## Abstract

Adiabatically rocked electron ratchets, defined by quantum confinement in semiconductor heterostructures, were experimentally studied in a regime where tunneling contributed to the particle flow. The rocking-induced electron flow reverses direction as a function of temperature. This result confirms a recent prediction of fundamentally different behavior of classical versus quantum ratchets. A wave-mechanical model reproduced the temperature-induced current reversal and provides an intuitive explanation.

In the absence of any net macroscopic forces, asymmetric potentials can be used to induce a particle flow when subjected to external fluctuations (1–5). Referred to as ratchets, their operation has been proposed as the underlying physical principle of molecular motors in biological systems (1, 2, 5–7), such as the myosin-actin system that affects muscle contraction (8, 9). Fluctuation-induced particle current has also been observed in artificial ratchets (10–16). When tunneling contributes to the particle flow in an adiabatically rocked ratchet, it has been predicted that the current direction will depend on temperature (17). Here we use quantum confinement to define an electron ratchet in a semiconductor nanostructure. We find that the two contributions to the rocking-induced current (tunneling through and excitation over the ratchet's energy barrier) flow in opposite directions. Thus, the net current direction depends on the electron energy distribution at a given temperature. This behavior can be compared to the energy-sorting task assigned to Maxwell's demon, and it also demonstrates a fundamental difference between classical and quantum ratchets (17–19).

A quantum ratchet can be established by confining electrons to an asymmetric conducting channel of a width comparable to the electron wavelength. Shown in the inset of Fig. 1A, the darker regions are etched trenches that laterally confine a two-dimensional sheet of electrons located parallel to the surface of a GaAs/AlGaAs heterostructure. The channel's periodic variation in width induces a corresponding variation in electron confinement energy that creates asymmetric energy barriers at each constriction. When a source-drain bias voltage*U* is applied, the resulting current *I* is determined by the barrier's reflection and tunneling coefficients and is sensitive to the precise shape of these barriers. Because of the geometric asymmetry, the electric field along the channel produced by the voltage *U* deforms the barriers in a way that depends on the polarity of *U* (Fig. 1B), such that*I*(*U*) ≠ −*I*(−*U*). Thus, a net current *I*
_{net} = (0.5)[*I*(*U*
_{0}) +*I*(−*U*
_{0})] is generated when a square-wave source-drain voltage of amplitude *U*
_{0}is applied, even though the time-averaged electric field is zero. In this way, current is generated in a rocked ratchet, where the term “rocking” refers to the effect of the voltage on the ratchet potential. Here we realize such an electron ratchet in the tunneling regime and experimentally confirm the theoretical prediction (17) that the current direction in quantum ratchets depends on temperature *T*.

Experimental data from two devices are presented. One device, denoted by R10, had a channel featuring 10 periodic ratchet cells (Fig. 1A). The other device, R1, consisted of a single ratchet cell of the same shape and size. The lateral dimensions of the ratchet cells, which were patterned with electron-beam lithography and shallow wet etching, were much smaller than the length scales for elastic (6 μm) and inelastic (>10 μm) scattering at the temperatures and voltages used here (energies *k*
_{B}
*T* and |*eU*| ≤ 1 meV, where *k*
_{B} is Boltzmann's constant and *e* is the electron charge). A net current was generated using a square-wave voltage of frequency 191 Hz. In order to enhance sensitivity, the square-wave voltage was 100% amplitude modulated, so that it was effectively applied in periodic bursts (repetition frequency of 17 Hz), and the two-terminal average current *I*
_{net} during each burst was measured by phase locking on the repetition frequency. None of the elements in the path of the current, such as ohmic contacts, caused substantial rectification (20). All of the measurement frequencies were much lower than any electron energy-relaxation processes in the electron gas (<1 ns), so that, electronically, the ratchet was in a stationary state at all times (so-called “adiabatic rocking”). The width of the conducting channel was tuned by using the electron sheet regions parallel to the channel as side gates. A lower (more negative) side-gate voltage *V*
_{g} corresponds to a narrower channel and therefore higher potential barriers.

In Fig. 1, we show *I*
_{net} versus*V*
_{g} for both devices in a range where the conductance of the devices (top axis in Fig. 1, A and B) was of the order of 0.2*e*
^{2}/*h* for R10 (Fig. 1A) and 2*e*
^{2}/*h* for device R1 (Fig. 1B) (where *h* is Planck's constant). This conductance corresponds to the transmission of only the lowest one-dimensional (1D) wave mode over each barrier (21), indicating that the barrier height *V*
_{0} approximately matches the Fermi energy μ_{F} of 11.8 meV. The rocking voltage was*U*
_{0} = 1 mV in both cases and corresponded to a voltage drop of 0.1 mV over each barrier in device R10 (<1% of μ_{F}) and a drop of 1 mV in device R1. The net current generated in this way corresponds to ∼1 to 5% rectification of the total current and, at 4 K, was initially positive (22) for low *V*
_{g} but reversed its direction as the barrier height was reduced (Fig. 1). At 0.4 K, additional structure emerged in the plots of *I*
_{net} versus*V*
_{g}. The current reversed direction several times as the barrier height was varied. We studied*I*
_{net}(*T*) in more detail by slowly warming the devices at fixed *V*
_{g} (at the values indicated by lettered arrows in Fig. 1) while recording*I*
_{net} (Fig. 2A). Confirming the prediction (17), the current reverses direction as a function of temperature while all other parameters, including the shape and height of the potential barriers, are kept constant.

To provide an intuitive understanding of this temperature-induced current reversal, we present a model of 1D transport over a single barrier of a shape based on the geometry of the device (see Fig. 1B). Electron-interference effects inside the individual ratchet cells (13) are not considered here. We assume that the transport is ballistic, no inelastic processes occur, and electrons in all but the lowest subband are fully reflected. Because backscattering is caused by a change in the potential, we assume a spatial distribution of the source-drain voltage drop *eU* that scales with the local gradient of the zero-voltage potential (Fig. 1B) (13, 23,24). At low temperatures (*k*
_{B}
*T* << μ_{F} − |*eU*
_{0}|), a square-wave voltage of amplitude*U*
_{0} generates an *I*
_{net} that can be written as (21)
where *e*, the electron charge, is negative (22). Δ*f* = {*f*[ɛ + (*eU*
_{0}/2) − μ_{F},*T*] − *f*[ɛ − (*eU*
_{0}/2) − μ_{F},*T*]} describes the range of the energy ɛ, in which electrons contribute to the current, and *f*(ɛ −μ_{F}, *T*) denotes the Fermi-Dirac distribution in the electron reservoirs. Δ*t* = {*t*[ɛ + (*eU*
_{0}/2), *U*
_{0}] −*t*[ɛ − (*eU*
_{0}/2), −*U*
_{0}]} is the difference between the transmission functions for positive and negative voltages, calculated by numerically solving the 1D time-independent Schrödinger equation. The current is zero in the linear response limit where Δ*t* = 0.

For *U*
_{0} = 0.5 mV, we find that Δ*t*(ɛ) is of the order of 10^{−3} for energies within a few millielectron volts of the barrier height*V*
_{0} = 12.0 meV (Fig. 3). Most important, however, and independent of the exact value of*U*
_{0} < *V*
_{0}, Δ*t* changes sign as a function of energy. The energy window Δ*f*(ɛ) is shown in Fig. 3 for*U*
_{0} = 0.5 mV, μ_{F} = 11.7 meV, and three different temperatures. At the lowest temperature, the electrons sample an energy region where Δ*t* is, on average, positive, resulting in a positive *I*
_{net} at this particular μ_{F} and *T*. With increasing*T*, however, more electrons contribute to*I*
_{net} in an energy range where Δ*t* is negative. Thus, *I*
_{net} reverses sign as a function of *T* (solid curve in the lower inset of Fig. 3). Temperature and rocking voltage have a similar energy-averaging effect, such that the temperature at which the current reversal occurs decreases with increasing voltage (Fig. 3, lower inset). This effect is also apparent in Fig. 2B, where *I*
_{net} in device R1 at*V*
_{g} = −0.735 V (point b in Fig. 1B) is shown as a function of the rocking voltage *U*
_{0} at different temperatures.

The crucial property of the model that leads to the current reversal is the sign change of Δ*t*(ɛ) around ɛ ≈*V*
_{0}. One can understand the physical origin of this feature intuitively by noting that a positive bias voltage deforms the potential in such a way that the barrier is thinner than it is at zero bias voltage and that a negative bias voltage results in a thicker barrier (see inset of Fig. 1B). Thus, the probability for tunneling at a given ɛ below the barrier top is greater for positive bias than for negative bias, such that Δ*t* > 0 below ɛ ≈*V*
_{0}. However, the thinner barrier also tends to be steeper and thus causes more wave reflection for electrons at energies above the barrier, so that Δ*t* < 0 for ɛ > *V*
_{0}. Classically, when the transmission function is simply a sharp step at ɛ = *V*
_{0}(Fig. 3, upper inset), no such sign change occurs.

At around ɛ ≈ *V*
_{0}, where Δ*t*(ɛ) changes sign in Fig. 3, the quantum transmission probability is ∼0.75 (Fig. 3, upper inset), which corresponds to a linear response conductance of about *G* = 1.5*e*
^{2}/*h* (21). This value agrees well with that of *G* for the single-barrier R1 at the*V*
_{g} at which *I*
_{net} changes direction in the measurement at 4 K (Fig. 1B). Similar features are observed in device R10 at about one-sixth the conductance value, a scaling factor that is within the range expected when ballistic transport between neighboring constrictions is considered (25,26). This agreement between experimental and theoretical data indicates that the positive net current at low*V*
_{g} is indeed related to tunneling. The qualitative similarity between the two devices shows that any coupling effects between individual ratchet cells do not have a strong effect on the qualitative behavior of the net current. The single-barrier model can therefore also be applied to the periodic ratchet R10. At low temperature (0.4 K), the behavior of the current observed in the experiment is more complex than in the model. The differences may be related to superimposed nonlinear rectifying behavior in the higher 1D subbands that are excluded in the model. Because the effect of temperature is simply to average Δ*t*(ɛ) over energy, the more complex behavior of the *I*
_{net} allows us to observe current reversals in both directions in the experiment (Fig. 2A). In close agreement between experiment and model, the temperature at which reversal occurs is typically a few kelvin, a value that is related to the energy scale of variations of Δ*t*(ɛ) (Fig. 3). The observed *I*
_{net} exceeds that found in the model, indicating that the voltage-induced modification of the ratchet potential is stronger than what is assumed in the calculation. Although a rigorous theoretical treatment will be of interest (24), the 1D single-barrier model used here yields intuitive, qualitative understanding of the sign and temperature dependence of the observed net current.

The observation that the rocking-induced flow of electrons with high and low energy is in opposite directions (Fig. 3) has a fundamentally interesting implication: A tunneling ratchet can perform the energy-sorting task assigned to Maxwell's demon (27), without violating the second law of thermodynamics, because it operates only at a finite voltage. Another important aspect is that artificial nanostructures can be used to test physical principles that may be of importance in living systems. Whether transport mechanisms based on quantum effects such as the one studied here play a role in biological motor proteins is an intriguing question.

↵* To whom correspondence should be addressed. E-mail: hl{at}phys.unsw.edu.au