Resonant Tunning Through Quantum Dot Array

CHAPTER TWO
RESONANT TUNNELING THROUGH QUANTUM DOT ARRAY PROCESS

E is incident on a potential barrier of height V0 . Classically the electron is reflected when E< V0, but quantum mechanically there is a certain probability that the electron is transmitted through the barrier.
Tunneling is a purely quantum mechanical phenomena which enables electrons to penetrate potential barriers even though it is classically forbidden. The scheme is illustrated in Figure above. Classically the electron would be reflected if E¿VO but due to tunneling there is a probability that the electron penetrates the barrier. On the other hand, classically, if the electron has an energy E¿VO it is certain to be transmitted through the barrier, but in quantum mechanics there is a probability of reflection even when the energy exceeds the barrier height. Tunneling through a potential barrier is characterized by a transmission coefficient T so that 0≤T≥1. The transmitted wave function T is thus given by T I where I is the wave function of the incident
particle. In a single barrier structure like the one described here the transmission coefficient is a monotonically increasing function of E when E¿VO (T(E1) ¿ T(E2) 8 E1 ¿E2\VO¿ E1).
Incident electron of energy E Transmitted electron To further study the tunneling through quantum dot arrays, two dots will be consider and resolved by Hamiltonian method in the next chapter

Figure 1.2: tunneling through a double barrier. If TI for both barriers the region between the two barriers will act as a quantum well with quantized energy levels. This gives rise to resonant tunneling.
A double barrier structure like the one shown in Figure 1.2 gives rise to a QM phenomena called resonant tunneling. If the transmission coefficients of the left and right barriers, TL and TR respectively, are both much smaller than unity a quantum welln arises in between the barriers. This means tha =t the energy levels in the well will be quantized. Strictly speaking this is not entirely true because TR and TL are in fact, of course, not equal to zero. This means that the energy levels are not clearly defined, there is some broadening of the levls. These energy levels will in the rest of the text be referred to as quasi-quantized energy levels and the bound states in the well will be referred to as quaasi-bound states. When an electron with an energy which is not coincident with one of the quasi-quantized levels in the well is incident on the barrier/well complex the global transmission coefficient TG is much smaller than unity. If however, the electron energy coincides with one of the energy levels in the well, resonance occurs and the electron can be transmitted with a transmission coefficient on the order of unity. This is the type of structure Which is utilized in resonant tunneling diodes.
Transport through a single quantum dot weakly coupled to two contacts has been the subject of much experimental and theoretical work, and a fairly clear picture has emerged. Relatively little work has been done on arrays of quantum dots, though its now seems feasible to fabricate such sytructures. 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 result for the conductance of coherent arrays as a function of the Fermi energy, G (EF).
A single quantum dot behaves as an artificial atom in its charge and energy quantization and is often describe by the Anderson Hamiltonian, 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 electron hop between nearest neighbor dots. It seems reasonable then to model and array of quantum dots using the Hubbar 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 contact) by exact diagonalization and then to incorporate the effect of the contact through a rate equation as done by Beenakker for single dots. This treatment of the contact 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( EF) form two distinct symmetric groups separated by U. thus even such short arrays exhibit properties reminiscent of the infinite Hubbard chain.
Interesting, we find that the inclusion of inclusion of inelastic process within the array does not significantly affect the results. It ight thus be possible tostudy various aspects of the Hubbard model using artificial quantum-dot arrays.