_{1}

^{*}

Apart from usual quantization steps on the ballistic conductance of quasi-one-dimensional conductor, an additional plateau-like feature appears at a fraction of about 0.7 below the first conductance step in GaAs-based quantum point contacts (QPCs). Despite a tremendous amount of research on this anomalous feature, its origin remains still unclear. Here, a unique model of this anomaly is proposed relying on fundamental principles of quantum mechanics. It is noticed that just after opening a quasi-1D conducting channel in the QPC a single electron travels the channel at a time, and such electron can be—in principle—observed. The act of observation destroys superposition of spin states, in which the electron otherwise exists, and this suppresses their quantum interference. It is shown that then the QPC-conductance is reduced by a factor of 0.74. “Visibility” of electron is enhanced if the electron spends some time in the channel due to resonant transmission. Electron’s resonance can also explain an unusual temperature behavior of the anomaly as well as its recently discovered feature: oscillatory modulation as a function of the channel length and electrostatic potential. A recipe for experimental verification of the model is given.

Short, narrow constrictions connecting two reservoirs of two-dimensional electron gas (2DEG) in semiconductor heterostructures, called quantum point contacts, exhibit at low temperature a quantization of the ballistic conductance in units of G_{0} = 2e^{2}/h (e and h are the elementary charge and the Planck’s constant, respectively, and the factor 2 arises from spin degeneracy). The constriction behaves like a quasi-one-dimensional conducting channel. The channel is defined by a voltage applied to a pair of finger-like gates deposited on the top of heterostructure. Upon widening the channel by tuning the gate voltage, one observes a staircase increase in the conductance, displaying distinct plateaus at integer multiples of G_{0} [_{0} to the conductance, whose number increases with the channel width.

Surprisingly, in clean channels of GaAs an additional plateau-like feature at about 0.7G_{0} appears, which is commonly known as the 0.7 anomaly [^{2}/h, which reveals its relation to the electron spin. In contrast to other plateaus, the 0.7 feature becomes less pronounced at lower temperatures, evolving from a distinct “plateau” at 4.2 K to a vague shoulder at 20 mK. At sufficiently low temperature the 0.7 feature is accompanied by the so- called zero bias anomaly (ZBA): appearing a maximum in nonlinear conductance while the bias voltage is swept through zero. That ZBA is characteristic of the Kondo effect in quantum dots.

Recently, structures supplied with three pairs of finger-like gates were studied, which allowed tuning the QPC length. It has been found in those structures that the 0.7 feature exhibits oscillatory modulation as a function of the channel length [_{0} as the mean value of the conductance anomaly [

It is commonly believed that the anomaly is a many-body effect, and accordingly numerous explanations have been proposed [_{0} characteristic of the anomaly and can explain all its main features.

Any spin state of the electron, _{i} and b_{i} that are complex numbers:

of this sphere. In the spherical coordinate system it can be represented as a spinor

Principle of quantum superposition claims that any physical system―such as electron―exists partly in all its possible states simultaneously, as long as it is not being observed. The state of linear coherent superposition can be here written as _{i} define contributions of different spin states to the superposition. In the absence of a magnetic field all spin states of the electron are equally probable and then

The second sum in parenthesis results from quantum interference between different spin states. Because of infinite number of those states, the discrete values Θ_{i} and Φ_{i} should be replaced by continuous variables, and the summations―by integration over the surface of the unit sphere, S, according to a transformation

Using the spinor representation of different spin states, we find after the integration

where the calculated interference term is

Similar procedure can also be applied in the case of spin polarization. Then, however, the coefficients c_{i} appearing in the superposition are different, resulting in a definite degree of spin polarization, given by a ratio

If an individual electron is being observed (detected), information is extracted that it is no more in the state of superposition. After the observation, electron must find itself in a definite spin state although we do not know in which one. The act of observation destroys interference, like that in the canonical double-slit experiment. Then, to get the probability, one has to sum partial probabilities of individual states instead of their probability amplitudes. The ratio, κ, of the probability of finding electron when it is subject to observation, P_{O}, to that when it exists in the state of superposition, P_{S}, is

Consider an electron travelling a QPC via the lowest 1D energy subband. The electron can exist there either in the state of superposition or―if it is being observed―in one of the possible spin states; the latter excludes the interference. Most importantly, it is not necessary to perform any real observation of the electron to suppress the interference. As demonstrated in the double-slit experiments, it is enough to create experimental conditions allowing such observation (see e.g. [

Essential condition allowing observation of an individual electron in the QPC is that no more than one electron travels the constriction region at the same time [

In fact, the 1D channel in QPC can behave as a resonant cavity for the electron wave. In the ballistic regime the two-probe resistance of QPC stems entirely from ”contact resistance” between 1D conducting channel and 2DEG reservoirs [

The width of quasi-1D channel in QPC varies with the position, x, along its length. Energy of the bottom of the lowest 1D subband,

where

resonance is met when

The quantity _{F} = 0.14 meV, and the Fermi velocity v_{F} = 2.7 × 10^{4} m/s (this and further numerical calculations concern n-GaAs). In equilibrium this E_{F} would determine the one- dimensional density of electrons which, under resonance condition, corresponded to two electrons in the channel.

Current considerations can be summarized as follows. Electron in a 2DEG reservoir, prior to its entry to QPC, finds itself in a state of superposition of all possible spin states. Probability of finding that electron, assumed further to be unity, contains a contribution originating from interference between different spin states. Individual electron entering the 1D-channel can be effectively detected if it is trapped in the channel for some time owing to resonant transmission. Detection of the electron suppresses the interference between different spin states. In other words: detection of the electron means its interaction with environment (that constitute here the gates connected with an external circuit) which causes decoherence of the state of superposition. The interference term I in Equation (3) becomes then equal to zero. Hence, probabilities of finding the electron after and before its entry to the channel are different; their ratio is 0.74. Obviously, the probability of finding the electron anywhere in the structure must be conserved. Here, this requirement comes down to conserving continuity of the probability current at the boundaries between 2DEG reservoirs and 1D channel. It can be met only if the electron wave- packet entering the channel is partly scattered back to the reservoir. That back-scattering contributes to an additional “contact resistance” which reduces the conductance of QPC just by a factor of 0.74.

Consider now the intriguing effect of modulation of 0.7 anomaly by the electrostatic potential, reported in [

When a negative electrostatic potential, U, is imposed on the QPC, the subband-edge energy, ε_{1}, is lifted up shutting the channel. In order to again open the channel, one has to adjust the gate voltage (making it less negative) to compensate the U-induced shift by widening the channel which lowers ε_{1}. However, by imposing the potential U an additional phase

This relationship predicts just the period of modulation of the anomaly. For L = 200 nm we find ΔU = 1.1 mV. This is a reasonable value to account for the observed modulation of the 0.7 anomaly while scanning negatively charged tip above the QPC [

In [_{g}_{2}, to that applied to the central ones, V_{g}_{1}. All these gate voltages contribute to the electrostatic potential acting on electrons in the channel. Their contribution manifests itself as more and more less negative gate voltage, V_{g}_{1}, required for opening the channel while the ratio V_{g}_{2}/V_{g}_{1} is being increased. In conclusion, we propose that tuning the electrostatic potential in the channel, and not its length, is the primary reason for modulation of 0.7 anomaly.

The 0.7 anomaly appears in the range of electron energies (that translates into a range of gate voltages) in which the resonance of single electron in the channel enhances its detectability. Accordingly, an extension of the 0.7 feature on the gate-voltage scale would be determined by the resonance linewidth. Here, we assume tentatively that the dominant mechanism of damping resonance is dephasing of the wave function. The phase-coherence length of electron,

Consider this issue in more detail. Uncertainty principle between momentum, p, and position, x, claims that

where E_{F} is the effective Fermi energy in resonance. This relation determines the spread of electron energies, ΔE, within the 0.7 anomaly appears. To have an idea about magnitudes of the quantities considered here, we put into Equation (7) the values of

The case of holes in the valence band of GaAs is much more complex than that of electrons in the conduction band mainly because of a strong spin-orbit interaction. Holes passing through p-type QPC just after opening the conducting channel are the heavy holes with relatively small wave numbers. Those holes behave as particles with effective spin J = 3/2, which have projections on the quantization axis J_{z} = ±3/2. Due to a size quantization, the holes coming out from a 2D reservoir have a quantization axis oriented in the epitaxial-growth direction (z-axis). It has been shown by Majorana [

At this aim we can simply use the same spinor representation as for spin-1/2 particle. However, the values of coefficients c_{i} in superposition are now diversified to favor the mean spin vector aligned along the quantization axis. Let us assume tentatively

The model proposed here is able to explain all main features of the 0.7 anomaly. In particular, it predicts a concrete value of the anomalous conductance, which has never been attempted by previous theories. It has been assumed here, in accord with Ref. [

This model can be verified experimentally. While tuning negative potential of the metallic tip placed above the QPC we expect to observe a periodic modulation of the anomaly whose period displays square dependence on the inverse channel length, described by Equation (6).

Thanks are due to Wlodek Zawadzki for the critical reading the manuscript and useful advices. This work was partly supported by the National Science Centre (Poland) under Grant No 2011/03/B/ST3/02457.

Tadeusz Figielski, (2016) The 0.7-Anomaly in Quantum Point Contact; Many-Body or Single-Electron Effect?. World Journal of Condensed Matter Physics,06,217-223. doi: 10.4236/wjcmp.2016.63021