Resonances in Conductance Through Tunable Attractors

We present the effects of resonant tunneling through doubly-coupled attractors in series in a quantum nanosystem. It is found that multiple split-tunneling peaks arise from resonant tunneling through multiple nondegenerate quasi-bound-states of the coupled attractors. We show that the tunability of coupling is achieved by modulating the strength of the attractor for a fixed separation length between the attractors, and by adjusting the geometrical separation length for a fixed intervening barrier strength.


INTRODUCTION
Current state-of-the-art techniques result in surface-independent probes or contacts that may be biased to introduce tunable scattering potentials ("impurities") in quantum nanosystems [1]. Many theoretical and experimental works then have studied the elastic scattering by impurities in these systems. In particular, for a quantum structure containing a finite-size attractive impurity, called an 'attractor', it is shown that an attractor introduces multiple quasi-bound-states (QBSs) in the channel for a sufficiently strong attractor, and these states give rise to multiple resonant peaks before the first plateau in the conductance [2].
These conductance peaks appear due to resonant tunneling through the QBSs.
In this paper, we study the splitting of multiple resonant peaks before the first plateau in the conductance for a quantum nanosystem with two identical attractors in series. The separation of the split-conductance peaks through both the first and the second QBSs depends strongly on the coupling between the two attractors, which can be tuned by modulating the intervening barrier strength [3] and/or changing the geometrical separation length.

MODEL AND METHOD
A sketch of the model two-dimensional nanosystem with two identical attractors in series is illustrated in Figure l( Our calculations of the conductance are based on a nearest-neighbor tight-binding (TB) Hamiltonian and a recursive Green's function technique. Electrons in this system are free to move in the xdirection but are confined in the y-direction, as By discretizing the system spatially with lattice constant a, the resulting TB Hamiltonian is exactly solved by using the recursive Green's function method [6]. The Green's function G of a system in the presence of a perturbation {z is related to the Green's function G in the absence of 9. The necessary recursion relations are obtained by taking the appropriate matrix elements of Dyson's equation, G G + G{ZG. The conductance is then obtained using the Landauer equation [7], G (2e2/h) P,, It,ll 2 (2e2/h) Tr (tt ), where t is the transmission matrix.

RESULTS AND DISCUSSIONS
We investigate conductance peak splitting due to resonant tunneling through multiple QBSs in a quantum system with two attractors in series. Figure   ing through E tt is mainly due to the coupling between E and E tt. Similarly, the resonant peak through E tt is due to the coupling between E2 and E tt. However, since the evanescent mode E2 is always energetically further away from the Fermi energy than El, the coupling between E and E tt is weaker than that between E1 and E tt. Therefore, when an electron moves from Ez to the E tt state, it stays a longer time in E tt and barely tunnels through it because of weak coupling. This indicates that the mean lifetime is larger and the width of the tunneling peak is narrower. Second, we discuss the splitting of tunneling peaks in the presence of coupling between attractors. For two identical series attractors, the single resonant peak through the first QBS of E tt splits into two well-separated peaks, shown in Figure 2 in the lower range of the Fermi energy for most values of Vatt displayed. [Here, the first half of the split peaks is not shown for l'att =-2.78 and -2.80]. In addition to these peaks, a conductance peak through the second QBS of E tt also splits into two sharp spikes for Vatt---2.75,-2.78 and -2.80. A more careful examination shows that each tunneling peak through E att 2 has a Lorentzian shape centered around the resonance energy [2]. We attribute this peak splitting to the elimination of the energy degeneracy due to the coupling between two attractors.
In other words, if the separation between identical attractors is small enough, then the aligned, uncoupled QBSs in each attractor overlap and produce new nondegenerate states of the coupled attractor system. For a simple two-attractor model, one coupled state is slightly higher (En,T)att and the other slightly below t'K'att 'n,J the uncoupled energy (Ett) [see Fig. l(b)], with a perfect tunneling rate indicated (i.e., transmittance through these states reaches one). Therefore, multiple split-conductance peaks arise from the resonant tunneling through multiple nondegenerate QBSs of the coupled attractors.
Next, we discuss the effect of the coupling between the attractors on conductance peak splitting. It is expected that the separation of the split-tunneling peaks in conductance is reduced, as the coupling between two attractors becomes weaker. This result is observed in the double and triple quantum dots, where a coupling (i.e., tunnel barrier) between the dots is controlled by separate gates [8]. Here, we show that the tunability of coupling can also be achieved by modulating (1) the strength of the attractor att for a fixed separation length between the attractors, and (2) the coupling length d (i.e., geometrical separation) for a fixed intervening barrier strength.
As the attractor depth att[ increases, in Figure   2, the separation of the split peaks through both the first and the second QBS becomes smaller. This is clearly seen in Figure 3, where the energy   makes the QBSs lower in energy and more localized in the attractor, and therefore results in less interactions or overlaps between them. As a consequence, the separation of peak splitting in conductance decreases as I?attl increases. We expect that the peak splitting will physically disappear when IVattl is large enough to produce strong localized QBSs in each attractor, i.e., for an extremely weak coupling case. Figure 4 shows the calculated conductance as a function of the Fermi energy for various coupling lengths. In order to see the effect of the geometrical coupling between the attractors on tunneling peak splitting, we focus on doubly split-tunneling peaks through the first two-nondegenerate QBSs (g att 1,T and ],att for afixed att -2.75. The geometrical separation length varies from d/Wa-0.5 (bottom curve) to d/Wa-3.0 (top curve) in steps of 0.1, and the origin of the curve for different coupling length is shifted vertically in order to make a clear comparison. As d/Wa increases (i.e., the attractors are weakly coupled), the level separation AE (= E E]) becomes narrower and hence the separation of the split-conductance peaks through these states is reduced in Figure 4. It is expected that when the attractors are far away from each other, each one has its original QBS E tt, and these degenerate states give rise to a single resonant peak before the first plateau. Notice that as d/Wa increases, the width of the peak through E att 1,T becomes narrower because the QBS 7'att is more 1,T bound and localized around the attractor. As expected, this feature can be seen. in the opposite way through the QBS E att Figure 5 shows how the separation of the split peaks changes with increasing coupling length d/W,. It appears that the energy separation of peak splitting, Asps, decreases exponentially as coupling length d/W, increases for att -2.75. Therefore, for the split-conductance peaks through both E tt and E tt, the coupling length d, in general, is related to/ksp as d -W In (Asps) for different ''att- 4 tractors in series in a quantum nanosystem. We have found that the doubly split-conductance peaks arise from the resonant tunneling through two nondegenerate QBSs of the coupled attrac-t6rs, and the separation of the split-tunneling peaks are strongly affected by the coupling length and the intervening barrier strength.