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Clocking misalignment tolerance of pipelined magnetic QCA architectures
Microelectronics Journal 43 (2012) 386–392
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Clocking misalignment tolerance of pipelined magnetic QCA architectures Xiaokuo Yang a,n, Li Cai a, Qiang Kang b, Xiaohui Zhao a a b
College of Science, Air Force Engineering University, Xi’an 710051, People’s Republic of China Department of Science Research, Air Force Engineering University, Xi’an 710051, People’s Republic of China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 December 2011 Received in revised form 18 February 2012 Accepted 28 February 2012 Available online 17 March 2012
In this paper, we present a simulation study on clocking misalignment tolerance of pipelined magnetic quantum-dot cellular automata (MQCA) architectures. By the three-phase pipelined clocking and deduced clocking misalignment model, a systematic evaluation of impacts of clocking misalignment on four fundamental MQCA architectures is performed at non-zero temperatures. It is found that for the fixed nanomagnet size, majority logic gate is the most reliable structure, while the corner is most susceptible to clocking misalignment. High temperature gives rise to a negative effect on allowable misalignment angles. The results also show that as the aspect ratio of nanomagnet increases, the ability that all the MQCA architectures tolerate clocking misalignment decreases. Moreover, we analyze potential reason of pipelined MQCA architecture failures by examining the energy profile of neighboring zone nanomagnets and conclude that various energy barrier difference accounts for failure of MQCA architectures under clocking misalignment defect. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Clocking misalignment Nanomagnet Magnetic quantum-dot cellular automata Aspect ratio Energy barrier
1. Introduction In traditional electronics, information is usually stored and processed using considerable electron charge, however, chargebased transistors are facing a worrisome power dissipation problem with scaling feature size [1]. Consequently, interest is very high in other logic technologies whose functionality does not rely on controlling vast electron charge. One promising technology among them is quantum-dot cellular automata (QCA) [2,3]. A magnetic implementation of the QCA devices (MQCA) was originally proposed by Cowburn and Welland [4], which has seen significant progress [5–8]. MQCA employs rectangular shapebased nanomagnet to build the device, which reaches the lowest energy state when the magnetization direction is aligned along the easy axis of a nanomagnet. Depending on the magnetization forces around elongated nanomagnet, its magnetization direction will point straightly up or down, corresponding to binary value ‘1’ or ‘0’, respectively. Recently, MQCA wire and majority logic gate have been fabricated at room temperature [5]. To date, some works have been done on non-pipelined MQCA interconnect wire, inverter and majority logic gate architectures to study their operation frequency, switching behavior and power gain and dissipation [9–13]. Moreover, switching characteristics of interconnect architecture with defective nanomagnets has also been investigated [14,15]. These theoretic researches continue to mature the MQCA concept and structure. However, since most of MQCA architectures are operated by hard axis magnetic field clocking
n
Corresponding author. E-mail address: [email protected] (X. Yang).
0026-2692/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2012.02.005
which erases logic state and re-evaluates magnetic ordering, during the re-evaluating process nanomagnets are very vulnerable to hard axis clocking misalignment: if magnetic field clocking departs away the hard axis direction (clocking misalignment) of nanomagnet, the magnetic ordering of MQCA architectures may fail. Clocking misalignment of MQCA device is succinctly discussed in Ref. [6]. Nanomagnet with a slanted edge subjected to non-pipelined clocking misalignment has also been explored [16]. However, these works are very rough to understand clocking misalignment issues of MQCA architectures, because some important aspects have not been considered or solved there, such as pipelined clocking misalignment, temperature and nanomagnet size effect. In the present work, we focus on clocking misalignment issue of pipelined MQCA architectures with detailed simulations and indepth consideration (size and temperature, etc.) of a few typical cases. We also further study the failure reason and mechanism of pipelining-clocked structures subjected to clocking misalignment. After presenting three MQCA-cell wide pipelined clocking and deducing the computational model, we investigate by two-state (‘0’ and ‘1’) system [17] and numerical method the impacts of clocking misalignment on the operation and defect tolerance of coupled nanomagnet architectures. The goal of this work is to provide a guide to deeply understand clocking misalignment induced error formation in the pipelined MQCA architectures.
2. Clocking misalignment of pipelined MQCA architectures Elongated single domain nanomagnet is used to achieve bistability and build basic MQCA device due to shape-induced anisotropy. At zero external magnetic field, the remanent magnetization
X. Yang et al. / Microelectronics Journal 43 (2012) 386–392
of nanomagnet usually points along the long axis [18]. Thus, the long axis is viewed as the ‘easy’ axis, while the short axis as the ‘hard’ axis. All MQCA architectures need external torque to help their switching [7,8,13,19,20]; this external torque is viewed as MQCA clocking. Previously, a magnetic field-based global clocking methodology was proposed and experimentally demonstrated using copper clad wire embedded underneath the nanomagnets [7,8]. One salient advantage of this global clocking is the simplicity and reduced lithographic burden because all the nanomagnets are clocked simultaneously with only a clocking circuitry. However, global clocking makes MQCA architectures non-pipelined and very slow. More serious is that the global clocking would aggravate destroying the whole point of nanomagnet logic if clocking field misalignment occurs. Recently, Refs. [15,16,19,20] investigate possible MQCA local clocking to obtain pipelined switching, such as two phases, three phases and four phases clocking. In the three phases (not clocked, null and switch) pipelined clocking [20] shown in Fig. 1, three MQCA cells are encapsulated in a clocking zone and clocking signal repeats every three zones. Here we introduce a gradually changing ‘switch’ field to realize stable switching. The detail information about three phases pipelined clocking can be found from Ref. [20]. Based on three phases pipelined clocking signals, all the investigated MQCA architectures are constructed. The sevennanomagnet wire is shown in Figs. 2(a), and (b) presents the six-nanomagnet inverter architecture. Moreover, to form more computation architecture, MQCA gates and wires need to be
connected. Here we consider the architecture of having a horizontal line of five nanomagnets drive a signal to a vertical line of three nanomagnets, namely MQCA corner shown in Fig. 2(c). The nanomagnets forming the majority logic gate are arranged as seen in Fig. 2(d). Three phases pipelined clocking are adopted in these MQCA architectures. The precondition for error-free operation of MQCA architectures lies in two aspects. One is to place nanomagnets in the accurate position using advanced lithography technology. The other is to generate accurate hard-axis direction clocking to clock the nanomagnets, which also requires the accuracy of current carrying wire placement. However, fabrication defects in the nanotechnology are expected to be quite high. The nanomagnet and clocking could be misaligned. Usually, there exist two phenomenas causing clocking misalignment in the fabrication of MQCA architectures.
The clocking field (zeeman field) is not exactly aligned along the
hard axis of the nanomagnet, but subtends an angle with it [5]. Here, we show the case of four nanomagnets where magnetic field clocking is oriented in a small subtended angle with hard axis direction in the lithography (real line arrowhead shown in Fig. 3(a) represents fabricated clocking field direction, while broken line arrowhead represents the hard axis direction of nanomagnets). Although the clocking field is exactly aligned along the hard axis of many nanomagnets, but some nanomagnets in the architecture are not deposited in the proper position during lithography
not clocked 1
387
switch
Zone 1 null
2
Zone 2
3
Zone 3
Zone 1
Zone 2
Zone 3
1
2
3
t Fig. 1. Three phases pipelined clocking of MQCA architectures. (a) Operation process; (b) Pipelined clocking signals.
out
Fig. 2. Investigated MQCA architectures and their clocking zones distribution. (a) Wire; (b) Inverter; (c) Corner; (d) Majority logic gate.
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y w
Hzeeman
j •
i •
h Hzeeman
l Hzeeman
k •
s
x
z
Fig. 4. Description of clocking misalignment in the pipelined MQCA array. Fig. 3. Characterizations of clocking misalignment. (a) Misalignment of zeeman field direction; (b) Clocking misalignment caused by nanomagnet deflection or rotation.
technology phase [6]. As seen from Fig. 3(b), the second nanomagnet in the array has been rotated, which can be understood from the case that nanomagnet become irregular in the manufacturing. Now the clocking misalignment defect still appears. Therefore, in magnetic field-based clocking scheme, misalignment is a very real and major issue. As we discussed before, global clocking field should be avoided because it is prone to cause circuits fail when the MQCA architectures suffer from clocking misalignment. Pipelined local clocking signal mitigates misalignment problem, hence it is more practical for MQCA technology. However, in order to develop a viable alternative technology to charge-based transistors, clocking misalignment effect and defect tolerance in the pipelining-clocked MQCA architectures must also be understood, especially when the nanomagnet size and temperature vary. Hence whether clocking misalignment and cell geometry would remarkably alter the performance of pipelined MQCA architectures and to which extent will be taken into account.
thermal magnetic field HT ðtÞ. HT ðtÞ accounts for the interaction effects of the magnetization with phonons and conducting electrons which cause magnetization fluctuations. After adding the thermal torque, we get new effective field X Heff ðtÞ ¼ Hzeeman ðtÞNM þ CM þHT ðtÞ ð2Þ where N is the demagnetization tensor (NM denotes the demagnetizing field), which takes into account three dimension sizes of a nanomagnet. For a cube nanomagnet considered in the paper, it is a diagonal matrix and the elements obey the relation Nx þ N y þ Nz ¼ 1. C is the coupling matrix between neighboring P nanomagnets and CM denotes the coupling field. Hzeeman represents applied external field, namely clocking in the MQCA architectures. Assume that (x(i), y(i), z(i)) are the coordinates of nanomagnet i and (x(j), y(j), z(j)) are the coordinates of nanomagnet j, then we can obtain the coordinate differences for these two nanomagnets as follows: ðjiÞ
ðjÞ
ðiÞ
ð3:aÞ
ðjiÞ
ðjÞ
ðiÞ
ð3:bÞ
ðjiÞ
ðjÞ
ðiÞ
dx ¼ dx dx ¼ l þs dy ¼ dy dy ¼ 0 dz ¼ dz dz ¼ 0
ð3:cÞ (i)
3. Modeling clocking misalignment of pipelined MQCA architectures Our first step will be to identify magnetization process of MQCA architectures undergoing defective pipelined clocking. We model the effects of clocking misalignment on the operation of MQCA architectures and focus on the clocking misalignment tolerance under various misalignment angles, nanomagnet sizes and temperatures. As pointed out in the Section 2, we adopt proposed three-nanomagnet-zone pipelined clocking in the array. The schematic diagram of such a six nanomagnets array subject to clocking misalignment (Fig. 3a) is shown in Fig. 4. To model clocking misalignment in the pipelined MQCA architectures, we need to start from basic Landau and Lifshitz equation [21], which in the Gilbert form reads dM a dM ¼ gM Heff þ ð1Þ M dt dt Ms where M¼(Mx, My, Mz) is time-related three dimension magnetization of a nanomagnet, Ms is saturation magnetization, while Heff is the effective field experienced by the magnetic moment, and it is the sum of externally applied field (zeeman field), field originating from nanomagnet itself (demagnetizing field), and coupling from nanomagnet’s neighbors. g ¼2.21 105 m/A s, and a is the Gilbert damping parameter. It is very important to note that effective numerical simulation result using this MQCA model has been reported [22]. Thermal torque is specifically taken into account in this work and it is introduced into the effective field Heff by a stochastic
In the mean time, assume V is the volume of nanomagnet i and the distance vector describing nanomagnets i and j is (ji) (ji) d(ji) ¼(d(ji) x ,dy ,dz ), then the third term of Eq. (2) can be calculated by CðjiÞ MðiÞ ¼
V ðiÞ 4pðd Þ5 ðjiÞ
ð3ðd ÞT d d IÞMðiÞ ðjiÞ
0 0 ¼
¼
lhw 4pðl þsÞ5
lhw 4pðl þsÞ4
ðjiÞ
0
ðjiÞ
1
B B C @3@ l þ s Að 0 l þ s 0 0 1 MðiÞ x B C B ð3l þ 3s1ÞM ðiÞ C y A @
10 M ðiÞ 1 x B ðiÞ C C 0 Þðl þ sÞIC AB @ My A ðiÞ Mz
ð4Þ
MðiÞ z
where l, h and w denote width, height and thickness of a nanomagnet, respectively, s represents nanomagnet spacing (see Fig. 4), I is the identity matrix. Note that we simplify the expression of CðjiÞ by the established coupling matrix with dipole-approximation [22]. Now we have presented the coupling energy calculation method of two nanomagnets arranged in a horizontal style like that appeared in Fig. 1. However, if nanomagnet i resides in the upside or downside of driven nanomagnet j, we can adopt similar method to calculate the vector CðjiÞ . As discussed before, magnetic field clocking may not be exactly aligned with the hard axis of a nanomagnet in the fabrication, but subtends an angle b with it. We call b misalignment angle (see Fig. 4). Now clocking field magnitude along hard axis direction is given by Hh ¼ Hzeeman ðtÞcos b, and the clocking
X. Yang et al. / Microelectronics Journal 43 (2012) 386–392
field magnitude along the easy axis direction is given by He ¼ Hzeeman ðtÞsin b. Consequently, the clocking vector has been changed from Hzeeman(t)¼[Hzeeman(t) 0 0] into Hzeeman ðtÞ ¼ ½Hh He 0
ð5Þ
Temperature fluctuation-induced torque [23] is obtained by the expression 1 HT ðtÞ ¼ pffiffiffiffiffiffiffiffiffi V Dt
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kB T a gðtÞ m0 gMs
ð6Þ
where g(t) is a Gaussian-distributed random vector, V is nanomagnet volume and Dt is the fixed time step used to solve the Landau–Lifshitz–Gilbert (LLG) equation. From the deduced computation model above, only the first term in the right hand side of Eq. (2) would be determined by clocking misalignment angle. After modeling the clocking misalignment, Eqs. (1)–(6) allow us to compute temporal evolution of magnetization vector of any nanomagnet in the pipeliningclocked MQCA architectures subject to clocking defect. The outputs magnetization polarization may vary with nanomagnet size, misalignment angle and temperature. To address their relations and interaction effects clearly, the following work will calculate how the cell geometry and defective pipelined clocking raise some impacts on the switching operation and signal propagation of MQCA architectures. In order to bring this work close to a fabrication reference, we specifically present a generation method of proposed pipelined clocking field. That is to adopt pulse current carrying Cu wire similar to that appeared in Ref. [8] to generate the magnetic field. A physically feasible clocking circuitry (comprised of Si substrate, oxide, carrying Cu wire and high permeability cladding, nanomagnets are placed on the top) to clock just 3 nanomagnets at a time is shown in Fig. 5. The clocking sequence is guaranteed by the phase of pulse current. This physical realization can ensure that clocking field is localized to three nanomagnets due to the following two reasons. One is that we use high permeability magnetic material to construct the cladding, which can effectively enhance and concentrate magnetic field on the three nanomagnets of one zone. The other is that the spacing between adjoining clocking circuitry is a bit large (yellow color), which greatly reduces, even vanishes the fringing fields. Moreover, although the dimensions of state-of-the-art carrying Cu wire are typically greater than the width of a nanomagnet device ( 50 nm), we fixed the width of clocking zone to three nanomagnets, this can be implemented in technology where the clocking wire diameter and the clocking wire pitch together do not exceed 3 50 nm ¼150 nm ( 200 nm if including spacing), which is feasible according to the International Technology Roadmap for Semiconductors [24]. At the same time, the horizontal oriented small nanomagnet (gray nanomagnet blocks in Fig. 2) serves as the input bias in all the pipelined MQCA architectures.
nanomagnet
carrying Cu wire
high permeability cladding oxide Si substrate Fig. 5. Schematic of a physically feasible three phases clocking circuitry.
389
4. Results and analysis In this section, the behaviors of four pipelined MQCA logic architecture under various misalignment angle and nanomagnet size are studied by applying the model presented in Section 3. In the simulations, a single clocking zone (except the input zone) misalignment defect is injected into every MQCA architectures, then we use fourth-order Runge–Kutta method to solve the system of coupled differential equations in Eq. (1) for the magnetic coupled nanomagnet arrays. For all the trials, nanomagnet spacing (horizontal and vertical direction) is chosen to be 15 nm—an optimal value [22] and the misalignment angle ranges from 01 to 51. The choice of this angle range is based on Refs. [6,16] where are two kinds of misalignment angles (11 and 21) have been studied. Moreover, Ref. [6] had stated that misalignment angle larger than 11 (several degrees) usually occurs in the MQCA although very advanced lithography is adopted. Therefore our misalignment angle range is in particular supported by this work. We do non-zero temperature simulations in the investigations. (i) The first set of experiments examines success rate Psuccess with respect to the misalignment angle factor m when the nanomagnets appear with a constant aspect ratio. (ii) The second set of experiments investigates the relation between allowable misalignment angle and nanomagnet size or aspect ratio (width/height). The geometry size of supermalloy nanomagnet is chosen to be 50 80 25 nm3 (aspect ratio is 0.625) and the saturation magnetization of supermalloy is 8 105 A/m. We adopt 95 mT hard axis clocking field magnitude. The default of damping constant of 0.5 is used. For every calculation, we considered MQCA architecture subjected to defective clocking field to be successful if the absolute value of normalized output magnetization polarization is larger than 0.8 [20]. Otherwise, we considered the architecture fail. In fact, once normalized output magnetization M=M s is larger than 0.5, switching would be correct due to signal restoration and power gain of nanomagnet clocking. M/Ms ¼0.8 in the present work is a more rigorous condition. To conduct a statistics analysis for clocking misalignment angle, a uniform distribution U (01, 51) is applied to model technology process. Suppose that bv satisfies this probability distribution, then the misalignment angle b, fabrication induced defect, is defined by
b ¼ bi þ mbv
ð7Þ
In Eq. (7), bi represents ideal direction (01) of clocking field and
m denotes a random number serially chosen in the interval (0,1), the latter can be used to characterize the amplitude of clocking misalignment angle spreading. In the present investigations, we use m as the misalignment angle factor and study the relation between successful operation and factor m. For each clocking zone at one m value, 500 trials (a batch of 500 misalignment angles satisfying uniform distribution) are used for the simulations of pipelined MQCA architectures. For single input devices (wire, inverter and corner), logic ‘0’ and ‘1’ are both used for one time, while for the majority logic gate, its eight input sequences are applied for the calculations. Numerical simulations for fabrication induced clocking misalignment defect are carried out and the success probabilities Psuccess are plotted as a function of the misalignment angle factor m for different MQCA architectures at two non-zero temperatures. Fig. 6 shows the success probabilities of four typical pipelined MQCA architectures with uniform distribution misalignment angle. We see that the operations of all the MQCA arrays are affected by clocking misalignment defect, and increasing
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1
1 0.9
T = 100K
0.8
0.8
0.7
0.7
(A) (A) Majority Gate (B) Inverter
0.6
(B) (C) (D)
(C) Wire
0.5
Psuccess
Psuccess
0.9
T = 300K
(D)
0.6 (A) Majority Gate (B) Inverter
0.5
(D) Corner
0.4
(C) Wire
0.4
0.3
(A) (B) (C)
(D) Corner
0.3
0
0.2
0.4 0.6 Misalignment factor μ
1
0.8
0
0.2
0.4 0.6 Misalignment factor μ
0.8
1
Fig. 6. Success probability as a function of misalignment angle factor m for different pipelined MQCA architectures. (a) T¼ 100K; (b) T¼ 300 K.
5
5
4.5
4.5 Wire Inverter
3.5
Corner(zone 3)
3
Majority gate
2.5 2
FAILURE
1.5 1
Inverter
3.5
Corner(zone 2)
T = 300K
Corner(zone 3)
3
Majority gate
2.5 2 FAILURE
1.5 1
SUCCESS
0.5
0.5 0
Wire
4
T = 100K
Corner(zone 2)
Misalignment Angle ( °)
Misalignment Angle ( °)
4
1
0.9
0.8
0.7 Aspect Ratio
0.6
0.5
0.4
0
SUCCESS
1
0.9
0.8
0.7 Aspect Ratio
0.6
0.5
0.4
Fig. 7. Simulated behavior of pipelined MQCA architectures at various misalignment angles and nanomagnet sizes. (a) T ¼100 K; (b) T¼300K.
temperature leads to a more unstable switching. However, the overall trends of all the plots are similar. Note the slight oscillations in the curve originate from the distribution interpolation technique and should be disregarded. The breakdowns of success probabilities for different MQCA architectures (do not show a success probability ‘1’) occur at different misalignment factors. All the detailed observations or conclusions can be drawn from the plots analysis. The success probability of all the pipelined MQCA architectures decreases with the large spreading (large m value) of clocking misalignment angle. From statistics analysis results, we find that in case of clocking misalignment the majority logic gate is the strongest (most reliable) architecture, while the corner is the weakest architecture at any thermal temperature if measured by the smallest misalignment factor that impacts success probabilities. For the wire and inverter architectures, the breakdowns of the success probabilities at T¼100 K are almost same (m ¼0.61), but at room temperature the inverter demonstrate a bit better performance than the wire. Regardless of misalignment factor, each MQCA architecture denotes a final fixed success probability value. Overall, it is evident that a misalignment error is a major threat to the pipelined MQCA architectures, especially for the clocking misalignment of corner at room temperature. Nevertheless, to our comfort, pipelined MQCA architectures can still tolerate moderate clocking misalignment, for example, the
majority logic gate demonstrates a complete correct operation for about three degrees misalignment at room temperature. For a detailed and comprehensive understanding of nanomagnet size and clocking misalignment impacts on the performance of four typical pipelined MQCA architectures, assume that defective clocking and nanomagnet size varies simultaneously, and then we can study the relation between MQCA size and allowable misalignment angle. For clarity, the aspect ratio of nanomagnet is used to describe variability of nanomagnet size. It should be pointed out that there is no certain simulation numbers for these experiments here, we cease simulation once we find out critical value of clocking misalignment angle according to the following simulation method. But for it to be declared successful, the threshold value of magnetization polarization is still 0.8. The experimental simulations are conducted by increasing aspect ratio of the nanomagnet in increment 0.1 (horizontal axis) and misalignment angle (vertical axis). The behaviors of pipelined MQCA architectures are plotted for various aspect ratios and clocking misalignment angles at two non-zero temperatures shown in Fig. 7, where region above every curve indicates that corresponding structure has an unsuccessful operation (FAILURE) when clocking misalignment occurs, while region below every curve indicates that the structure has a successful operation (SUCCESS). Note that ‘Corner(zone 2)’ indicates a clocking
X. Yang et al. / Microelectronics Journal 43 (2012) 386–392
391
misalignment happens to the clocking zone 2 in the corner architecture. From Fig. 7, we can see four important points.
E () 240 kT
The first is that as the aspect ratio of nanomagnet increases, all
the pipelined MQCA architectures tolerate decreased misalignment angle and their performances deteriorate. In other words, the operation of defective clocking array significantly depends on nanomagnet size. The second is that the ‘Corner(zone 2)’ demonstrates worse function than the ‘Corner(zone 3)’. In fact, when the aspect ratio of nanomagnets in the corner decreases to 0.8, clocking defect occurring in the zone 3 can tolerate larger misalignment angle than that occurring in the zone 2. We think the difference comes from the way that the corner architecture is arranged. Note that the zone 2 in the corner represents an antiferromagnetic coupling (AF coupling), while the zone 3 in the corner represents a ferromagnetic coupling (F coupling). Generally speaking, clocking misalignment toward northeast direction favors binary ‘1’, while misalignment toward southeast direction favors binary ‘0’. In the AF coupling case, misalignment of the applied clocking field direction does not favor a particular logic state since binary ‘0’ and binary ‘1’ appear by turns. Thus the corner architecture is prone to fail when the zone 2 subjects to a misalignment defect. Conversely, in the F coupling case, the frustrated signal propagation arised by the clocking misalignment direction may accord with the expected transferring logic value, and the information transferring can be forced out of local sink, frustration is thus diminished. The third is that square-like nanomagnet (aspect ratio is 1) almost shows a zero tolerance to clocking misalignment for all the architectures. In fact, for larger aspect ratio value, the smaller nanomagnet coercive field and the clocking misalignment direction decrease the nanomagnet’s energy barrier to resist environmental or neighboring scattering effect, leading to unsuccessful operation of MQCA architecture. From this, we can conclude that circular nanomagnet with no barrier separating the ‘up’ and ‘down’ states does be more sensitive to misalignment than rectangle nanomagnet. Finally, room temperature case denotes smaller success region than 100 K case. Thus as the temperature increases, the operations of pipelined MQCA architectures are more unreliable.
Overall, these results are helpful for the fabrication scientists to guide the fabrication processes of MQCA architectures and three phases pipelined clocking.
5. Discussions In this section, we qualitatively analyze why clocking misalignment may cause pipelined MQCA architectures fail and provide theoretical explanation to preceding simulation results. The unreliable switching of the MQCA architectures with defective pipelined clocking can be explained by examining nanomagnets energy barrier profile. Take the wire as an example, the schematic of two nanomagnets in the neighboring zone is shown in the inset of Fig. 8. j is the in-plane magnetization angle relative to the easy axis of the left fixed nanomagnet and the two nanomagnets have the same size and spacing as that used before. The energy landscape of total magnetostatic energy of a system of two coupled single-domain nanomagnets containing fourfold magnetocrystalline anisotropy [25] is shown Fig. 8. The ‘dip’ in the center of the energy landscape represents the case when j is 01, which comes from the position of the first nanomagnet (zone 1) and the second nanomagnet (zone 2). From Fig. 8, one can see that the two nanomagnets experience three critical energy stages in the switching
A •
•B
•C
120 kT
−
−/2
0
/2
ϕ
Fig. 8. Total magnetostatic energy landscape of two interactional nanomagnets.
process, which corresponds to three energy barrier states. The first critical energy state appears when j is 01, now the two nanomagnets stay in a parallel state (metastable state). The second critical energy state appears when j is 901 (highest energy barrier), it shows an unstable manner of the two nanomagnets. The third critical energy state (stable state) appears when the two nanomagnets display in an antiparallel phenomenon (j is 1801). Normally, three phases pipelined clocking is accurately placed along the hard axis in a special sequence, three nanomagnets in one zone, for example zone 2, would stand in the point A of the energy landscape at the end of switching. After the clocking is removed, all the three nanomagnets would be re-evaluated to a correct antiparallel state driving by the left input. However, now a significant clocking misalignment happens to clocking zone 2, namely the direction of magnetic field points to southeast or northeast direction. Consequently, the nanomagnets in the zone 2 would be magnetized toward one of these two directions, the final energy state between the last nanomagnet in the zone 1 and the first nanomagnet in the zone 2 would not be centered at point A (901) and has an excursion with ideal magnetization direction, corresponding to point B or point C. If the energy state stays in point C after the clocking is removed, the nanomagnet in the zone 1 would magnetize the first nanomagnet in the zone 2 to an antiparallel arrangement as expected. This is also called the self-correction of pipelined clocking since one clocking zone only manages three nanomagnet, However, if the clocking misalignment makes the energy state stay in point B, the interaction from the last nanomagnet in the zone 1 cannot make the first nanomagnet in the zone 2 to move past the middle critical point A, then it would return the first critical energy state (j is 01). Moreover, the last nanomagnet in the zone 2 would also be flipped by the first nanomagnet in the zone 3, so the frustrated switching occurs. Overall, clocking misalignment would introduce signal blockade or error into pipelined MQCA architectures. In order to describe the failure process clearly, magnetization evolution image of pipelined MQCA wire with defective clocking by OOMMF tool [26] is shown in Fig. 9. Fig. 9(a) denotes an initial state. In Fig. 9(b), new input (logic ‘1’) has been written into the wire, now a northeast direction clocking misalignment occurs in the zone 2. In Fig. 9(c), the error has been formed due to misalignment effect since the fourth nanomagnet should be logic ‘0’ according to the new input in Fig. 9(b). At last, we obtain wrong logic ‘0’, not expected wire function and logic ‘1’ output (see two dashed ellipses). From this image, we can see that how the signal
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Shaanxi Provincial Natural Science for Basic Research (Grant no. 2011JZ015). References
Fig. 9. Magnetization evolution image of frustrated nanomagnets wire by OOMMF simulation due to clocking misalignment.
propagation fails and the MQCA architecture demonstrates an obvious error under defective clocking.
6. Conclusions Summarizing, we present three phases pipelined clocking with three MQCA-cell width and deduce a computation model of clocking misalignment to perform a systematic evaluation of the impacts of clocking misalignment on four typical pipelined MQCA architectures. It is found that clocking misalignment introduces signal blockade into the pipelined MQCA architectures and put some limitations on the nanomagnet size and operational temperature. In detail, the majority logic gate tolerates the biggest misalignment angle, while the corner tolerates the smallest misalignment angle for the fixed nanomagnet size. The results also show that as the aspect ratio of nanomagnet increases, the ability that all the MQCA architectures tolerate clocking misalignment decreases. Moreover, we find out that various energy barrier difference accounts for the failure of pipelined MQCA architectures. In general, the results presented in this paper would develop an important guide for how to improve pipelined MQCA architectures defect tolerance. Overall, clocking misalignment is a ‘show stopper’ for pipelined MQCA architectures. However, the error rates are manageable. Finally, it should be pointed out that very recently there is a nanomagnet logic device variant that are completely insensitive to misalignment [27]. But unlike the MQCA device presented in this paper is well established, application and practicability of this device variant still needs deep investigation.
Acknowledgment We acknowledge the support of the National Natural Science Foundation of China (Grant no. 61172043) and the Key Program of
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Contents lists available at SciVerse ScienceDirect
Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
Clocking misalignment tolerance of pipelined magnetic QCA architectures Xiaokuo Yang a,n, Li Cai a, Qiang Kang b, Xiaohui Zhao a a b
College of Science, Air Force Engineering University, Xi’an 710051, People’s Republic of China Department of Science Research, Air Force Engineering University, Xi’an 710051, People’s Republic of China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 December 2011 Received in revised form 18 February 2012 Accepted 28 February 2012 Available online 17 March 2012
In this paper, we present a simulation study on clocking misalignment tolerance of pipelined magnetic quantum-dot cellular automata (MQCA) architectures. By the three-phase pipelined clocking and deduced clocking misalignment model, a systematic evaluation of impacts of clocking misalignment on four fundamental MQCA architectures is performed at non-zero temperatures. It is found that for the fixed nanomagnet size, majority logic gate is the most reliable structure, while the corner is most susceptible to clocking misalignment. High temperature gives rise to a negative effect on allowable misalignment angles. The results also show that as the aspect ratio of nanomagnet increases, the ability that all the MQCA architectures tolerate clocking misalignment decreases. Moreover, we analyze potential reason of pipelined MQCA architecture failures by examining the energy profile of neighboring zone nanomagnets and conclude that various energy barrier difference accounts for failure of MQCA architectures under clocking misalignment defect. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Clocking misalignment Nanomagnet Magnetic quantum-dot cellular automata Aspect ratio Energy barrier
1. Introduction In traditional electronics, information is usually stored and processed using considerable electron charge, however, chargebased transistors are facing a worrisome power dissipation problem with scaling feature size [1]. Consequently, interest is very high in other logic technologies whose functionality does not rely on controlling vast electron charge. One promising technology among them is quantum-dot cellular automata (QCA) [2,3]. A magnetic implementation of the QCA devices (MQCA) was originally proposed by Cowburn and Welland [4], which has seen significant progress [5–8]. MQCA employs rectangular shapebased nanomagnet to build the device, which reaches the lowest energy state when the magnetization direction is aligned along the easy axis of a nanomagnet. Depending on the magnetization forces around elongated nanomagnet, its magnetization direction will point straightly up or down, corresponding to binary value ‘1’ or ‘0’, respectively. Recently, MQCA wire and majority logic gate have been fabricated at room temperature [5]. To date, some works have been done on non-pipelined MQCA interconnect wire, inverter and majority logic gate architectures to study their operation frequency, switching behavior and power gain and dissipation [9–13]. Moreover, switching characteristics of interconnect architecture with defective nanomagnets has also been investigated [14,15]. These theoretic researches continue to mature the MQCA concept and structure. However, since most of MQCA architectures are operated by hard axis magnetic field clocking
n
Corresponding author. E-mail address: [email protected] (X. Yang).
0026-2692/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2012.02.005
which erases logic state and re-evaluates magnetic ordering, during the re-evaluating process nanomagnets are very vulnerable to hard axis clocking misalignment: if magnetic field clocking departs away the hard axis direction (clocking misalignment) of nanomagnet, the magnetic ordering of MQCA architectures may fail. Clocking misalignment of MQCA device is succinctly discussed in Ref. [6]. Nanomagnet with a slanted edge subjected to non-pipelined clocking misalignment has also been explored [16]. However, these works are very rough to understand clocking misalignment issues of MQCA architectures, because some important aspects have not been considered or solved there, such as pipelined clocking misalignment, temperature and nanomagnet size effect. In the present work, we focus on clocking misalignment issue of pipelined MQCA architectures with detailed simulations and indepth consideration (size and temperature, etc.) of a few typical cases. We also further study the failure reason and mechanism of pipelining-clocked structures subjected to clocking misalignment. After presenting three MQCA-cell wide pipelined clocking and deducing the computational model, we investigate by two-state (‘0’ and ‘1’) system [17] and numerical method the impacts of clocking misalignment on the operation and defect tolerance of coupled nanomagnet architectures. The goal of this work is to provide a guide to deeply understand clocking misalignment induced error formation in the pipelined MQCA architectures.
2. Clocking misalignment of pipelined MQCA architectures Elongated single domain nanomagnet is used to achieve bistability and build basic MQCA device due to shape-induced anisotropy. At zero external magnetic field, the remanent magnetization
X. Yang et al. / Microelectronics Journal 43 (2012) 386–392
of nanomagnet usually points along the long axis [18]. Thus, the long axis is viewed as the ‘easy’ axis, while the short axis as the ‘hard’ axis. All MQCA architectures need external torque to help their switching [7,8,13,19,20]; this external torque is viewed as MQCA clocking. Previously, a magnetic field-based global clocking methodology was proposed and experimentally demonstrated using copper clad wire embedded underneath the nanomagnets [7,8]. One salient advantage of this global clocking is the simplicity and reduced lithographic burden because all the nanomagnets are clocked simultaneously with only a clocking circuitry. However, global clocking makes MQCA architectures non-pipelined and very slow. More serious is that the global clocking would aggravate destroying the whole point of nanomagnet logic if clocking field misalignment occurs. Recently, Refs. [15,16,19,20] investigate possible MQCA local clocking to obtain pipelined switching, such as two phases, three phases and four phases clocking. In the three phases (not clocked, null and switch) pipelined clocking [20] shown in Fig. 1, three MQCA cells are encapsulated in a clocking zone and clocking signal repeats every three zones. Here we introduce a gradually changing ‘switch’ field to realize stable switching. The detail information about three phases pipelined clocking can be found from Ref. [20]. Based on three phases pipelined clocking signals, all the investigated MQCA architectures are constructed. The sevennanomagnet wire is shown in Figs. 2(a), and (b) presents the six-nanomagnet inverter architecture. Moreover, to form more computation architecture, MQCA gates and wires need to be
connected. Here we consider the architecture of having a horizontal line of five nanomagnets drive a signal to a vertical line of three nanomagnets, namely MQCA corner shown in Fig. 2(c). The nanomagnets forming the majority logic gate are arranged as seen in Fig. 2(d). Three phases pipelined clocking are adopted in these MQCA architectures. The precondition for error-free operation of MQCA architectures lies in two aspects. One is to place nanomagnets in the accurate position using advanced lithography technology. The other is to generate accurate hard-axis direction clocking to clock the nanomagnets, which also requires the accuracy of current carrying wire placement. However, fabrication defects in the nanotechnology are expected to be quite high. The nanomagnet and clocking could be misaligned. Usually, there exist two phenomenas causing clocking misalignment in the fabrication of MQCA architectures.
The clocking field (zeeman field) is not exactly aligned along the
hard axis of the nanomagnet, but subtends an angle with it [5]. Here, we show the case of four nanomagnets where magnetic field clocking is oriented in a small subtended angle with hard axis direction in the lithography (real line arrowhead shown in Fig. 3(a) represents fabricated clocking field direction, while broken line arrowhead represents the hard axis direction of nanomagnets). Although the clocking field is exactly aligned along the hard axis of many nanomagnets, but some nanomagnets in the architecture are not deposited in the proper position during lithography
not clocked 1
387
switch
Zone 1 null
2
Zone 2
3
Zone 3
Zone 1
Zone 2
Zone 3
1
2
3
t Fig. 1. Three phases pipelined clocking of MQCA architectures. (a) Operation process; (b) Pipelined clocking signals.
out
Fig. 2. Investigated MQCA architectures and their clocking zones distribution. (a) Wire; (b) Inverter; (c) Corner; (d) Majority logic gate.
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y w
Hzeeman
j •
i •
h Hzeeman
l Hzeeman
k •
s
x
z
Fig. 4. Description of clocking misalignment in the pipelined MQCA array. Fig. 3. Characterizations of clocking misalignment. (a) Misalignment of zeeman field direction; (b) Clocking misalignment caused by nanomagnet deflection or rotation.
technology phase [6]. As seen from Fig. 3(b), the second nanomagnet in the array has been rotated, which can be understood from the case that nanomagnet become irregular in the manufacturing. Now the clocking misalignment defect still appears. Therefore, in magnetic field-based clocking scheme, misalignment is a very real and major issue. As we discussed before, global clocking field should be avoided because it is prone to cause circuits fail when the MQCA architectures suffer from clocking misalignment. Pipelined local clocking signal mitigates misalignment problem, hence it is more practical for MQCA technology. However, in order to develop a viable alternative technology to charge-based transistors, clocking misalignment effect and defect tolerance in the pipelining-clocked MQCA architectures must also be understood, especially when the nanomagnet size and temperature vary. Hence whether clocking misalignment and cell geometry would remarkably alter the performance of pipelined MQCA architectures and to which extent will be taken into account.
thermal magnetic field HT ðtÞ. HT ðtÞ accounts for the interaction effects of the magnetization with phonons and conducting electrons which cause magnetization fluctuations. After adding the thermal torque, we get new effective field X Heff ðtÞ ¼ Hzeeman ðtÞNM þ CM þHT ðtÞ ð2Þ where N is the demagnetization tensor (NM denotes the demagnetizing field), which takes into account three dimension sizes of a nanomagnet. For a cube nanomagnet considered in the paper, it is a diagonal matrix and the elements obey the relation Nx þ N y þ Nz ¼ 1. C is the coupling matrix between neighboring P nanomagnets and CM denotes the coupling field. Hzeeman represents applied external field, namely clocking in the MQCA architectures. Assume that (x(i), y(i), z(i)) are the coordinates of nanomagnet i and (x(j), y(j), z(j)) are the coordinates of nanomagnet j, then we can obtain the coordinate differences for these two nanomagnets as follows: ðjiÞ
ðjÞ
ðiÞ
ð3:aÞ
ðjiÞ
ðjÞ
ðiÞ
ð3:bÞ
ðjiÞ
ðjÞ
ðiÞ
dx ¼ dx dx ¼ l þs dy ¼ dy dy ¼ 0 dz ¼ dz dz ¼ 0
ð3:cÞ (i)
3. Modeling clocking misalignment of pipelined MQCA architectures Our first step will be to identify magnetization process of MQCA architectures undergoing defective pipelined clocking. We model the effects of clocking misalignment on the operation of MQCA architectures and focus on the clocking misalignment tolerance under various misalignment angles, nanomagnet sizes and temperatures. As pointed out in the Section 2, we adopt proposed three-nanomagnet-zone pipelined clocking in the array. The schematic diagram of such a six nanomagnets array subject to clocking misalignment (Fig. 3a) is shown in Fig. 4. To model clocking misalignment in the pipelined MQCA architectures, we need to start from basic Landau and Lifshitz equation [21], which in the Gilbert form reads dM a dM ¼ gM Heff þ ð1Þ M dt dt Ms where M¼(Mx, My, Mz) is time-related three dimension magnetization of a nanomagnet, Ms is saturation magnetization, while Heff is the effective field experienced by the magnetic moment, and it is the sum of externally applied field (zeeman field), field originating from nanomagnet itself (demagnetizing field), and coupling from nanomagnet’s neighbors. g ¼2.21 105 m/A s, and a is the Gilbert damping parameter. It is very important to note that effective numerical simulation result using this MQCA model has been reported [22]. Thermal torque is specifically taken into account in this work and it is introduced into the effective field Heff by a stochastic
In the mean time, assume V is the volume of nanomagnet i and the distance vector describing nanomagnets i and j is (ji) (ji) d(ji) ¼(d(ji) x ,dy ,dz ), then the third term of Eq. (2) can be calculated by CðjiÞ MðiÞ ¼
V ðiÞ 4pðd Þ5 ðjiÞ
ð3ðd ÞT d d IÞMðiÞ ðjiÞ
0 0 ¼
¼
lhw 4pðl þsÞ5
lhw 4pðl þsÞ4
ðjiÞ
0
ðjiÞ
1
B B C @3@ l þ s Að 0 l þ s 0 0 1 MðiÞ x B C B ð3l þ 3s1ÞM ðiÞ C y A @
10 M ðiÞ 1 x B ðiÞ C C 0 Þðl þ sÞIC AB @ My A ðiÞ Mz
ð4Þ
MðiÞ z
where l, h and w denote width, height and thickness of a nanomagnet, respectively, s represents nanomagnet spacing (see Fig. 4), I is the identity matrix. Note that we simplify the expression of CðjiÞ by the established coupling matrix with dipole-approximation [22]. Now we have presented the coupling energy calculation method of two nanomagnets arranged in a horizontal style like that appeared in Fig. 1. However, if nanomagnet i resides in the upside or downside of driven nanomagnet j, we can adopt similar method to calculate the vector CðjiÞ . As discussed before, magnetic field clocking may not be exactly aligned with the hard axis of a nanomagnet in the fabrication, but subtends an angle b with it. We call b misalignment angle (see Fig. 4). Now clocking field magnitude along hard axis direction is given by Hh ¼ Hzeeman ðtÞcos b, and the clocking
X. Yang et al. / Microelectronics Journal 43 (2012) 386–392
field magnitude along the easy axis direction is given by He ¼ Hzeeman ðtÞsin b. Consequently, the clocking vector has been changed from Hzeeman(t)¼[Hzeeman(t) 0 0] into Hzeeman ðtÞ ¼ ½Hh He 0
ð5Þ
Temperature fluctuation-induced torque [23] is obtained by the expression 1 HT ðtÞ ¼ pffiffiffiffiffiffiffiffiffi V Dt
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kB T a gðtÞ m0 gMs
ð6Þ
where g(t) is a Gaussian-distributed random vector, V is nanomagnet volume and Dt is the fixed time step used to solve the Landau–Lifshitz–Gilbert (LLG) equation. From the deduced computation model above, only the first term in the right hand side of Eq. (2) would be determined by clocking misalignment angle. After modeling the clocking misalignment, Eqs. (1)–(6) allow us to compute temporal evolution of magnetization vector of any nanomagnet in the pipeliningclocked MQCA architectures subject to clocking defect. The outputs magnetization polarization may vary with nanomagnet size, misalignment angle and temperature. To address their relations and interaction effects clearly, the following work will calculate how the cell geometry and defective pipelined clocking raise some impacts on the switching operation and signal propagation of MQCA architectures. In order to bring this work close to a fabrication reference, we specifically present a generation method of proposed pipelined clocking field. That is to adopt pulse current carrying Cu wire similar to that appeared in Ref. [8] to generate the magnetic field. A physically feasible clocking circuitry (comprised of Si substrate, oxide, carrying Cu wire and high permeability cladding, nanomagnets are placed on the top) to clock just 3 nanomagnets at a time is shown in Fig. 5. The clocking sequence is guaranteed by the phase of pulse current. This physical realization can ensure that clocking field is localized to three nanomagnets due to the following two reasons. One is that we use high permeability magnetic material to construct the cladding, which can effectively enhance and concentrate magnetic field on the three nanomagnets of one zone. The other is that the spacing between adjoining clocking circuitry is a bit large (yellow color), which greatly reduces, even vanishes the fringing fields. Moreover, although the dimensions of state-of-the-art carrying Cu wire are typically greater than the width of a nanomagnet device ( 50 nm), we fixed the width of clocking zone to three nanomagnets, this can be implemented in technology where the clocking wire diameter and the clocking wire pitch together do not exceed 3 50 nm ¼150 nm ( 200 nm if including spacing), which is feasible according to the International Technology Roadmap for Semiconductors [24]. At the same time, the horizontal oriented small nanomagnet (gray nanomagnet blocks in Fig. 2) serves as the input bias in all the pipelined MQCA architectures.
nanomagnet
carrying Cu wire
high permeability cladding oxide Si substrate Fig. 5. Schematic of a physically feasible three phases clocking circuitry.
389
4. Results and analysis In this section, the behaviors of four pipelined MQCA logic architecture under various misalignment angle and nanomagnet size are studied by applying the model presented in Section 3. In the simulations, a single clocking zone (except the input zone) misalignment defect is injected into every MQCA architectures, then we use fourth-order Runge–Kutta method to solve the system of coupled differential equations in Eq. (1) for the magnetic coupled nanomagnet arrays. For all the trials, nanomagnet spacing (horizontal and vertical direction) is chosen to be 15 nm—an optimal value [22] and the misalignment angle ranges from 01 to 51. The choice of this angle range is based on Refs. [6,16] where are two kinds of misalignment angles (11 and 21) have been studied. Moreover, Ref. [6] had stated that misalignment angle larger than 11 (several degrees) usually occurs in the MQCA although very advanced lithography is adopted. Therefore our misalignment angle range is in particular supported by this work. We do non-zero temperature simulations in the investigations. (i) The first set of experiments examines success rate Psuccess with respect to the misalignment angle factor m when the nanomagnets appear with a constant aspect ratio. (ii) The second set of experiments investigates the relation between allowable misalignment angle and nanomagnet size or aspect ratio (width/height). The geometry size of supermalloy nanomagnet is chosen to be 50 80 25 nm3 (aspect ratio is 0.625) and the saturation magnetization of supermalloy is 8 105 A/m. We adopt 95 mT hard axis clocking field magnitude. The default of damping constant of 0.5 is used. For every calculation, we considered MQCA architecture subjected to defective clocking field to be successful if the absolute value of normalized output magnetization polarization is larger than 0.8 [20]. Otherwise, we considered the architecture fail. In fact, once normalized output magnetization M=M s is larger than 0.5, switching would be correct due to signal restoration and power gain of nanomagnet clocking. M/Ms ¼0.8 in the present work is a more rigorous condition. To conduct a statistics analysis for clocking misalignment angle, a uniform distribution U (01, 51) is applied to model technology process. Suppose that bv satisfies this probability distribution, then the misalignment angle b, fabrication induced defect, is defined by
b ¼ bi þ mbv
ð7Þ
In Eq. (7), bi represents ideal direction (01) of clocking field and
m denotes a random number serially chosen in the interval (0,1), the latter can be used to characterize the amplitude of clocking misalignment angle spreading. In the present investigations, we use m as the misalignment angle factor and study the relation between successful operation and factor m. For each clocking zone at one m value, 500 trials (a batch of 500 misalignment angles satisfying uniform distribution) are used for the simulations of pipelined MQCA architectures. For single input devices (wire, inverter and corner), logic ‘0’ and ‘1’ are both used for one time, while for the majority logic gate, its eight input sequences are applied for the calculations. Numerical simulations for fabrication induced clocking misalignment defect are carried out and the success probabilities Psuccess are plotted as a function of the misalignment angle factor m for different MQCA architectures at two non-zero temperatures. Fig. 6 shows the success probabilities of four typical pipelined MQCA architectures with uniform distribution misalignment angle. We see that the operations of all the MQCA arrays are affected by clocking misalignment defect, and increasing
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1
1 0.9
T = 100K
0.8
0.8
0.7
0.7
(A) (A) Majority Gate (B) Inverter
0.6
(B) (C) (D)
(C) Wire
0.5
Psuccess
Psuccess
0.9
T = 300K
(D)
0.6 (A) Majority Gate (B) Inverter
0.5
(D) Corner
0.4
(C) Wire
0.4
0.3
(A) (B) (C)
(D) Corner
0.3
0
0.2
0.4 0.6 Misalignment factor μ
1
0.8
0
0.2
0.4 0.6 Misalignment factor μ
0.8
1
Fig. 6. Success probability as a function of misalignment angle factor m for different pipelined MQCA architectures. (a) T¼ 100K; (b) T¼ 300 K.
5
5
4.5
4.5 Wire Inverter
3.5
Corner(zone 3)
3
Majority gate
2.5 2
FAILURE
1.5 1
Inverter
3.5
Corner(zone 2)
T = 300K
Corner(zone 3)
3
Majority gate
2.5 2 FAILURE
1.5 1
SUCCESS
0.5
0.5 0
Wire
4
T = 100K
Corner(zone 2)
Misalignment Angle ( °)
Misalignment Angle ( °)
4
1
0.9
0.8
0.7 Aspect Ratio
0.6
0.5
0.4
0
SUCCESS
1
0.9
0.8
0.7 Aspect Ratio
0.6
0.5
0.4
Fig. 7. Simulated behavior of pipelined MQCA architectures at various misalignment angles and nanomagnet sizes. (a) T ¼100 K; (b) T¼300K.
temperature leads to a more unstable switching. However, the overall trends of all the plots are similar. Note the slight oscillations in the curve originate from the distribution interpolation technique and should be disregarded. The breakdowns of success probabilities for different MQCA architectures (do not show a success probability ‘1’) occur at different misalignment factors. All the detailed observations or conclusions can be drawn from the plots analysis. The success probability of all the pipelined MQCA architectures decreases with the large spreading (large m value) of clocking misalignment angle. From statistics analysis results, we find that in case of clocking misalignment the majority logic gate is the strongest (most reliable) architecture, while the corner is the weakest architecture at any thermal temperature if measured by the smallest misalignment factor that impacts success probabilities. For the wire and inverter architectures, the breakdowns of the success probabilities at T¼100 K are almost same (m ¼0.61), but at room temperature the inverter demonstrate a bit better performance than the wire. Regardless of misalignment factor, each MQCA architecture denotes a final fixed success probability value. Overall, it is evident that a misalignment error is a major threat to the pipelined MQCA architectures, especially for the clocking misalignment of corner at room temperature. Nevertheless, to our comfort, pipelined MQCA architectures can still tolerate moderate clocking misalignment, for example, the
majority logic gate demonstrates a complete correct operation for about three degrees misalignment at room temperature. For a detailed and comprehensive understanding of nanomagnet size and clocking misalignment impacts on the performance of four typical pipelined MQCA architectures, assume that defective clocking and nanomagnet size varies simultaneously, and then we can study the relation between MQCA size and allowable misalignment angle. For clarity, the aspect ratio of nanomagnet is used to describe variability of nanomagnet size. It should be pointed out that there is no certain simulation numbers for these experiments here, we cease simulation once we find out critical value of clocking misalignment angle according to the following simulation method. But for it to be declared successful, the threshold value of magnetization polarization is still 0.8. The experimental simulations are conducted by increasing aspect ratio of the nanomagnet in increment 0.1 (horizontal axis) and misalignment angle (vertical axis). The behaviors of pipelined MQCA architectures are plotted for various aspect ratios and clocking misalignment angles at two non-zero temperatures shown in Fig. 7, where region above every curve indicates that corresponding structure has an unsuccessful operation (FAILURE) when clocking misalignment occurs, while region below every curve indicates that the structure has a successful operation (SUCCESS). Note that ‘Corner(zone 2)’ indicates a clocking
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misalignment happens to the clocking zone 2 in the corner architecture. From Fig. 7, we can see four important points.
E () 240 kT
The first is that as the aspect ratio of nanomagnet increases, all
the pipelined MQCA architectures tolerate decreased misalignment angle and their performances deteriorate. In other words, the operation of defective clocking array significantly depends on nanomagnet size. The second is that the ‘Corner(zone 2)’ demonstrates worse function than the ‘Corner(zone 3)’. In fact, when the aspect ratio of nanomagnets in the corner decreases to 0.8, clocking defect occurring in the zone 3 can tolerate larger misalignment angle than that occurring in the zone 2. We think the difference comes from the way that the corner architecture is arranged. Note that the zone 2 in the corner represents an antiferromagnetic coupling (AF coupling), while the zone 3 in the corner represents a ferromagnetic coupling (F coupling). Generally speaking, clocking misalignment toward northeast direction favors binary ‘1’, while misalignment toward southeast direction favors binary ‘0’. In the AF coupling case, misalignment of the applied clocking field direction does not favor a particular logic state since binary ‘0’ and binary ‘1’ appear by turns. Thus the corner architecture is prone to fail when the zone 2 subjects to a misalignment defect. Conversely, in the F coupling case, the frustrated signal propagation arised by the clocking misalignment direction may accord with the expected transferring logic value, and the information transferring can be forced out of local sink, frustration is thus diminished. The third is that square-like nanomagnet (aspect ratio is 1) almost shows a zero tolerance to clocking misalignment for all the architectures. In fact, for larger aspect ratio value, the smaller nanomagnet coercive field and the clocking misalignment direction decrease the nanomagnet’s energy barrier to resist environmental or neighboring scattering effect, leading to unsuccessful operation of MQCA architecture. From this, we can conclude that circular nanomagnet with no barrier separating the ‘up’ and ‘down’ states does be more sensitive to misalignment than rectangle nanomagnet. Finally, room temperature case denotes smaller success region than 100 K case. Thus as the temperature increases, the operations of pipelined MQCA architectures are more unreliable.
Overall, these results are helpful for the fabrication scientists to guide the fabrication processes of MQCA architectures and three phases pipelined clocking.
5. Discussions In this section, we qualitatively analyze why clocking misalignment may cause pipelined MQCA architectures fail and provide theoretical explanation to preceding simulation results. The unreliable switching of the MQCA architectures with defective pipelined clocking can be explained by examining nanomagnets energy barrier profile. Take the wire as an example, the schematic of two nanomagnets in the neighboring zone is shown in the inset of Fig. 8. j is the in-plane magnetization angle relative to the easy axis of the left fixed nanomagnet and the two nanomagnets have the same size and spacing as that used before. The energy landscape of total magnetostatic energy of a system of two coupled single-domain nanomagnets containing fourfold magnetocrystalline anisotropy [25] is shown Fig. 8. The ‘dip’ in the center of the energy landscape represents the case when j is 01, which comes from the position of the first nanomagnet (zone 1) and the second nanomagnet (zone 2). From Fig. 8, one can see that the two nanomagnets experience three critical energy stages in the switching
A •
•B
•C
120 kT
−
−/2
0
/2
ϕ
Fig. 8. Total magnetostatic energy landscape of two interactional nanomagnets.
process, which corresponds to three energy barrier states. The first critical energy state appears when j is 01, now the two nanomagnets stay in a parallel state (metastable state). The second critical energy state appears when j is 901 (highest energy barrier), it shows an unstable manner of the two nanomagnets. The third critical energy state (stable state) appears when the two nanomagnets display in an antiparallel phenomenon (j is 1801). Normally, three phases pipelined clocking is accurately placed along the hard axis in a special sequence, three nanomagnets in one zone, for example zone 2, would stand in the point A of the energy landscape at the end of switching. After the clocking is removed, all the three nanomagnets would be re-evaluated to a correct antiparallel state driving by the left input. However, now a significant clocking misalignment happens to clocking zone 2, namely the direction of magnetic field points to southeast or northeast direction. Consequently, the nanomagnets in the zone 2 would be magnetized toward one of these two directions, the final energy state between the last nanomagnet in the zone 1 and the first nanomagnet in the zone 2 would not be centered at point A (901) and has an excursion with ideal magnetization direction, corresponding to point B or point C. If the energy state stays in point C after the clocking is removed, the nanomagnet in the zone 1 would magnetize the first nanomagnet in the zone 2 to an antiparallel arrangement as expected. This is also called the self-correction of pipelined clocking since one clocking zone only manages three nanomagnet, However, if the clocking misalignment makes the energy state stay in point B, the interaction from the last nanomagnet in the zone 1 cannot make the first nanomagnet in the zone 2 to move past the middle critical point A, then it would return the first critical energy state (j is 01). Moreover, the last nanomagnet in the zone 2 would also be flipped by the first nanomagnet in the zone 3, so the frustrated switching occurs. Overall, clocking misalignment would introduce signal blockade or error into pipelined MQCA architectures. In order to describe the failure process clearly, magnetization evolution image of pipelined MQCA wire with defective clocking by OOMMF tool [26] is shown in Fig. 9. Fig. 9(a) denotes an initial state. In Fig. 9(b), new input (logic ‘1’) has been written into the wire, now a northeast direction clocking misalignment occurs in the zone 2. In Fig. 9(c), the error has been formed due to misalignment effect since the fourth nanomagnet should be logic ‘0’ according to the new input in Fig. 9(b). At last, we obtain wrong logic ‘0’, not expected wire function and logic ‘1’ output (see two dashed ellipses). From this image, we can see that how the signal
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Shaanxi Provincial Natural Science for Basic Research (Grant no. 2011JZ015). References
Fig. 9. Magnetization evolution image of frustrated nanomagnets wire by OOMMF simulation due to clocking misalignment.
propagation fails and the MQCA architecture demonstrates an obvious error under defective clocking.
6. Conclusions Summarizing, we present three phases pipelined clocking with three MQCA-cell width and deduce a computation model of clocking misalignment to perform a systematic evaluation of the impacts of clocking misalignment on four typical pipelined MQCA architectures. It is found that clocking misalignment introduces signal blockade into the pipelined MQCA architectures and put some limitations on the nanomagnet size and operational temperature. In detail, the majority logic gate tolerates the biggest misalignment angle, while the corner tolerates the smallest misalignment angle for the fixed nanomagnet size. The results also show that as the aspect ratio of nanomagnet increases, the ability that all the MQCA architectures tolerate clocking misalignment decreases. Moreover, we find out that various energy barrier difference accounts for the failure of pipelined MQCA architectures. In general, the results presented in this paper would develop an important guide for how to improve pipelined MQCA architectures defect tolerance. Overall, clocking misalignment is a ‘show stopper’ for pipelined MQCA architectures. However, the error rates are manageable. Finally, it should be pointed out that very recently there is a nanomagnet logic device variant that are completely insensitive to misalignment [27]. But unlike the MQCA device presented in this paper is well established, application and practicability of this device variant still needs deep investigation.
Acknowledgment We acknowledge the support of the National Natural Science Foundation of China (Grant no. 61172043) and the Key Program of
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