spacingSVbetween the bands of suppression allows one to determine that the charge velocity is larger than the spin velocity by a factor of 1.3, which is consistent with other estimates for this system. Detailed calculations, in which we have generalized the Luttinger liquid theory to a situation of two coupled wires, with a varying density along the length of one wire, give good agreement with the observations. We believe that these are the most precise measurements to date of separate spin and charge velocities in a tunneling measurement of one-dimensional wires.

In summary, the experiments, under interpretation, give detailed information about the confinement at the ends of the wire, as well as a measurement of the charge and spin velocities in the coupled system.

This work has been supported by the National Science Foundation through grants DMR-02-33773, and by the US-Israel BSF.