ar X iv : c on d - m at / 9 50 70 58 v 1 1 5 Ju l 1 99 5 Spatial Coherence of Tunneling in Double Wells

Tunneling between two 2D electron gases in a weak magnetic field is of resonance character, and involves a long lifetime excitonic state of an electron and hole uniformly spread over cyclotron orbits. The tunneling gap is linear in the field, in agreement with the experiment, and is anomalously sensitive to the electron density mismatch in the wells. The spatial coherence of tunneling along the orbit can be probed by magnetic field parallel to the plane, which produces an Aharonov-Bohm phase of the tunneling amplitude, and leads to an oscillatory field dependence of the current. Electrons in GaAs quantum wells have very high mobility, and at low temperature form an almost ideal Fermi liquid. Strong Coulomb interaction and sensitivity to external magnetic fields make the physics of this system rich and interesting. Recently, several phenomena have been discovered in tunneling experiments in quantum wells, including: a tunneling gap induced by a magnetic field [1]; resonance peaks of conductivity near zero bias [2]; and excitonic effects [3]. Typically, one has two wells containing 2D Fermi liquids with an electron density of the order of 10cm, separated by an oxide barrier of few tens nanometers thick. It is characteristic that the barrier thickness is very uniform, so that tunneling is coherent in lateral dimensions. This results in conservation of both momentum and energy. Basically, at a zero magnetic field, only tunneling between identical plane wave states can occur, which restricts the phase space of final states and leads to an unusual I − V curve with a sharp peak near zero bias [2]. Finite width of the peak is determined by elastic scattering. One 1 can probe spatial coherence by applying a magnetic field parallel to the barrier. Having practically no effect on the dynamics in the plane, the vector potential of the parallel field shifts electron momenta in one plane, which creates a mismatch of the Fermi surfaces of different planes. The effect on the tunneling rate can be easily accounted for by the free electron picture [2]. A magnetic field applied perpendicular to the plane makes the situation more interesting, especially in high fields, when the system is in the Quantum Hall state [1]. Compared to the I−V curve at zero field, the current peak is shifted away from zero bias to some finite voltage. Also, the peak broadens as the field increases. The current is almost entirely suppressed below the lower edge of the peak, called a “tunneling gap.” Recently the tunneling gap has been intensively studied because it is believed that it can be used to probe the QH state. The gap depends on the field linearly at weak field (ν ≫ 1) [4], and saturates at higher field (ν ≃ 1) [1]. To summarize, one can say that the I − V curve can be interpreted in terms of a resonant tunneling mechanism, involving some intermediate state with the lifetime given by inverse peak width, and of the work needed to create this state corresponding to the gap. The gap at high field is quite well understood [5–7]: its energy scale is of the order of e/ǫa, where a = n is the interparticle distance, and ǫ is the dielectric constant. Until recently, the low field gap has received less attention. The only available theory is by Aleiner, Baranger, and Glazman [8], who developed a hydrodynamical picture by treating the system as an ideal compressible conducting liquid. They wrote down classical electrodynamics equations in terms of charge and current densities, and derived the tunneling gap ∆ = (h̄ωc/ν) ln(νe /ǫh̄vF ). At constant electron density, they predict quadratic dependence of the gap on the magnetic field, which is different from the linear dependence found experimentally [4]. The reason for the disagreement, in our opinion, is that in a clean metal, such that ωcτ ≫ 1, one can use classical electrodynamics only on a scale much bigger than the cyclotron radius Rc = vF/ωc. However, we will argue that the important scale of the problem is of the order of Rc, and thus one has to have a Fermi liquid in magnetic field. On this scale, 2 the state formed at tunneling has a non-trivial spatial structure, which makes the physics very different from that of the gap in high fields. Also, we will find that at weak field a large simplification occurs, because the problem is semiclassical, and one can use the classical Fermi liquid equation to describe the dynamics in terms of Fermi surface fluctuations. Another point is that in two dimensions the energy and momentum conservation prescribes that all quantum numbers of the final and initial tunneling states coincide. In a magnetic field, this implies that the radius of an electron orbit as well as the guide center of the orbit are conserved at tunneling. In a weak field, this results in spatial coherence of tunneling over a large distance of the order of Rc, and leads to interesting effects. Summary of results. Our goal is to explain the linear field dependence of the gap in a weak magnetic field, and to propose experiments that will reveal spatial structure of the intermediate tunneling state. Although at the end we are going to do a rigorous Fermi liquid calculation, it is instructive to begin with a semiclassical picture of electron states localized near classical cyclotron orbits and weakly interacting with each other. At tunneling, an electron hops from the orbit of radius Rc in one layer to the identical orbit in the other layer, and leaves a hole on the first orbit. This creates an electrostatic configuration of two oppositely charged rings of radius Rc separated by the barrier of width d ≪ Rc. The energy of this charge distribution is ∆ = e ǫπRc ln d lB , (1) where ǫ is the dielectric constant, and the magnetic length lB = √ h̄c/eB characterizes the ring’s “thickness.” Basically, we are saying that, in order to transfer an electron, one has to charge the “two ring capacitor,” and its charging energy e/2C constitutes the tunneling gap. The result (1) holds for lB ≤ d, i.e., fields which are not too low, and fall in the experimental range [4]. The gap ∆ dependence on magnetic field is nearly linear, since the log term is roughly constant. The dependence on the barrier width is in agreement with the excitonic picture [3,7]. By the order of magnitude the gap (1) agrees with the experiment [4], which raises the question of why there is no gap suppression due to Coulomb screening. Basically, 3 the reason is that he charge istribution is localized in a very thin ring, of the thickness lB comparable to the screening length, which makes the screeing effects not too dramatic: they simply change the log in Eq.(1) by a constant of the order of one. That the tunneling is coherent along the cyclotron orbit can be easily verified by applying a magnetic field parallel to the barrier, in addition to the perpendicular field. The parallel field flux “captured” between electron and hole trajectories will give an Aharonov-Bohm phase to the tunneling amplitude, and make it proportional to