Detection of Quantum Cellular Automaton Action in Silicon-on-insulator Cells

We present a proposal for an experiment to demonstrate QCA (Quantum Cellular Automaton) functionality for a cell fabricated with silicon-on-insulator technology. The fundamental feature of a working QCA cell consists in the anticorrelated transition of electrons in the two pairs of dots forming the cell: we show how such a phenomenon can be detected from the appearance of a "locking" effect between the Coulomb Blockade current peaks relative to each pair. The proposed approach allows the detection ofQCA action without the need for additional noninvasive charge detectors probing each dot. We have performed detailed numerical simulations on the basis of interdot capacitance values obtained from experimental data and determined the range of parameters within which the effect should be detectable.


INTRODUCTION
The Quantum Cellular Automaton concept, first proposed by Lent et al. [1], is an interesting approach to nanoelectronics, based on two-dimen- sional arrays of bistable cells, which allow the implementation of arbitrary combinatorial logic circuits.The basic cell is made up of four quantum dots and contains two electrons: if confinement in each dot is strong enough, the electrons will repel each other, and will tend to align along one of the diagonals of the square cell.In the absence of external electric fields or of other nearby cells, alignment along either diagonal is equally probable.If we place, next to the cell we are considering, another cell with a well defined polarization (indicated as "driver cell" in the fol- lowing), the electrons in the "driven cell" will tend to align parallel to those in the driver cell.If a logic value is associated with each polarization configuration, it is apparent that such a value can be propagated along a line of cells (binary wire) in a domino-like fashion and it has been shown that combinatorial logic functions can be easily ob- tained [2] by properly assembling two-dimensional arrays of cells.This approach would have the advantage of very limited power dissipation, since charges move only within a cell and there is no net charge flow across the array, and significant per- spectives for extreme miniaturization, down to the molecular level.However, many complex technical problems need to be solved before this technology can be applied to practical purposes; in particular viable solutions must be found to the issues of fabrication tolerances [3] and of detecting the occupancy of each dot without perturbing the cell state.
The first experimental demonstration of a QCA cell was obtained [4,5] with metallic tunnel junc- tions with A1/A1Ox/A1 tunnel barriers, with a setup in which electrons are supplied to the cell via external leads and polarization was measured by means of charge amplifiers based on single-elec- trons transistors capacitively coupled to the metal islands.
A different approach is represented by the fabrication of a 4-dot cell with GaAs/A1GaAs tech- nology, defining the dots by means of properly shaped metal gates, deposited on top of a hetero- structure.In this case it is possible to fabricate minimally invasive charge detectors by means of quantum point contacts, whose resistance is influ- enced by the charge stored in the quantum dots.
The QCA cell implementation on which we are currently focusing is based on silicon dots [6] de- fined by electron beam lithography and successive size reduction by means of controlled oxidation on silicon-on-insulator material.This approach has the advantage of allowing the fabrication of extremely small dots (in the 10 nm range) and of improving the strength of the electrostatic inter- action in comparison to materials such as GaAs, due to the reduced dielectric permittivity of silicon oxide.It would be possible to add also charge detectors, in the form, for example, of single- electron transistors, but this would translate into added complexity for a structure that already requires extremely challenging tuning procedures to obtain simultaneous operation of the four dots.
We propose a measurement procedure that can provide a clear demonstration of correlated electron switching between the two pairs of quantum dots forming a cell from the observation of the relationship between the position of the Coulomb Blockade current peaks through the dot pairs.A Monte Carlo code has been developed for the vali- dation of the proposed procedure and the deter- mination of the appropriate parameter values.

PRINCIPLE OF OPERATION
The proposed experiment will be described with reference to the layout represented in Figure 1, corresponding to the structure fabricated by Single et al. [6] in silicon-on-insulator material.There are four lithographically defined islands, with four adjustment gates.The barriers separating the dots belonging to the same pair and those separating the dots from the leads are obtained as geometrical constrictions.Such a device can be represented, for the sake of investigating its electrical behavior, with the equivalent circuit of Figure 2, where the TE's and Tz's are tunneling capacitors.In this cell, tunneling can occur only between dots belonging to the same pair, contrary to "classical" QCA cells, in which all dots are connected via tunneling barriers.This does not represent a significant FIGURE Sketch of the Si-SiO2 QCA cell.Oxidized silicon is represented in grey.Tunnel capacitances correspond to the six constrictions.difference from the operational point of view, since charge configurations are the very same in both cases.In Figure 2 we did not include the cap- acitances between the adjustment leads and the diagonally opposite dots, for the sake of simplicity.We have included them in our calculations and noticed no relevant contribution.The same can be said for the capacitances between the dots and the backgate that is usually present under the structure.
In order to understand the proposed procedure, it is convenient to focus first on the operation of a single pair of dots, considering all voltage sources connected to the other pair of dots as deactivated and the corresponding electrodes as grounded.In the presence of a small voltage applied between the left and right lead, a current will flow through the three tunneling junctions in a dot pair only if the chemical potentials of the three dots are aligned with those of the leads, which corresponds to having both dots on the edge between two stable occupancies.
The chemical potentials in the dots can be varied by means of the voltage applied to the adjustment gates: if we apply a positive voltage ramp to the gate controlling the left dot and a negative ramp to that connected to the right dot, and adjust the relative position of the ramps in such a way as to synchronize the transition from M+ to M electrons in the right dot with that from N-to N in the left dot, the Coulomb Blockade will be lifted and a current peak will be observed through the dots.The same will happen in correspondence with any other synchronous change of the occupancy in the two dots.We assume that the rate of variation for the applied voltages is very slow compared to the tunneling rates and to the RC time constants, so that a quasi-static treatment is in order.
The experiment can then be repeated for the upper pair of dots (grounding the voltage source connected to the lower pair), applying a ramp with positive slope to the adjustment gate for the right dot and a ramp with negative slope to the other gate, in order to obtain transitions in dot occu- pancy that are opposite to those achieved in the lower pair of dots.A shift is also included with respect to the ramps for the lower pair, in order to displace the peaks of I2 with respect to those of 11, as shown in Figure 3.
If both the upper and lower section are operat- ed simultaneously, the electrostatic interaction be- tween the dots will determine a synchronization between opposite occupancy variation phenomena, thereby yielding a "locking" effect between the peaks in the two currents (see Fig. 4), despite the presence of the above mentioned shift.(solid line) and through the bottom dots (dashed line) when the other semicell is switched off. (solid line) and through the bottom dots (dashed line) when both semicells are biased.The synchronization of current peaks implies QCA operation.
This will demonstrate correlated switching and, therefore, cell operation.

SIMULATION CODE
Our initial checks on the feasibility of the proposed experiment were performed with the public domain version of the well-known SIMON [7] single elec- tron circuit simulator, but we then developed our specifically devised Monte Carlo simulator in order to obtain a better numerical precision and a more direct control of internal parameters.The computational strategy is quite straightforward and analogous to what has appeared in the literature for single electron circuits.The simulation is subdivided into a number of steps, each corresponding to a given value of the gate voltages.Within each step a stationary Monte Carlo calculation is performed, letting the elec- trons cross the tunneling junctions according to the following rules: the variation of free energy AE associated with the crossing of each junction is computed, then the tunneling rates are evaluated according to the "orthodox" Coulomb blockade theory: AE e2RT --e-/XE/(k.) where Rv is the tunneling resistance, e is the electron charge, kB the Boltzmann constant and T the temperature.A random number is generated, uniformly distributed in an interval which is partitioned into sections with a width proportional to the rate for each possible transition: the tran- sition corresponding to the section containing the generated random number is then chosen.Dot charges are updated as a result of the transition and a new iteration is performed, after generating the time elapsed during the current iteration as an exponentially distributed random number with average equal to the inverse of the total tunneling rate (obtained as the sum of the rates for all possible transitions).The procedure is repeated a number of times sufficient to get reliable estimators of the quantities of interest: the charge in each dot, obtained by averaging the values at each iteration, and the current flowing through each junction, determined by taking the ratio of the total charge that has traversed it to the sum of the times cor- responding to each iteration.
The whole sequence is repeated for all the gate voltage values into which the simulation has been subdivided and results for the dot charges and junction currents are collected.Our code has been tested on the structures of Ref. [6], and has yielded results that are in extremely good agreement with the experimental data.

NUMERICAL RESULTS AND FEASIBILITY ASSESSMENT
The values of the geometrical capacitances be- tween dots have been evaluated by means of the FASTCAP [8] program, modeling the various electrodes as parallelepipeds approximating the actual geometries.We have assumed a dot size of 60 60 60nm, and a separation of 16nm between dots and of 100nm between each dot and the corresponding adjustment gate.Separation between dots belonging to different pairs must be very small, otherwise coupling is too weak to allow proper operation.As a rule of thumb, we have found out that values of the coupling capacitances between the upper and lower dots must be of the same order of magnitude as that of the tunneling junctions.With this procedure the following results have been obtained: CD= 0.94 aF, Cv-1.65 aF, CB--1.36 aF, Cc-1.0 aF, Ca 0.94 aF.The capacitances of the tunnel junctions cannot be computed from simple geome- trical considerations, since, as it turns out from the experiments, localization in the dots and crea- tion of the barriers between them is substantially the consequence of the potential fluctuations due to the random distribution of dopants.We have therefore resorted to recent measurement results [6] yielding the values: TE-9aF and TD--10aF.
As a consequence of their relatively large value, operation is possible only at very low tempera- tures, in order to avoid excessive thermal broad- ening that would prevent observation of the "locking" effect.We have performed calculations for a temperature of 0.2K, with external bias voltages V1-V4 V5-V8-1.7 mV.We start with V5, V6, V7, V8 turned off, i.e., with the upper part of the cell disabled, and apply the previously described voltage ramps to the adjust- ment gates.The resulting current I1 flowing through the lower dots is reported in Figure 3 with a dashed line.The time evolution of the adjustment voltages is reported in the inset.The whole procedure is then repeated for the upper section, disabling the lower section.The voltage sources connected to V6 and V7 deliver voltage ramps corresponding to those previously supplied by V3 and V2, respectively, but with a 7 mV shift, which determines a displacement of the curve for I2 (reported with a solid line in Fig. 3) with respect to that for I.
The final phase consists in operating both the upper and lower section of the cell at the same time: the peaks in 11 and I2 are now locked, as shown in Figure 4, which represents evidence of the correlated switching between the two halves of the cell and therefore of QCA action.The locking effect is very sensitive to the choice ofparameters, in particular to the already mentioned coupling cap- acitances between the upper and lower dots, and to the temperature.If the coupling capacitances are reduced by a factor 10, no locking occurs, due to the insufficient electrostatic interaction between the upper and lower halves.Operation with the con- sidered layout is disrupted also if temperature is increased above about K, due to thermal broad- ening of the current peaks, and the ensuing im- possibility to distinguish the actual separation between the peaks for 12 and I1, unless this is so wide that locking cannot take place even with relatively large coupling capacitances.

CONCLUSIONS
We have proposed an experiment for the demon- stration of QCA action with silicon-on-insulator technology, based on the detection of the electro- static interaction between the two halves of a cell made up of four silicon quantum dots embedded in silicon oxide.A numerical Monte Carlo simulator has been devised, which has allowed the determination of the range of circuit parameters suitable for the experimental observation of correlated switching between the two cell halves.The major difficulties, from the point of view of the actual experimental implementation, consist in obtaining large enough coupling capacitances between the upper and lower dots and in per- forming measurements at temperatures below K, with the associated decrease in current levels, due to the reduced number of carriers.Overall, however, the experiment appears to be feasible, and its implementation is currently being pursued.

FIGURE 2
FIGURE 2 Equivalent capacitance network of the QCA cell.

FIGURE 3
FIGURE 3 Time dependent current through the upper dots

FIGURE 4
FIGURE 4 Time dependent current through the upper dots