Resonant Tunning Through Quantum Dot Array

CHAPTER FOUR
DISCUSSION OF RESULTS
From the matrices gotten in the previous chapter, it is seen that the diagonal four-by-four matrices are proportional to the partition function Z.

In Eq. (2), En, is the energy of many-body state (n, i), the ith of the n-particle states, RL ,®ij are the transition rates between state (n,i) and (n-I,j) by losing or getting one electron through the left (right) lead, and peqn,I is the occupation probability of state (n,i) at equilibrium

Equation (2) is basically the same as those used in ref. 3 for single dots. The only difference is that the eigenstates of an array have a more complicated spatial structure and this has to be taken into account when calculating the transition rates RL,®ij in terms of the transition rate R out of a single dot into a lead.
Equation (2) is valid if there is no inelastic scattering within the array. For comparison, we also evaluated the conductance in the opposite limit when the inelastic scattering is so strong that the array is in local equilibrium. no significant differences were found. For this reason, we believe that inelastic process within the dot has no effect on our results. it is important to mention that the approach we use here is valid only when the lead coupling is small compared to the thermal energy, otherwise one cannot neglect the leads when calculating the many-body eigenstate of the array.
To start with we assume that the array is uniform, i.e Eia=Eo, Ui =U, ti=t, VLka=VR , and no interdot repulsion Wi=0 (i=1,2,.., N).first we examine a two-dot system, and assume low temperature and small interdot coupling (U>>T), so that excited states and second order interdot coupling O (t2 ) can be neglected. Consider what happens as we raise the Fermi energy starting with zero electrons in the array. The first electron coming into the system occupies the bounding state, which has an energy o-t when a second electron comes in, the two electrons tend to localize at different sites to avoid the intradot repulsion u; therefore the ground state energy is roughly 2eo =O (t2). After half filling, the third electron has to overcome the repulsion U, and with the same bounding scheme, the system is at energy, 3ec + U-t.
Finally, the system with four electronics has an energy 4ec+2U. as one sweeps up the Fermi energy, the two-dot system undergoes 0-1,1-2,2-3, and 3-4 particle transitions at Eo-t,Eo+t+O (t 2), and Eo = U+t repectively, according to Eq. (3). At the transition points, the system is able to fluctuate between n and n t 1 (n=0,1,2,3) particle states, allowing electrons to tunnel in and out of the array, giving rise to a peak in the conductance. For a two-dot array the eigenstates can be expressed analytically, and the second order interdot-coupling o(t2) turns out to be -2t2 /U. for longer chains, the many-body states are solved numerically, and the peack positions can be determine by physical arguments similar to those given above.
The conductance spectrum for chains containing two to six dots in the atomic limit is shown in figs. 1(a)-1(e) (U=5 mev, t=1 me v). figure 1(a) (dashed curve) also shows the number of dots in the array. The conductance has two symmetric groups separated by roughly the intradot repulsion U, and each group has a number of peaks equal to dots in the array. The conductance spectrum starts to refect the formation of Hubbard bands as the chain gets longer.