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Quantum-dot Cellular Automata (QCA) represents a new technology at the nanotechnology level. It was first introduced by Lent et al. which offered a new potential paradigm shift in computing. Unlike conventional technologies which use voltage or current to represent binary values, QCA uses the position of the electrons in a cell to represent binary values. QCA technology has inherent features like high device density (10Â¹Â² devices/cmÂ²), higher operating speed (high clock frequency, usually in the range of Tera Hz) and lower power consumption (100W/cmÂ²).
2.2 Quantum Cells
The standard QCA cells have four quantum dots and two electrons. There are other various kinds of QCA cells proposed, however in the research we have used the standard QCA cells. Figure 2.1 illustrates the standard QCA cell with four dots and two electrons. The two electrons within each cell are free to tunnel within the cell but they are not allowed to leave the cell. These two electrons within each cell repel each other to occupy diagonally opposite corners of the cell. This therefore results in only two stable states to represent the binary logic values. Logic '0' is represented when the two electrons occupy the upper-left dot and lower-right dot. Logic '1' is represented when
the two electrons occupy the upper-right dot and lower-left dot. The transition of the cell polarization is shown in Figure 2.2.
QCA Cell Polarizations and Representations: (a) Binary 1. (b) Binary 0.
QCA Cell Polarizations transition.
The information transmission and processing in QCA is purely due to coulombic mechanism unlike other conventional technologies where information is transferred by electric current [11 in Midwest v1.4]. QCA is an array of nano-electronic device cells. Polarization can be defined as a quantity which measures the extent to which the charge distribution is aligned along one of the two diagonals. If Pi denotes the electron charge at dot i, the polarization is given in eqn. 1:
Quantum dots are small semi-conductor or metal islands with a diameter that is small enough to make their charging energy greater than kBT (where kB is Boltzmann's constant and T is the operating temperature) [3 sowmya thesis]. Exactly two mobile electrons are loaded in the cell and can move to different quantum dots in the QCA cell by means of electron tunneling. Electron tunneling is assumed to be controllable by potential barriers (that would exist underneath the cell) that can be raised and lowered between adjacent QCA cells by means of capacitive plates. Apart from the two states discussed above, there is another 'unpolarized' state in which the cell has little or no polarization. In such a state, the inter-dot potential barriers are lowered which reduces the confinement of electrons in the quantum dots [11 sowmya thesis].
Thus, QCA cells perform computation by coulombic interactions with neighboring cells to influence each other's polarization. The following section reviews some simple, yet essential, QCA logical devices: QCA "wires", a majority gate and inverters.
2.3 Clocking in QCA
Timing is controlled through a reference signal (i.e., a clock) and is mostly required for sequential circuits. Timing in QCA is accomplished by clocking in four distinct and periodic phases [Lombardi 28] and is needed for both combinational and sequential circuits. Clocking provides not only control of information flow but also true power gain in QCA [Lombardi 29]. Signal energy lost to the environment is restored by the clock. Clocking is an important parameter in QCA design. Clocking is usually used in sequential circuits for data flow. But in QCA, clocking is used for both combinational and sequential circuits. In QCA, clocking not only provides the control of information flow but also provides true power gain in QCA.
Abrupt switching and adiabatic switching are the two types of switching methods in the operation of QCA.
Abrupt Switching: The inputs to the QCA circuit change suddenly and the circuit can be in some excited state. The QCA circuit is relaxed to ground state by dissipating energy. This inelastic relaxation is uncontrolled and the QCA circuit may enter a metastable state that is determined by a ground state.
Adiabatic Switching: The system is always kept in its instantaneous ground state. A clock signal is introduced to ensure adiabatic switching. This is the preferred method of switching.
For QCA, the clock signals are generated through an electric field, which is applied to the cells to either raise or lower the tunneling barrier between dots within a QCA cell. This electric field can be supplied by CMOS wires, or CNTs  buried under the QCA circuitry. Depending upon the strength of the electric field applied to a QCA cell, the QCA cell has three distinct polarizations. One of the methods for clocking was proposed by K. Hennessy et. al. where conducting wires were used below the plane of the QCA molecules to generate a suitable electric field [clocking in thesis folder]. The schematic of this idea is shown in Figure 2.3. The QCA cells are spread across the x y plane and the y plane consists of a series of conducting wires below the plane of the QCA cells. The conducting wires are excited to produce electric fields by applying suitable voltage. A conductor placed above the QCA cells is grounded so that it draws the electric field in the z direction. The state of the QCA cell is affected only by the
component of the electric field. The component does not affect the state of the QCA cell and the Å· is zero due to symmetry. Time varying voltages are applied to the conducting wires in such a way that two adjacent wires have a phase shift of Ð¿/2 between them. This configuration ensures that every fourth wire will have the same applied signal. When the barrier is low, the cells are in a non-polarized state; when the barrier is high, the cells are not allowed to change state. Adiabatic switching is achieved by lowering the barrier, removing the previous input, applying the current input and then raising the barrier . If transitions are gradual, the QCA system will remain close to the ground state.
Modified Schematic Model to Represent Clocked QCA Array [clocking in thesis folder] .
In a clocked QCA circuit, information is transferred and processed in a pipelined fashion   and allows multi-bit information transfer for QCA through signal latching. All cells within the same zone are allowed to switch simultaneously, while cells in different zones are isolated. This pipelined information transfer is illustrated in the Figure 2.3.
Information Transfer in a QCA Binary Wire.
QCA clock has four phases (switch, hold, release, and relax) and each phase can be thought of as a potential that modules the inter-dot barriers of all the cells in that phase. For effective information flow, array of QCA cells can be divided into sub-arrays of different clock phases. Figure 2.4 illustrates the polarizations in the QCA cell during the four phases of the clock. During the 'switch' phase of the clock, the unpolarized QCA cells polarize in accordance with the driver cell, the cells maintain their polarization for the duration of the 'hold' phase. The 'release' phase of the clock releases the inter-dot barriers and the cells lose their polarization, and continue to stay in an unpolarized state in the 'relax' phase.
Phases of a QCA Clock.
2.4 QCA Logical Devices
A driver of a QCA cell could be an input device such as a nanotube, a very thin wire or a tip of a scanning tunneling microscope (STM). In semiconductor QCA, a standard technique called "plunger electrode" has been used to alter the electron occupancy of the input cell (Lombardi text   ). Reading the output state of a QCA cell is difficult, because the required measurement process must not change the charge of the output cell. Electrometers made from ballistic point-contacts (Lombardi text  ), the STM method (Lombardi text ), and SET electrometer have been used to read the output. Refer to Lombardi text. There are two configurations for fabricating the QCA cells. As the information in QCA is transferred due to coulombic interactions between the two corresponding QCA cells, the state of one cell influences the state of the other corresponding cell. The various kinds of QCA logic devices are:
2.4.1 Binary Wires
The QCA cells tend to align to the polarization of its neighbors. Therefore a linear arrangement of cells can be used as a wire to transmit information. An example of QCA wire is shown in the Figure 2.5(a) and its simulation is shown in Figure 2.5(b). Here as we can see, the binary wire is divided into various clock zones. This is done to ensure that the signal strength carried by the wire is not degraded, as the signal strength tends to deteriorate with a long chain of QCA cells in the same clock zone.
Binary Wire (a) QCA layout of Binary Wire (b) QCA Simulation.
The other kind of QCA wire is termed as the "Inverter chain". In Inverter chain, the standard QCA cells are rotated by 45 in vertical or horizontal orientation. The polarizations in such QCA cells tend to align opposite to the corresponding QCA cell. The Figure 2.6(a) shows the Inverter chain and Figure 2.6(b) shows the simulation.
Inverter Chain (a) QCA layout of Inverter Chain (b) QCA Simulation.
2.4.2 Majority Voter
Majority Voter (MV) is one of the logical gates in QCA. The logic function of MV is MV (A, B, C) = A.B +B.C + C.A. The logic of the MV is implemented by the five QCA cells shown in the Figure 2.7. The center QCA cell is called the decision making cell which tends to polarize to the majority of the inputs. The QCA implementation and simulation of the MV is as shown below.
QCA layout of Majority Voter.
QCA Simulation of Majority Voter.
By fixing the value of one of the three inputs to logic '0' or logic '1',the MV can be programmed as an AND gate or OR gate. By fixing one of the inputs of the MV to logic '0', MV functions as an AND gate. Likewise, it functions as an OR gate when one of the inputs to MV is fixed at logic '1'. The logical functions of AND and OR gate are shown in eqn. 2 and eqn. 3.
MV (A, B, C) = AB when C = 0 (2)
MV (A, B, C) = A + B when C = 1 (3)
Figure 2.9 and Figure 2.11 show the QCA implementation of AND and OR gate respectively. Figure 2.10 and Figure 2.12 show the simulation of AND and OR gate respectively.
QCA Layout of AND Gate.
QCA Simulation of AND Gate
. QCA Layout of OR Gate.
. QCA Simulation of OR Gate.
Analogous to NOT gate, the basic functionality of the Inverter is to output an inverted value of the input. There are many ways of implementing an inverter in QCA. The standard cells in a diagonal orientation tend to align to the opposite polarization due to the electron repulsion as shown in Figure 2.13. The respective simulation is shown in Figure 2.14.
. QCA Layout of the Inverter using Standard QCA Cells.
. QCA Simulation of the Inverter using Standard QCA Cells.
The rotated cells when configured as shown in the Figure 2.15 can also be used as an inverter when there are 2n QCA cells. The simulation for the rotated cells is shown in Figure 2.16.
. QCA Layout of the Inverter using Rotated QCA Cells.
. QCA Simulation of the Inverter using Rotated QCA Cells.
The most commonly used inverter is shown in Figure 2.17. The simulation for the inverter is shown in Figure 2.18. The signal comes in from the left on a binary wire and splits into two parallel wires as illustrated in the figure. The two corner cells on the parallel wires polarize the cell diagonal (at the right end) to them in the opposite direction. This causes the signal to be inverted. This anti-aligning behavior of standard cells in diagonal orientation can be used to "tap" the signal passing through an inverter chain [Sowmya thesis 8]. Placing a standard cell and aligning it halfway between an even and odd numbered rotated cell will obtain an inverted signal and placing it halfway between an odd and even numbered cell will give a buffered value.
. QCA Layout of the Inverter.
. QCA Simulation of the Inverter.
This tapping of signal is useful when implementing large circuits where crossover of wires in the same plane is essential.
2.5 Cross-Wiring in QCA
Couloumbic interaction between two cells influences the data flow in the QCA circuits. The kink energy Eki,j represents the energy cost of cells i and j having opposite polarization. In other words, the kink energy can be thought of as the essential energy required for a successful switching. The parameters that govern this kink energy in QCA are the distance between the corresponding cells and the angle Î¸ as shown in the Figure 2.19.
The relationship between kink energy, Î¸ and D is:
. Relation between Distance D and Angle Î¸.
The cells interact through a quadrupole-quadrupole interaction which decays inversely as a power of five of the distance between cells. Therefore the kink energy will decay rapidly with distance and thus indicates that every cell has an effective neighborhood and its capacity to polarize those in its neighborhood.
There are two types of crossovers in QCA.
In QCA, two binary wires carrying information can cross over each other without interacting. This unique feature in QCA is used in coplanar crossover. The main principle is that the standard QCA cells do not get influenced by cells rotated by 45áµ’ in horizontal or vertical axis. This is because the energy between the standard QCA cells and the rotated QCA cells cancels out to have kink energy of 0 J. As discussed in Section 2.4, the kink energy of a cell influences the polarization of the corresponding cell, Ekink = 0 J means that there is no influence of one QCA cell on the other QCA cell. Figure 2.20 shows two binary wires having different polarization and successful crossover. Figure 2.21 shows its simulation.
. QCA Layout of Coplanar Crossover.
. QCA Simulation of Coplanar Crossover.
The disadvantages of coplanar crossover are:
The fabrication process becomes more complex and the costs increases as there are different orientation of QCA cells.
The cells have to be properly aligned. Any misalignment of the cells results in a non-zero value of kink energy resulting in the interaction of the cells in the region of crossover.
Multi-Layer crossover [14 in sowmya thesis], are be considered a reliable method of crossover.
QCA circuits can cross signals effectively by passing them in more than one layer, using a vertical interconnect. Multi-layer crossover is illustrated in Figure 2.22 [15 sowmya thesis]. The figure shows a wire polarized to logic '0', call it 'A'. Another wire is polarized to logic '1', call it 'B'. If wire B has to crossover wire A, it can be sent to another layer and transmitted horizontally. Vertical interconnect cells are stacked on top of each other and the wire B is send to the second layer. In the second layer, B can cross over any number of wires lying in the first layer. Wire B can be brought back to the first layer by using the same vertical interconnect cells one below each other as shown in Figure 2.23.
. QCA Simulation of Multilayer Crossover.
. Crosssectional View of Various Layers in MultiLayer Crossover.
The vertical separation between the cells is tuned to match the interaction energy Ekink of the horizontal cells to reduce crosstalk. Several intermediate layers called 'vias' can be introduced to reduce interaction among the cells between the first and second layers.
The advantages of Multi-Layer crossover over coplanar crossover are:
It eliminates the need of rotated cells in any design, thereby reducing the overhead of fabricating two different orientations of QCA cells.
Also, the extra layers created during the crossover can be used to implement circuits, providing better integration.
The designs in this thesis use the multilayer crossover technique when crossover is required.
There are 3 types QCA devices proposed for physically implementing QCA from past research. They are:
QCA cells can also be built from metallic tunnel junctions and very small capacitors. In this implementation, the device consists of four aluminum islands (dots) connected with aluminum oxide tunnel junctions and capacitors. The major difference between semiconductor and the metal QCA is the use of capacitive coupled metal islands instead of the conventional coulombically coupled quantum dots and the presence of many conduction band electrons in the metal island unlike the quantum dot.
In case of metal QCA, a combination of two series dots can either be coupled to other islands through bias power supply and ground or can be left floating. In the former case, current may flow during the switching event of adding or removing an electron where as in the later case, the dots connected by tunnel junction may only exchange electrons keeping the total number of electrons to be a constant. The capacitance of the island is determined by the area of the tunnel junction. This determines the operating temperature of the device. The device area is approximately 60Ã-60 nmÂ² and is mounted on a surface at 10mK temperature.
The electrostatic energy of a configuration can be expressed in terms of the voltage and charges on gate electrodes and metal islands:
where C is the capacitance matrix for the islands and electrodes, v is a column vector if voltage on the gate electrodes, q and q' are the column vectors o the island charges and the lead charges, respectively. The first term in the above expression calculates the electrostatic energy stored in the capacitors and tunnel junctions. The second term calculates the work done to transfer the charge from the source to the leads.
One of the ways to fabricate this device is by using Electron Beam Lithography (EBL) and dual shadow evaporation on an oxidized silicon wafer. In Figure 2.24 the aluminum dots are located at D1 through D4, coupled by tunnel junctions. The two dots (E1 and E2) are SET electrometers for sensing the output. The operating temperature of these devices is 70mK.
. QCA Layout of Coplanar Crossover.
A majority voter designed using this implementation is shown in Figure 2.25. Differential signals A (between gates 1Â and 3), B (between gates 1Â and 2), and C (between gates 2Â and 4) constitute the inputs to the central cell. The negative (positive) bias on a gate, Ñ„- (Ñ„+), mimics the presence (absence) of an electron in the input dots, as shown by the shaded regions in Figure 2.26. The amplitudes of Ñ„+ and Ñ„- are carefully chosen to mimic the potentials due to the polarization of an input cell while they remain small enough not to change the number of excess electrons in the cell.
. QCA Layout of Coplanar Crossover.
. QCA Layout of Coplanar Crossover.
In Figure 2.25, as dots D1 and D2 are coupled to only one gate electrode each, voltages corresponding to inputs A and B on gate 1,Â and inputs B and C on gate 2,Â are added in order to mimic the effect of two input dots. For instance, the input configuration (ABCÂ =Â 111) is achieved by setting V1Â =Â 2 Ñ„-, V2Â =Â 2 Ñ„+, V3Â =Â Ñ„+, and V4Â =Â Ñ„-. With inputs A, B, and C traced as a function of time, the differential potential between dots D4 and D3, Ñ„D4 - Ñ„D3 , is measured using the electrometers E1 and E2. The transient characteristics are determined by the time constant of our electrometer circuitry.
Another Metal dot QCA implementation of a shift register is shown below in Figure 2.27.
QCA Layout of Coplanar Crossover
Metal QCA devices have certain advantages that motivate the use of such devices. Firstly, such devices are easy to fabricate and more reliable. Secondly, these devices are easier to analyze and model. On the other hand, these metal dot devices have several downsides that have to be considered. The metal dots were in the order of 1Âµm in dimension hence the devices had to be cooled to low temperatures (4K at most) for the observation of electron switching. Additionally, there have been fabrication defects for some of the tested metal dot devices where special additional circuitry had to be used to balance the defects. These defects lead to the experiments that revealed the potential of a molecular QCA system.
A simpler representation of QCA logic gates is by implementing them using molecular QCA which not only yields greater performance but can also operate at room temperature. This one of the major reasons to continue developing logic gates using molecular QCA. In molecular QCA, cells are structurally homogeneous down to the atomic level. The molecular QCA cells are implemented using a class of compounds called 'mixed - valence compounds' which exhibit a unique property of having multiple redox centers in different oxidation states. Each molecule functions as a QCA cell and the redox centers function as quantum dots where the information is encoded with charge configuration and the tunneling junction provided by bridging ligands as shown in Figure 2.28. The transfer of electronic charge from one molecule to its neighboring molecule is through quadrupole - quadrupole interactions. The coulombic energies should be in the range of 0.2 - 0.5 eV for the model to operate at room temperature.
. QCA Layout of Coplanar Crossover.
To find the response functions of a model, the quadrupole moment of a single molecule should be interpolated between the states '0' and '1'. The Scrodinger's equation is used to evaluate the value of the output molecule.
. QCA Layout of Coplanar Crossover.
The logic states can be defined using the allyl group shown in Figure 2.30. The state with a positive dipole moment is represented as logic '1' and the state with a negative dipole moment are represented as logic '0'. The alignment for logic '1' is such that an unpaired electron is on the bottom allyl group and a positive charge is on the top allyl group. On the other hand, the alignment for logic '0' is such that the bottom allyl group has a positive charge and the top allyl group has an electron.
Molecular QCA devices offer certain advantages over the other implementation models with respect to area, power dissipation and information transfer rates. Implementation of molecular QCA devices reduces the cell size to approximately 1nm x 1nm. Experiments conducted by Lent and Timler  have proved that, if clocking is considerably slowed down, the reversible processes can be implemented with ultra low power dissipation. For the information to be transferred without any loss of data, the molecular QCA devices should be able to switch between '0' and '1' fast enough to cope up with the clock. It has been reported that the transfer rates of the mixed valence compounds used in molecular QCA have switching times between 10s and 13s.
Molecular QCA presents unique challenges such as bonding of the array surface which requires complexes by stereoscopic and electro-chemical techniques, treating the existent strongly bound, chemically robust, and mixed valence complexes in chemistry.
Mobile chargers are created by chemical oxidation or reduction as shown below in Figure 2.29.
. QCA Layout of Coplanar Crossover.
Molecules are suitable for the implementation because they act as natural and uniformly small quantum dots with high density that are suitable for room temperature operations.
This concept was introduced by Cowburn and Welland who demonstrated the operations of magnetic QCA using an array of disk shaped particles, with a diameter of 110nm, that exhibit collaborative behavior. In magnetic QCA, magneto static interactions between nano particles ensure that the system is bi-stable. The moments of nano-particles are either parallel or anti parallel with the axis of the chain. The information is propagated via magnetic interactions as opposed to the electrostatic interactions in metal and molecular interactions.
In this system, the representation of the binary information as well as the information propagation (magnetization reversal) is primarily determined by the coupling-induced magnetic anisotropy in the chain. The quantum mechanical interactions in an MQCA networks are due to the interactions between the spins within a single dot which form a single strong classical spin.
The logic '1' is signaled when the dots magnetization vector points in the upward direction and the logic '0' is signaled when the vector points in the downward direction. Figure 2.30(a) and (b) show how two magnets can be assigned logic states '1' or '0', and can couple either in a ground state, or higher-energy meta stable state.
Most magnetic thin films, such as perm alloy or cobalt, display in-plane magnetization. That is, the preferred domain orientation is parallel to the plane of the film. Other films, such as carefully constructed CoPt multi-layers, can exhibit out-of-plane magnetization. Coupling of magnets in either of these configurations can, in principle, be used for QCA.
. QCA Layout of Coplanar Crossover.
The result of the magnetization pattern for a chain of nano-magnets depends on their alignment. The magnets might be aligned to enter two different states depending on their axis of alignment. A collinear alignment of the magnets along their long axis reinforces the magnetization in the same direction. This state is called the ferromagnetically ordered state. On the other hand, placing the magnets side-by-side in parallel will result in a line that favors anti-parallel alignment of the electric dipoles. This state is called the anti-ferromagnetically ordered state. In MQCA, these ordering phenomena drive the computation of the patterns. It is not hard to see how a chain of narrowly spaced nano-magnets as shown in Figure 2.32(c) could be used as a QCA 'wire'.
A NAND or NOR gate can be designed using the simplest arrangement of five nano-magnets (a central magnet surrounded by four other magnets). The top, left and bottom magnets can be used as inputs driven by additional driver magnets oriented in the x-direction. The fourth magnet on the right acts as the output. Eight different logic combinations can be achieved by varying the states of the driver nano-magnets as shown in Figure 2.33. Table 1 summarizes the states of the central magnet and the output magnet.
. (a - h) Alignment of Magnetic Dipoles to Demonstrate the Correct Functionality of Majority Voter.
Summary of The Alignment of the magnetic dipoles.
Logic State of Input Magnets
Logic State of Central Magnet
Logic state of Output magnet
The three input majority gate discussed above can be programmed as a two input NAND or NOR depending upon the state of any one of the three input magnets and the inversion at the output magnet. Thus any Boolean logic function can be regenerated using a network of arrangements.
With improvements in the current fabrication techniques, MQCA has the potential to replace CMOS technology in the coming years. The device packing of MQCA is intense as compared to CMOS technology. Going by Moore's law the CMOS technology faces serious physical limitations for further downsizing but on the other hand MQCA seems to successfully continue abiding by this law. According to , integration density of 5500 million cm-2 is comparable with 6.6 million cm-2 in today's CMOS technology.