Power Dissipation Bounds and Models for Quantum-dot Cellular Automata CircuitsSaket Srivastava*, Sudeep Sarkar# and Sanjukta Bhanja*

* Electrical Engineering, # Computer Science and EngineeringUniversity of South Florida, Tampa, Florida.

Quantum-Dot Cellular Automata (QCA) Overview

P=+1 P=-1

In a QCA cell two electrons occupy diagonally opposite dots in the cell due to mutual repulsion of like charges. Hence a QCA cell can be in any one of the two possible states depending on the polarization of charges in the cell.

Purpose:• Non-Adiabatic Power Estimation Model for QCA Designs at Circuit Level.• Variation of Total Power Dissipated in a QCA Circuit with respect to Clock Energy.• Design Analysis in terms of Maximum Power and Total Power Dissipated.

NON ADIABATIC POWER MODEL: UPPER BOUND

RESULTS: Single Cell Switching Dynamics

•For this work, we have taken the value of high clock to be 0.7EK.

•Fig. 4 shows the power loss at each cell of an inverter when the polarization of the input cell is (a) kept constant 0→0 and (b) changed 0→1. Fig. 5 shows the power dissipated in the cells of a majority gate whenever its inputs are (a) not switched 000→000 and (b) switched 000→011.

•As can be predicted when there is no change in input polarization (0→0 transition and 000→000 ), the total power dissipated each cell will be only due to the raising and lowering of clock barriers.

•We can see clearly from the power dissipation graphs of inverter and majority gate, the total power dissipation in an inverter is much larger as compared to that of a majority gate. Also the value maximum power dissipated in single cell in an inverter is larger than that of a majority gate.

•For thermal stability of designs, it would be important to consider the maximum power dissipated at any cell in a design.

RESULTS: Upper Bound of Non-Adiabatic Power Dissipated

Fig. 5 Total power dissipated in each cell of a QCA majority gate for (a) 000→000 and (b) 000→011 input transition.

(a) (b)

Fig. 4 Total power dissipated in each cell of a QCA inverter for (a) 0→0 and (b) 0→1 input transition.

(a) (b)

where

and

Using the above equations we arrive at the upper bounds of power and energy dissipated during a non-adiabatic switching event in a QCA cell, given by:

P is the instantaneous power dissipated in a QCA cell.E is the expectation value of cell energy.Γ and λ are the coherence and Hamiltonian vectors respectively.Ek is the maximum kink energy.

γ is the tunneling energy. is the weighted sum of neighborhood polarizations.Ћ is Planck's constant.ρ is density matrix and σ is the Pauli's spin operator.pn,po are the new and old polarizations of the QCA cell.

We build the power model based on quantum mechanical formulation of the QCA cell operation [1]. Following Timler et al. [6], we use the Bloch formulation of the Schrodinger equation that expresses the evolution of quantum systems in operator spaces. The term P1 includes power from clock introduced into the cell Pclock and power gain from input to output Pin-Pout. We are concerned with P2 , Pdiss which represents the instantaneous dissipated power. Energy dissipated during switching can thus be calculated by integrating Pdiss over time.

Fig. 2 Polarization change (top plot) and power loss (bottom plot) in a single cell when its polarization changes from -1 to 1 (or 0 to 1 logic) during non-adiabatic clocking scheme.

Fig. 3 Polarization change (top plot) and power loss (bottom plot) in a single cell when its polarization changes from -1 to 1 (or 0 to 1 logic) during a quasi-adiabatic clocking scheme

As we see in Fig. 1, at higher clock energies the total dissipated energy also increases; this is due to the contribution of dissipative event associated with clock transitions (See Fig. 2).

Fig. 1. Dependence the total energy dissipated in a cell with clock energy for different clock transitions (a) 00 (b) 01 (c) 10 and (d) 11.

As we can see from the graphs in Fig. 2 and Fig. 3, the steady state polarization

λss of the cell follows the driver

polarization. We can also see that the total power dissipated by the cell occurs not only when its polarization changes, but a significant amount of power loss also occurs when the clock energy barriers are raised and lowered.

Each cell in a QCA circuit sees three types of events: (i) clock going from low to high so as to “depolarize” a cell, (ii) input or cells in previous clock zone switching states, and (iii) clock changing from high to low, latching and holding the cell state to the new state. Each of these events are associated with power loss. In our analysis we will consider the non-adiabatic case, which will provide us with an upper bound on the power dissipation.