A study on energy optimized 4 dot 2 electron two dimensional quantum dot cellular automata logical reversible flip-flops

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Abstract

In the present scope, new design methodologies for reversible flip flops are proposed and the results are analyzed by the QCADesigner tool. To the best of our knowledge such methodologies are reported for the first time in the literature. In this paper, we provide few formalisms also. The first one is for the system energy derived using Hamiltonian paradigm and provides internal energy of cell electrons. The second formalism provides the minimum energy requirement for execution of a QCA architecture. This procedure reduces wastage of clock energy. Two very interesting parameters are identified playing crucial role in this context: (i) The electron quantum number n which indicates quantum energy level and (ii) intermediate quantum number for an electron lying between 1 and (n−1). It is established that the incident energy frequency is directly proportional to the number of cells and quadratic function of electron quantum number and intermediate quantum number. The dissipated energy frequency is also directly proportional to the product of number of cells and quadratic function of electron quantum number. This paper, reports some remarkable results. The relaxation time is observed being inversely proportional to the product of number of cells in the architecture and quadratic function of quantum number as well as intermediate quantum number. Apart from these, differential frequency is found directly proportional to the number of cells in the architecture and quadratic function of intermediate quantum number. Few major observations are also indicated: (i) There is always a probability of reflection even if the system energy exceeds barrier energy. (ii) On the contrary, there is always a probability of transmission even though system energy is dominated by the barrier energy.

Introduction

CMOS technology has consistently ruled the VLSI technological field, because it works at high speed and consumes low power. The International Technology Road map for Semiconductor (ITRS) [1] had a prediction about some of the limitations of CMOS technology while to be integrated at nano level. It has also been predicted that, because of these limitations this technology is about to saturate within next few years. Naturally, there is an all pervading effort to achieve an efficient alternative to CMOS. Quantum Dot Cellular Automata (QCA) has all the technical ingredients to replace CMOS technology in days to come. QCA deals with binary information. The information is coded according to the antipodal arrangement of electrons within square cells [2], [3], [4]. QCA has the potential that any digital model can be converted into its QCA counterpart. There is no movement of electrons between the cells and is confined within cells instead. As a result, no electron flow and hence no power dissipation due to electron flow takes place. Accordingly, this paradigm has the advantage of very low power generation and consumption. In this paper, logical reversibility of various flip flops (SR, JK, D and T) is thoroughly studied and analyzed. To the best of our knowledge, there is hardly any reporting on reversible 2D 4 Dot 2 electron QCA circuit for flip flops. Accordingly, there is practically no scope of comparison with existing works in this respect. One study [5] focuses on the dynamism of confined electrons within cell channels. This paper presents the general derivation of system energy electrons within cells. In fact, our main interest here is to determine the exact clock energy required to function a reversible flip flop architecture unit consisting of N number of cells. This results in reduction of wastage of clock energy. To the best of our knowledge this effort happens to be unique in this respect. There are few existing works on the analysis of energy and power of QCA architecture. Blair et al. [5] presented heuristic model based on cell–cell interactions. Timler et al. [6] offered theoretical approach for the analysis of QCA power and energy based on density matrix formalism. It has discussed the approach to analyze the energy flow within QCA architecture and also discussed about the energy relaxation time. However, it is devoid of any derivation and analysis. In this paper, we start with developing a formalism regarding the system energy. It is the internal energy of electrons within channels. If the system energy is high enough to surpass the channel barrier, then the electron should have a flow as per the classical formulation. But quantum mechanically, even if the system energy exceeds the barrier energy, there is a probability of reflection. On the other hand, if system energy is less than the barrier energy, there is a probability of transmission. Basically, this paper proposes a methodology of regulating the clock signal to achieve minimum wastage of energy. The rest of this paper is organized as follows, Section 2 offers preliminary description of QCA. Section 3 presents discussions on reversible computing along with analysis of various reversible flip flops as case study. Section 4 presents formalisms for energy requirement for flow of electron through a channel within a cell. Further ratio of output to input power and its analysis are discussed in Section 4. These derivations are for the minimum energy required for running an architecture comprising N cells. Conclusion is drawn in Section 5.

Section snippets

Preliminaries

In Quantum-Dot Cellular Automata paradigm, the logic states are represented by position of individual electron. The beauty of the paradigm is the absence of any current flow through the architecture. Hence, power consumption is very less as compared to existing VLSI counterparts [7]. QCA architecture encompasses the use of the primitives such as QCA cell, QCA wire, and clocking in their realization. This section considers the description of these primitives.

Reversible computing

In recent times, reversible logic paradigm has received great attention due to its ability to reduce the dissipation of power to a great extent. This is particularly useful in ultra-low power logic circuit design. Energy dissipation results in irreversible computation due to information loss. The amount of energy dissipation for every bit operation in case of irreversible logic is at least kT ln 2 J, where k=1.3806505×1023J/Kis Boltzmann׳s constant and T is the operating temperature [9]. In

Energy analysis of reversible QCA architecture

We assume that a reversible QCA flip flop circuit receives signal power say, Pcv. This input signal power will reach the output without any loss. Input signals flow from environment to the architecture with power Pin, power flows from clock to architecture Pclock and from cells to environment Penv. Let Ei and Pi respectively be the instantaneous signal energy and average power for input signals xi(t), getting into the architecture from outside. Also let the power and energy are calculated over

Conclusion

Logical Reversible design of flip flop circuits has been proposed and justified by means of QCADesigner tool. Their block diagrams, function tables are presented. All flip flops input/output matrices support logical reversibility here. Thermodynamic aspects have been kept out of the present purview. To the best of our knowledge, there is hardly any reporting on reversible 2D 4 Dot 2 electron QCA circuit for flip flops. Accordingly there is practically no scope of comparison with existing works

Acknowledgment

We gratefully acknowledge the stimulating conversations with the members of Bengal Institute of technology and Management, specially Prof S. P. Kuila and Prof D. P. Sinha. We are also thankful to both the Department of Computer and System Sciences, Visva-Bharati University, Santiniketan and the Department of Computer Science and Engineering, Bengal Institute of Technology and Management, Santiniketan.

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