Transport through a single quantum dot weakly coupled to two contacts has been the subject of much experimental and theoretical work, i 4 and a fairly clear picture has emerged. Relatively little work has been done on arrays of quantum dots, though it now seems feasible to fabricate such structures. 5 Most theoretical work on arrays has been based on the RC model which neglects coherence between individual dots in the array. However, it is expected that in semiconductor quantum-dot arrays such interdot coherence will play an important role in determining the transport properties. The purpose of this paper is to present theoretical results for the conductance of coherent arrays as a function of the Fermi energy, G(E~). A single quantum dot behaves as an artificial atom in its charge and energy quantizations and is often described by the Anderson Hamiltonian, 4 in which there is a finite Coulomb repulsion between any two electrons on the dot. For an array of quantum dots with phase coherence, each dot can be viewed as an artificial atom with intradot Coulomb repulsion (as in Anderson model), and electrons hop between nearest neighbor dots. It seems reasonable then to model an array of quantum dots using the Hubbard Hamiltonian~ characterized mainly by two parameters: the intradot charging energy (U) and the interdot coupling matrix element (t). Our approach is to calculate the many-body eigenstates of the array (isolated from the contacts) by exact diagonalizations and then to incorporate the effect of the contacts through a rate equation as done by Beenakker for single dots. This treatment of the contacts should be accurate as long as the temperature is higher than the Kondo temperature. We have studied arrays containing N = 2, 3, ... up to six dots, and find that (1) in the atomic limit (t ( U) the peaks in the conductance G(E~) form two distinct symmetric groups separated by U and (2) in the band limit (t) U) the peaks occur in pairs separated by order of U. Thus even such short arrays exhibit properties reminiscent of the infinite Hubbard chain. Interestingly, we find that the inclusion of inelastic processes within the array does not significantly affect the results. It might thus be possible to study various aspects of the Hubbard model using such artificial quantum-dot arrays. Real arrays can be expected to have two main deviations from the ideal Hubbard model. First, in addition to the intradot Coulomb repulsion, there will exist a certain degree of interdot repulsion. Second, individual dots will be invariably "detuned" kom each other to some extent. Both these aspects are readily incorporated in our model, and we find that they have a noticeable effect on the conductance spectrum.
Interdot repulsion destroys the symmetry between the two groups of peaks which we identified as the upper and the lower Hubbard bands. Detuning tends to localize the electron states, thus suppresses the conduction peaks. The effects of detuning have been presented in a separate publication. Consider a one-dimensional (1D) array of N coupled dots, indexed &om left to right as 1—N, described by the Hamiltonian H, N N kaCI a ka + CiaCiaCia + g .~i&'t'%Z t a;leal, R a;i=1 N—1 N—1 + (ttc. C~+io + C.C.) + ) W'n '6 +i@ a,j9;i=1 + ) (Vi ci cl +c.c.) a;A;&L + ) (Vgaciv cg~ + c.c.). a;It:GR In Eq. (1), es and e; are energy levels in leads and the ith dot of the array, respectively, with o. being the spin index. U; is the intradot repulsion of the ith dot, while W; and ti are the interdot repulsion and the interdot coupling between the ith dot and its right neighbor [the (i+ 1)th dot]. The tunneling matrix element V&L (VP) connects dot 1 (dot N) to the left (right) lead. We assume two spin-degenerate levels on each dot. We treat the whole array as a 8ingle quantum system and calculate its many-body eigenstates by exact diagonalization.
The demand on computing power grows factorially with the number of state
