Simulation of Random Cell Displacements in QCA
Simulation of Random Cell Displacements in QCA GABRIEL SCHULHOF University of Calgary and KONRAD WALUS University of British Columbia and GRAHAM. A. JULLIEN University of Calgary
In this work, we analyze the behaviour of quantum-dot cellular automata (QCA) building blocks in the presence of random cell displacements. The QCA cells are modeled using the coherence vector description and simulations are performed using the QCADesigner tool. We evaluate various fundamental circuit elements, including the wire, the inverter, the majority gate, and the two approaches to wire crossing: the coplanar crossover and the multi-layer crossover. We show that the different building blocks have different tolerances to such displacements. The coplanar crossover and inverter show the weakest tolerance, while the wire is the most robust. Our results also show that the tolerance to random displacements is not a function of cell size, they depend instead on but rather of the circuit layout. Categories and Subject Descriptors: B.7.3 [Fault Tolerance Analysis]: Statistical Approach General Terms: Reliability Additional Key Words and Phrases: quantum-dot cellular automata, QCA, fabrication variances, fault tolerance
1.
INTRODUCTION
Quantum-dot cellular automata is an interesting nano-scale paradigm capable of general purpose computation. Since it was first introduced by C. Lent et al. in 1993 at the University of Notre Dame [Lent 1993], many researchers have become increasingly interested in its simple concept and its many potential applications in future computer technology. Although the practical realization of QCA cells and circuits is still being developed, many architectural features of QCA circuits such as majority reduction of logic [Walus et al. 2004], and the full adder [Wang et al. 2003] have been investigated. A QCA computer-aided design (CAD) and simulation tool now exists in the form of QCADesigner [Walus and Schulhof 2001], developed at the University of Calgary ATIPS Laboratory. A review of QCA circuits and systems is available in [Walus and Jullien 2006]. The tolerance of QCA building blocks to manual displacements of cells has been previously studied[Tahoori et al. 2004; Huang et al. 2004]. However, such studies were not able to capture the statistical nature of manufacturing variations because the tools for such analysis were not developed. In this work, we present the results of using recently developed statistical simulation tools on the operation of QCA building blocks in the presence of random four-dot cell displacements. The rest of the paper is as follows: Section 2 describes the QCA model and ACM Journal Name, Vol. V, No. N, January 2007, Pages 1–0??.
2
·
Gabriel Schulhof et al.
dynamical equation used throughout the simulations. Section 3 describes the particular choices of simulation parameters and their relationship. Section 4 describes the majority gate and simulation results, Section 5 the inverter, Section 6 the wire, and Section 7 describes the coplanar crossover and provides an alternative layout, the multi-layer crossover, which has been found to be more displacement tolerant than the coplanar crossover. Finally, Section 8 provides some conclusions. 2.
MODEL
The standard QCA cell is shown in Fig. 1.
Fig. 1.
Standard 4-dot QCA cell. The two ground states of the cell are shown.
The Coulombic interaction between two cells can be described by the so-called kink energy, Ekink , associated with the energetic cost of two cells having opposite polarization [Lent and Tougaw 1997]. The electrostatic interaction between charges in two four-dot cells, i and j, is E i,j =
4 4 j 1 X X qni qm , j 4π0 r n=1 m=1 |rni − rm |
(1)
where r is the dielectric constant, qni is the charge in dot n of cell i, and rni is the position of dot n in cell i. The kink energy is the difference in energy between two cells which have opposite polarization and those same two cells having the same polarization; i.e, m,n m,n m,n Ekink = Eopposite polarization − Esame polarization .
(2)
Background charges of + 21 e are included in each dot, to ensure that the overall cell remains charge neutral. These positive charges are included in the calculation of the kink energy. In general, the interaction between cells can be described by a quadrupole-quadrupole interaction in which the kink energy decays as the fifth power of the distance between cells. QCA dynamics are modeled using the so-called coherence vector description[Timler and Lent 2003; 2002], using the Hartree-Fock approximation. The two-state Hamiltonian for a single cell is P i,j − 12 Ekink Pj −γi j∈Ωi ˆ (i) = P i,j H (3) , 1 −γi E P j kink 2 j∈Ωi
ACM Journal Name, Vol. V, No. N, January 2007.
Simulation of Random Cell Displacements in QCA
·
3
where the neighborhood, Ωi , of cell i, is defined by the “radius of effect” parameter in the simulation tool. The coefficients of the coherence vector, ~λ = [λx , λy , λz ]T , are determined from the expectation value of their associated Pauli spin matrix; i.e., λx = hˆ σx i. The polarization of a cell is defined by the expectation value of σ ˆz . ˆ is given by In the Heisenberg picture, the time dependence of an operator O ∂ ˆ h ˆ ˆi = O, H . (4) i~ O ∂t P i,j replacing O with each of the spin matrices and letting K = Ekink Pj results in j∈Ωi
~
σ ˆ 0 −K 0 σ ˆx d x ˆy . σ ˆy = K 0 2γ σ dt σ ˆz 0 −2γ 0 σ ˆz
Taking the expectation value of both sides we ˆ λ 0 −K d ˆx ~ λy = K 0 dt ˆ 0 −2γ λz
(5)
get ˆ λx 0 ˆy , 2γ λ ˆz 0 λ
which can be rewritten in the vector form as d ~ ~λ = ~Γ × ~λ, dt where −2γ 0 ~Γ = P i,j . Ekink Pj
(6)
(7)
(8)
j∈Ωi
Equation (7) is the dynamical equation of motion for the coherence vector. In order to introduce dissipative coupling to the environment, we use a relaxation time constant τ that incorporates all the available dissipative channels. In this relaxation-time approximation, the dynamical equation of motion is d 1 ~ ~ ~ ~λ = ~Γ × ~λ − λ − λss (9) dt τ The steady-state coherence vector is ~ ~λss = − Γ tan ∆, |~Γ|
(10)
where the temperature ratio ∆ is defined as ∆= 3.
~|~Γ| . 2kB T
(11)
SIMULATION
In the following sections, we examine the ability of the fundamental QCA building blocks to maintain correct logical operation in the presence of random cell displacements through simulation. Although analytical descriptions of the displacement tolerance would be highly useful, they have proven to be difficult to construct. ACM Journal Name, Vol. V, No. N, January 2007.
4
·
Gabriel Schulhof et al.
For each cell in the simulated circuits, the cell displacements are uniformly distributed in a circular area centered at the original location of each cell and with a diameter 0 ≤ R ≤ rmax . We did not to include random rotations of cell dots with respect to cell centers in our simulation. For each data point, 200 unique simulations are performed for a single value of rmax using a batch simulation mode in QCADesigner and the percentage of circuits that successfully complete the exhaustive vector simulation is recorded. For each simulation, all the cells in the design are randomly displaced in randomly chosen directions within this radial range. In order to determine the effect of cell scaling, we have performed simulations for 1 nm, 5 nm, 10 nm, and 20 nm cells. We scaled dot diameters along with cell sizes to preserve the overall proportions of our cells. The spacing between cells is equal to the cell size. Except for the 1 nm cells, we have chosen a value of the dielectric constant to be that of GaAs; i.e., r = 12.9. For the 1 nm cells we have chosen the dielectric constant to be unity, consistent with molecular QCA cells. The remaining simulation parameters are based on the kink energy. In this way, the simulation parameters are scaled based on the kink energy for each cell size. The calculated kink energies are given in Table I. Cell Type Molecular QCA (r = 1) Self-Assembled Semiconductor Lithographically Defined Lithographically Defined Table I.
Cell Size 1nm 5nm 10nm 20nm
Kink Energy 588.6 meV 9.13 meV 4.56 meV 2.28 meV
Kink energy between adjacent cells
The relaxation time, τ , is chosen as τ = 5~/Ekink , time step is ∆t = τ /1000, and the simulation duration is T = 1000τ . The time step was set to three orders of magnitude smaller than the relaxation time in order to prevent divergence during our simulations. The simulation temperature was chosen to be 1 K for all simulations. The maximum radial displacements (rmax ) for each simulation were chosen to range from 0 to 50% of the cell size. The maximum tunneling energy was γmax = 2Ekink and the minimum was γmin = 0.3Ekink . The simulation engine parameters (the time step and the relaxation time) are further explained in [Walus and Schulhof 2001]. We gauge the successful operation of each circuit by comparing the simulation output of the randomly modified circuit with the simulation output of a reference circuit that we have previously verified to be logically correct. We do not compare the output waveforms themselves. Instead, we interpret the waveforms using a simple threshold system: polarization < lower threshold (usually −0.5) ⇒ logic 0 and polarization > upper threshold (usually 0.5) ⇒ logic 1, otherwise the state is indeterminate. We then compare the logical output of the circuit. 4.
MAJORITY GATE
The majority gate (Fig. 2) is the fundamental logic primitive available with this technology. Previous analysis has shown that some of the cells of the majority gate are more sensitive to displacement errors than others[Tahoori et al. 2004]. However, we expect that in a typical manufacturing environment, there will be no ACM Journal Name, Vol. V, No. N, January 2007.
Simulation of Random Cell Displacements in QCA
·
5
control over which cells are displaced and therefore our analysis randomly displaces all cells. The results of our simulations are shown in Fig. 3 for all four cell sizes. A
Ouput
B
C Fig. 2.
Fig. 3.
Layout of the simulated QCA majority gate.
Displacement simulation results for the majority gate.
In all cases, it is evident that up to a maximum radial displacement of 2.5% of the cell size, the majority continues to work without error. The failures for this building block arise in large part when one of the three inputs is displaced sufficiently close to the device cell such that its polarization dominates the output of the gate. This error results in a gate that behaves as a simple wire and performs no logical function. These and subsequent results indicate that, when scaling cells while preserving their proportions, the displacement tolerance is not a function of the cell size, since the curves scale without change. As a result, it is evident that the tolerance to displacements is primarily a function of the layout and geometry of the circuit. 5.
INVERTER
Another important QCA building block is the inverter. Although signal inversion can be realized in a number of ways, we have chosen to examine the most common ACM Journal Name, Vol. V, No. N, January 2007.
6
·
Gabriel Schulhof et al.
layout (the “C”-shaped inverter, as in Fig. 4). Fig. 5 shows the simulation results for the inverter. Input
Fig. 4.
Fig. 5.
Ouput
Layout of the simulated QCA inverter.
Displacement simulation results for the inverter.
The simulation results for the inverter indicate that this building block begins to fail at any displacement beyond 0%. It is suspected that the failures arise from the weaker interaction between the output cells of the inverter and the two legs of the “C”. The simulation results are consistent across the four different cell sizes, again indicating that the displacement tolerance is a function of circuit layout. The robustness of this building block can be improved by adding parallel inverter legs, such as an “E” shaped inverter. 6.
WIRE
A wire is implemented in QCA with a linear array of cells. A simple, 5-cell QCA wire is shown in Fig. 6. Input Fig. 6.
Output 5-cell QCA wire.
Fig. 7 shows our simulation results for the QCA wire. The wire is by far the most robust circuit element, with correct operation in the presence of a displacement of up to 12.5%. These results indicate that the simple geometry of the wire has a beneficial effect on the fault tolerance. The primary ACM Journal Name, Vol. V, No. N, January 2007.
Simulation of Random Cell Displacements in QCA
Fig. 7.
·
7
Displacement simulation results for the QCA wire.
fault with such a wire results from the addition of unwanted inversion as cells are displaced to the point where the kink energy that couples them goes from a positive to a negative value. Each cell that is coupled to its neighbour with a negative kink energy essentially acts as an inverter. On the other hand, two inverters in series cancel the fault. 7.
COPLANAR AND MULTI-LAYER CROSSOVER
We have also investigated the tolerance of the so-called coplanar crossover, which is a signal crossing building block used in QCA circuits that operate on a four phase clock. The coplanar crossover can, under some circumstances, cross signals on one planar layer by taking advantage of certain symmetries between rotated and nonrotated cells. If these cells are perfectly realized, it can be shown that when placed adjacent to each other there will be no crosstalk between them, as shown in Fig. 8.
Fig. 8. A crossover shown in a, assumes that there is very little interaction between the vertical rotated cell wire and the horizontal wire as shown in b.
Previous work has also examined the possibility of multi-layer QCA [Gin et al. 1999]. Using these multi-layer QCA circuits we can effectively cross signals over on another layer. To do this, we require a vertical interconnect. By stacking cells one on top of another we can transmit the signal to another layer where the signal is again transmitted horizontally. The multi-layer crossover is shown in Fig. 9. ACM Journal Name, Vol. V, No. N, January 2007.
8
·
Gabriel Schulhof et al.
There are two intermediate layers of cells to prevent any possible crosstalk between the two interconnects. For the multi-layer crossover simulations, the layer-to-layer separation was chosen to be 0.575 nm, 2.875 nm, 5.75 nm, and 11.5 nm for the 1 nm, 5 nm, 10 nm, and 20 nm cells respectively. The Coulombic interaction between stacked cells forces the cells in between to tend to the opposite polarization of the layers above and below.
Fig. 9.
Multi-layer crossover.
Unlike present CMOS integrated circuits, where metal layers are used to connect discontinuous sections of a circuit and cannot perform any logic functions, the extra layers of QCA are no different from the base layer. Cells lying in the extra layers are governed by the same Coulombic interactions as those lying in the base layer. Thus, extra layers can accomodate computational circuit elements, not just wires. In this way, we believe that multi-layer QCA circuits can be made to consume significantly less area as compared to planar circuits. The layout for the coplanar crossover circuit is shown in Fig. 10, whereas the multi-layer crossover circuit is shown in Fig. 11. Normal Input
Crossover Input
Crossover Ouput
Normal Output
Fig. 10.
The coplanar crossover circuit used in our simulations.
The simulation results for the coplanar crossover are shown in Fig. 12 and the multi-layer crossover in Fig. 13. Our simulation results indicate that each multi-layer circuit is significantly more fault tolerant than the equivalent coplanar crossover. The very poor performance of the coplanar crossover can be understood in terms of crosstalk between the vertical and horizontal wires. Given a perfect layout, the crosstalk between the two ACM Journal Name, Vol. V, No. N, January 2007.
Simulation of Random Cell Displacements in QCA
·
9
Normal Input
Crossover Input
Crossover Ouput
Normal Output
Fig. 11. The multi-layer crossover circuit used in our simulations. Conventionally, cells marked “X” denote cells lying in a crossover layer, and cells marked “O” denote cells forming a vertical interconnect. All other cells are unchanged.
Fig. 12.
Displacement simulation results for the coplanar crossover.
Fig. 13.
Displacement simulation results for the coplanar crossover. ACM Journal Name, Vol. V, No. N, January 2007.
10
·
Gabriel Schulhof et al.
interconnects is zero. The single cell gap in the crossover wire drops the overall coupling between the two horizontal sections by a factor of 1/3232. The weak coupling renders the nearly floating output section of the horizontal interconnect highly sensitive to crosstalk. Even the smallest displacement can result in sufficient crosstalk between the two interconnects to cause the circuit to operate incorrectly. The robustness of the multi-layer crossover arises from the fact that there are no gaps anywhere along the length of the interconnects, and therefore none of the sections of interconnect are loosely coupled to each other. Although this building block presents a solution to crossing signals, it may be very difficult to realize in practice. Our objective in this work is not to evaluate the feasibility of such a building block. We are primarily interested in demonstrating that there are options available that will increase the overall displacement tolerance of the coplanar crossover. 8.
CONCLUSION
In this paper, we have investigated the behaviour of various QCA building blocks in the presence of random radial displacements of their constituent cells. We have shown that the various building blocks have different tolerances to cell displacements. The coplanar crossover’s performance decays most rapidly of all circuits, having very high failure rates at very small simulated displacements (2.5%). This is a result of the significant crosstalk developed between the two interconnects. By comparison, the multi-layer crossover has superior displacement tolerance when compared to the coplanar crossover. Although such a building block may be difficult to realize in practice, it provides a displacement tolerant alternative to the coplanar crossover if 3D QCA circuits are realized. The best performance was given by the simple wire. Such high performance can be attributed to its simple geometry. In all cases, we have shown that the displacement tolerances are independent of cell size. They depend instead on the layout and geometry of the circuit. We can also conclude that any practical QCA realization which has some potential for cell displacements cannot be based on the coplanar crossover layout described here. Acknowledgment The authors acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada, Micronet R&D (a Network of Centres of Excellence), iCORE (Alberta), and workstations and tools from the Canadian Microelectronics Corporation. REFERENCES Gin, A., Tougaw, P. D., and Williams, S. 1999. An alternative geometry for quantum-dot cellular automata. J. Appl. Phys. 85, 12 (June), 8281–8286. Huang, J., Momenzadeh, M., Tahoori, M. B., and Lombardi, F. 2004. Defect characterization for scaling of QCA devices. In Proc. of the 19th IEEE International Symposium on Defect and Fault Tolerance in VLSI Systems. Lent, C. S. 1993. Quantum cellular automata. Nanotechnology 4, 49–57. Lent, C. S. and Tougaw, P. D. 1997. A device architecture for computing with quantum dots. Proc. IEEE 85, 4 (April), 541–557. Tahoori, M. B., Huang, J., Momenzadeh, M., and Lombardi, F. 2004. Testing of quantum cellular automata. IEEE Trans. Nano. 3, 4 (December), 432–442. ACM Journal Name, Vol. V, No. N, January 2007.
Simulation of Random Cell Displacements in QCA
·
11
Tahoori, M. B., Momenzadeh, M., Huang, J., and Lombardi, F. 2004. Defects and faults in quantum cellular automata at nano scale. In Proc. of the 22nd IEEE VLSI Test Symposium, Washington, DC, USA, pp. 291. IEEE Computer Society. Timler, J. and Lent, C. S. 2002. Power gain and dissipation in quantum-dot cellular automata. J. Appl. Phys. 91, 2 (January), 823–831. Timler, J. and Lent, C. S. 2003. Maxwell’s demon and quantum-dot cellular automata. J. Appl. Phys. 94, 2 (July), 1050–1060. Walus, K. and Jullien, G. A. 2006. Design tools for an emerging soc technology: quantum-dot cellular automata. Proc. IEEE 94, 6 (June), 1225–1244. Walus, K. and Schulhof, G. 2001. QCADesigner Homepage. [Online] http://www.qcadesigner.ca/ . Walus, K., Schulhof, G., Zhang, R., Wang, W., and Jullien, G. A. 2004. Circuit design based on majority gates for applications with quantum-dot cellular automata. In Proc. of IEEE Asilomar Conference on Signals, Systems, and Computers. Wang, W., Walus, K., and Jullien, G. A. 2003. Quantum-dot cellular automata adders. In Proc. of IEEE Conference on Nanotechnology.
ACM Journal Name, Vol. V, No. N, January 2007.
In this work, we analyze the behaviour of quantum-dot cellular automata (QCA) building blocks in the presence of random cell displacements. The QCA cells are modeled using the coherence vector description and simulations are performed using the QCADesigner tool. We evaluate various fundamental circuit elements, including the wire, the inverter, the majority gate, and the two approaches to wire crossing: the coplanar crossover and the multi-layer crossover. We show that the different building blocks have different tolerances to such displacements. The coplanar crossover and inverter show the weakest tolerance, while the wire is the most robust. Our results also show that the tolerance to random displacements is not a function of cell size, they depend instead on but rather of the circuit layout. Categories and Subject Descriptors: B.7.3 [Fault Tolerance Analysis]: Statistical Approach General Terms: Reliability Additional Key Words and Phrases: quantum-dot cellular automata, QCA, fabrication variances, fault tolerance
1.
INTRODUCTION
Quantum-dot cellular automata is an interesting nano-scale paradigm capable of general purpose computation. Since it was first introduced by C. Lent et al. in 1993 at the University of Notre Dame [Lent 1993], many researchers have become increasingly interested in its simple concept and its many potential applications in future computer technology. Although the practical realization of QCA cells and circuits is still being developed, many architectural features of QCA circuits such as majority reduction of logic [Walus et al. 2004], and the full adder [Wang et al. 2003] have been investigated. A QCA computer-aided design (CAD) and simulation tool now exists in the form of QCADesigner [Walus and Schulhof 2001], developed at the University of Calgary ATIPS Laboratory. A review of QCA circuits and systems is available in [Walus and Jullien 2006]. The tolerance of QCA building blocks to manual displacements of cells has been previously studied[Tahoori et al. 2004; Huang et al. 2004]. However, such studies were not able to capture the statistical nature of manufacturing variations because the tools for such analysis were not developed. In this work, we present the results of using recently developed statistical simulation tools on the operation of QCA building blocks in the presence of random four-dot cell displacements. The rest of the paper is as follows: Section 2 describes the QCA model and ACM Journal Name, Vol. V, No. N, January 2007, Pages 1–0??.
2
·
Gabriel Schulhof et al.
dynamical equation used throughout the simulations. Section 3 describes the particular choices of simulation parameters and their relationship. Section 4 describes the majority gate and simulation results, Section 5 the inverter, Section 6 the wire, and Section 7 describes the coplanar crossover and provides an alternative layout, the multi-layer crossover, which has been found to be more displacement tolerant than the coplanar crossover. Finally, Section 8 provides some conclusions. 2.
MODEL
The standard QCA cell is shown in Fig. 1.
Fig. 1.
Standard 4-dot QCA cell. The two ground states of the cell are shown.
The Coulombic interaction between two cells can be described by the so-called kink energy, Ekink , associated with the energetic cost of two cells having opposite polarization [Lent and Tougaw 1997]. The electrostatic interaction between charges in two four-dot cells, i and j, is E i,j =
4 4 j 1 X X qni qm , j 4π0 r n=1 m=1 |rni − rm |
(1)
where r is the dielectric constant, qni is the charge in dot n of cell i, and rni is the position of dot n in cell i. The kink energy is the difference in energy between two cells which have opposite polarization and those same two cells having the same polarization; i.e, m,n m,n m,n Ekink = Eopposite polarization − Esame polarization .
(2)
Background charges of + 21 e are included in each dot, to ensure that the overall cell remains charge neutral. These positive charges are included in the calculation of the kink energy. In general, the interaction between cells can be described by a quadrupole-quadrupole interaction in which the kink energy decays as the fifth power of the distance between cells. QCA dynamics are modeled using the so-called coherence vector description[Timler and Lent 2003; 2002], using the Hartree-Fock approximation. The two-state Hamiltonian for a single cell is P i,j − 12 Ekink Pj −γi j∈Ωi ˆ (i) = P i,j H (3) , 1 −γi E P j kink 2 j∈Ωi
ACM Journal Name, Vol. V, No. N, January 2007.
Simulation of Random Cell Displacements in QCA
·
3
where the neighborhood, Ωi , of cell i, is defined by the “radius of effect” parameter in the simulation tool. The coefficients of the coherence vector, ~λ = [λx , λy , λz ]T , are determined from the expectation value of their associated Pauli spin matrix; i.e., λx = hˆ σx i. The polarization of a cell is defined by the expectation value of σ ˆz . ˆ is given by In the Heisenberg picture, the time dependence of an operator O ∂ ˆ h ˆ ˆi = O, H . (4) i~ O ∂t P i,j replacing O with each of the spin matrices and letting K = Ekink Pj results in j∈Ωi
~
σ ˆ 0 −K 0 σ ˆx d x ˆy . σ ˆy = K 0 2γ σ dt σ ˆz 0 −2γ 0 σ ˆz
Taking the expectation value of both sides we ˆ λ 0 −K d ˆx ~ λy = K 0 dt ˆ 0 −2γ λz
(5)
get ˆ λx 0 ˆy , 2γ λ ˆz 0 λ
which can be rewritten in the vector form as d ~ ~λ = ~Γ × ~λ, dt where −2γ 0 ~Γ = P i,j . Ekink Pj
(6)
(7)
(8)
j∈Ωi
Equation (7) is the dynamical equation of motion for the coherence vector. In order to introduce dissipative coupling to the environment, we use a relaxation time constant τ that incorporates all the available dissipative channels. In this relaxation-time approximation, the dynamical equation of motion is d 1 ~ ~ ~ ~λ = ~Γ × ~λ − λ − λss (9) dt τ The steady-state coherence vector is ~ ~λss = − Γ tan ∆, |~Γ|
(10)
where the temperature ratio ∆ is defined as ∆= 3.
~|~Γ| . 2kB T
(11)
SIMULATION
In the following sections, we examine the ability of the fundamental QCA building blocks to maintain correct logical operation in the presence of random cell displacements through simulation. Although analytical descriptions of the displacement tolerance would be highly useful, they have proven to be difficult to construct. ACM Journal Name, Vol. V, No. N, January 2007.
4
·
Gabriel Schulhof et al.
For each cell in the simulated circuits, the cell displacements are uniformly distributed in a circular area centered at the original location of each cell and with a diameter 0 ≤ R ≤ rmax . We did not to include random rotations of cell dots with respect to cell centers in our simulation. For each data point, 200 unique simulations are performed for a single value of rmax using a batch simulation mode in QCADesigner and the percentage of circuits that successfully complete the exhaustive vector simulation is recorded. For each simulation, all the cells in the design are randomly displaced in randomly chosen directions within this radial range. In order to determine the effect of cell scaling, we have performed simulations for 1 nm, 5 nm, 10 nm, and 20 nm cells. We scaled dot diameters along with cell sizes to preserve the overall proportions of our cells. The spacing between cells is equal to the cell size. Except for the 1 nm cells, we have chosen a value of the dielectric constant to be that of GaAs; i.e., r = 12.9. For the 1 nm cells we have chosen the dielectric constant to be unity, consistent with molecular QCA cells. The remaining simulation parameters are based on the kink energy. In this way, the simulation parameters are scaled based on the kink energy for each cell size. The calculated kink energies are given in Table I. Cell Type Molecular QCA (r = 1) Self-Assembled Semiconductor Lithographically Defined Lithographically Defined Table I.
Cell Size 1nm 5nm 10nm 20nm
Kink Energy 588.6 meV 9.13 meV 4.56 meV 2.28 meV
Kink energy between adjacent cells
The relaxation time, τ , is chosen as τ = 5~/Ekink , time step is ∆t = τ /1000, and the simulation duration is T = 1000τ . The time step was set to three orders of magnitude smaller than the relaxation time in order to prevent divergence during our simulations. The simulation temperature was chosen to be 1 K for all simulations. The maximum radial displacements (rmax ) for each simulation were chosen to range from 0 to 50% of the cell size. The maximum tunneling energy was γmax = 2Ekink and the minimum was γmin = 0.3Ekink . The simulation engine parameters (the time step and the relaxation time) are further explained in [Walus and Schulhof 2001]. We gauge the successful operation of each circuit by comparing the simulation output of the randomly modified circuit with the simulation output of a reference circuit that we have previously verified to be logically correct. We do not compare the output waveforms themselves. Instead, we interpret the waveforms using a simple threshold system: polarization < lower threshold (usually −0.5) ⇒ logic 0 and polarization > upper threshold (usually 0.5) ⇒ logic 1, otherwise the state is indeterminate. We then compare the logical output of the circuit. 4.
MAJORITY GATE
The majority gate (Fig. 2) is the fundamental logic primitive available with this technology. Previous analysis has shown that some of the cells of the majority gate are more sensitive to displacement errors than others[Tahoori et al. 2004]. However, we expect that in a typical manufacturing environment, there will be no ACM Journal Name, Vol. V, No. N, January 2007.
Simulation of Random Cell Displacements in QCA
·
5
control over which cells are displaced and therefore our analysis randomly displaces all cells. The results of our simulations are shown in Fig. 3 for all four cell sizes. A
Ouput
B
C Fig. 2.
Fig. 3.
Layout of the simulated QCA majority gate.
Displacement simulation results for the majority gate.
In all cases, it is evident that up to a maximum radial displacement of 2.5% of the cell size, the majority continues to work without error. The failures for this building block arise in large part when one of the three inputs is displaced sufficiently close to the device cell such that its polarization dominates the output of the gate. This error results in a gate that behaves as a simple wire and performs no logical function. These and subsequent results indicate that, when scaling cells while preserving their proportions, the displacement tolerance is not a function of the cell size, since the curves scale without change. As a result, it is evident that the tolerance to displacements is primarily a function of the layout and geometry of the circuit. 5.
INVERTER
Another important QCA building block is the inverter. Although signal inversion can be realized in a number of ways, we have chosen to examine the most common ACM Journal Name, Vol. V, No. N, January 2007.
6
·
Gabriel Schulhof et al.
layout (the “C”-shaped inverter, as in Fig. 4). Fig. 5 shows the simulation results for the inverter. Input
Fig. 4.
Fig. 5.
Ouput
Layout of the simulated QCA inverter.
Displacement simulation results for the inverter.
The simulation results for the inverter indicate that this building block begins to fail at any displacement beyond 0%. It is suspected that the failures arise from the weaker interaction between the output cells of the inverter and the two legs of the “C”. The simulation results are consistent across the four different cell sizes, again indicating that the displacement tolerance is a function of circuit layout. The robustness of this building block can be improved by adding parallel inverter legs, such as an “E” shaped inverter. 6.
WIRE
A wire is implemented in QCA with a linear array of cells. A simple, 5-cell QCA wire is shown in Fig. 6. Input Fig. 6.
Output 5-cell QCA wire.
Fig. 7 shows our simulation results for the QCA wire. The wire is by far the most robust circuit element, with correct operation in the presence of a displacement of up to 12.5%. These results indicate that the simple geometry of the wire has a beneficial effect on the fault tolerance. The primary ACM Journal Name, Vol. V, No. N, January 2007.
Simulation of Random Cell Displacements in QCA
Fig. 7.
·
7
Displacement simulation results for the QCA wire.
fault with such a wire results from the addition of unwanted inversion as cells are displaced to the point where the kink energy that couples them goes from a positive to a negative value. Each cell that is coupled to its neighbour with a negative kink energy essentially acts as an inverter. On the other hand, two inverters in series cancel the fault. 7.
COPLANAR AND MULTI-LAYER CROSSOVER
We have also investigated the tolerance of the so-called coplanar crossover, which is a signal crossing building block used in QCA circuits that operate on a four phase clock. The coplanar crossover can, under some circumstances, cross signals on one planar layer by taking advantage of certain symmetries between rotated and nonrotated cells. If these cells are perfectly realized, it can be shown that when placed adjacent to each other there will be no crosstalk between them, as shown in Fig. 8.
Fig. 8. A crossover shown in a, assumes that there is very little interaction between the vertical rotated cell wire and the horizontal wire as shown in b.
Previous work has also examined the possibility of multi-layer QCA [Gin et al. 1999]. Using these multi-layer QCA circuits we can effectively cross signals over on another layer. To do this, we require a vertical interconnect. By stacking cells one on top of another we can transmit the signal to another layer where the signal is again transmitted horizontally. The multi-layer crossover is shown in Fig. 9. ACM Journal Name, Vol. V, No. N, January 2007.
8
·
Gabriel Schulhof et al.
There are two intermediate layers of cells to prevent any possible crosstalk between the two interconnects. For the multi-layer crossover simulations, the layer-to-layer separation was chosen to be 0.575 nm, 2.875 nm, 5.75 nm, and 11.5 nm for the 1 nm, 5 nm, 10 nm, and 20 nm cells respectively. The Coulombic interaction between stacked cells forces the cells in between to tend to the opposite polarization of the layers above and below.
Fig. 9.
Multi-layer crossover.
Unlike present CMOS integrated circuits, where metal layers are used to connect discontinuous sections of a circuit and cannot perform any logic functions, the extra layers of QCA are no different from the base layer. Cells lying in the extra layers are governed by the same Coulombic interactions as those lying in the base layer. Thus, extra layers can accomodate computational circuit elements, not just wires. In this way, we believe that multi-layer QCA circuits can be made to consume significantly less area as compared to planar circuits. The layout for the coplanar crossover circuit is shown in Fig. 10, whereas the multi-layer crossover circuit is shown in Fig. 11. Normal Input
Crossover Input
Crossover Ouput
Normal Output
Fig. 10.
The coplanar crossover circuit used in our simulations.
The simulation results for the coplanar crossover are shown in Fig. 12 and the multi-layer crossover in Fig. 13. Our simulation results indicate that each multi-layer circuit is significantly more fault tolerant than the equivalent coplanar crossover. The very poor performance of the coplanar crossover can be understood in terms of crosstalk between the vertical and horizontal wires. Given a perfect layout, the crosstalk between the two ACM Journal Name, Vol. V, No. N, January 2007.
Simulation of Random Cell Displacements in QCA
·
9
Normal Input
Crossover Input
Crossover Ouput
Normal Output
Fig. 11. The multi-layer crossover circuit used in our simulations. Conventionally, cells marked “X” denote cells lying in a crossover layer, and cells marked “O” denote cells forming a vertical interconnect. All other cells are unchanged.
Fig. 12.
Displacement simulation results for the coplanar crossover.
Fig. 13.
Displacement simulation results for the coplanar crossover. ACM Journal Name, Vol. V, No. N, January 2007.
10
·
Gabriel Schulhof et al.
interconnects is zero. The single cell gap in the crossover wire drops the overall coupling between the two horizontal sections by a factor of 1/3232. The weak coupling renders the nearly floating output section of the horizontal interconnect highly sensitive to crosstalk. Even the smallest displacement can result in sufficient crosstalk between the two interconnects to cause the circuit to operate incorrectly. The robustness of the multi-layer crossover arises from the fact that there are no gaps anywhere along the length of the interconnects, and therefore none of the sections of interconnect are loosely coupled to each other. Although this building block presents a solution to crossing signals, it may be very difficult to realize in practice. Our objective in this work is not to evaluate the feasibility of such a building block. We are primarily interested in demonstrating that there are options available that will increase the overall displacement tolerance of the coplanar crossover. 8.
CONCLUSION
In this paper, we have investigated the behaviour of various QCA building blocks in the presence of random radial displacements of their constituent cells. We have shown that the various building blocks have different tolerances to cell displacements. The coplanar crossover’s performance decays most rapidly of all circuits, having very high failure rates at very small simulated displacements (2.5%). This is a result of the significant crosstalk developed between the two interconnects. By comparison, the multi-layer crossover has superior displacement tolerance when compared to the coplanar crossover. Although such a building block may be difficult to realize in practice, it provides a displacement tolerant alternative to the coplanar crossover if 3D QCA circuits are realized. The best performance was given by the simple wire. Such high performance can be attributed to its simple geometry. In all cases, we have shown that the displacement tolerances are independent of cell size. They depend instead on the layout and geometry of the circuit. We can also conclude that any practical QCA realization which has some potential for cell displacements cannot be based on the coplanar crossover layout described here. Acknowledgment The authors acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada, Micronet R&D (a Network of Centres of Excellence), iCORE (Alberta), and workstations and tools from the Canadian Microelectronics Corporation. REFERENCES Gin, A., Tougaw, P. D., and Williams, S. 1999. An alternative geometry for quantum-dot cellular automata. J. Appl. Phys. 85, 12 (June), 8281–8286. Huang, J., Momenzadeh, M., Tahoori, M. B., and Lombardi, F. 2004. Defect characterization for scaling of QCA devices. In Proc. of the 19th IEEE International Symposium on Defect and Fault Tolerance in VLSI Systems. Lent, C. S. 1993. Quantum cellular automata. Nanotechnology 4, 49–57. Lent, C. S. and Tougaw, P. D. 1997. A device architecture for computing with quantum dots. Proc. IEEE 85, 4 (April), 541–557. Tahoori, M. B., Huang, J., Momenzadeh, M., and Lombardi, F. 2004. Testing of quantum cellular automata. IEEE Trans. Nano. 3, 4 (December), 432–442. ACM Journal Name, Vol. V, No. N, January 2007.
Simulation of Random Cell Displacements in QCA
·
11
Tahoori, M. B., Momenzadeh, M., Huang, J., and Lombardi, F. 2004. Defects and faults in quantum cellular automata at nano scale. In Proc. of the 22nd IEEE VLSI Test Symposium, Washington, DC, USA, pp. 291. IEEE Computer Society. Timler, J. and Lent, C. S. 2002. Power gain and dissipation in quantum-dot cellular automata. J. Appl. Phys. 91, 2 (January), 823–831. Timler, J. and Lent, C. S. 2003. Maxwell’s demon and quantum-dot cellular automata. J. Appl. Phys. 94, 2 (July), 1050–1060. Walus, K. and Jullien, G. A. 2006. Design tools for an emerging soc technology: quantum-dot cellular automata. Proc. IEEE 94, 6 (June), 1225–1244. Walus, K. and Schulhof, G. 2001. QCADesigner Homepage. [Online] http://www.qcadesigner.ca/ . Walus, K., Schulhof, G., Zhang, R., Wang, W., and Jullien, G. A. 2004. Circuit design based on majority gates for applications with quantum-dot cellular automata. In Proc. of IEEE Asilomar Conference on Signals, Systems, and Computers. Wang, W., Walus, K., and Jullien, G. A. 2003. Quantum-dot cellular automata adders. In Proc. of IEEE Conference on Nanotechnology.
ACM Journal Name, Vol. V, No. N, January 2007.
