Abstract
In the quest for novel, scalable and energyefficient computing technologies, many noncharge based logic devices are being explored. Recent advances in multiferroic materials have paved the way for electric field induced low energy and fast switching of nanomagnets using the magnetoelectric (ME) effect. In this paper, we propose a voltage driven logicdevice based on the ME induced switching of nanomagnets. We further demonstrate that the proposed logicdevice, which exhibits decoupled read and write paths, can be used to construct a complete logic family including XNOR, NAND and NOR gates. The proposed logic family shows good scalability with a quadratic dependence of switching energy with respect to the switching voltage. Further, the proposed logicdevice has better robustness against the effect of thermal noise as compared to the conventional current driven switching of nanomagnets. A devicetocircuit level coupled simulation framework, including magnetization dynamics and electron transport model, has been developed for analyzing the present proposal. Using our simulation framework, we present energy and delay results for the proposed MagnetoElectric Spin Logic (MESL) gates.
Introduction
CMOS technology has been the driving force behind the ever improving computing efficiency for the past few decades^{1,2}. However, as the miniaturization of CMOS devices continues, issues like the leakage power consumption, short channel effects, increased variability etc. have necessitated exploration of novel devices^{3,4}. Further, with the current emphasis on smart sensors and Internet of Things (IoT), low leakage nonvolatile computing devices have became more attractive than ever before. Such beyondCMOS logic devices are expected to augment/complement the existing CMOS technology^{5}.
Spin based logic devices are a promising candidate for beyondCMOS technologies due to 1) nonvolatility (ability to retain data in absence of power supply) and hence low leakage power consumption and 2) areaefficiency. As such, many proposals for logical operations using spin devices can be found in the literature. All spin logic (ASL)^{6,7} is one of the widely studied logic families based on nonlocal spin currents. The dependence of ASL logic on nonlocal spin currents presents a major drawback due to short spinflip lengths in metallic channels. On the other hand, nonvolatile logic based on magnetic tunnel junctions (MTJs) embedded within CMOS logic circuits were explored in ref. 8. In addition, ME based logic devices have been explored in refs 9,10. The ME logic presented in ref. 9 suffers from the requirement of a complex DC (direct current) bridge including three resistors for cascading. In ref. 10, an XOR device was proposed, however details of cascading and the complexity of the required cascading circuits is missing. Recently, a spin logic based on magnetoelectric switching and the Inverse Rashba Edelstein effect was proposed in ref. 11. In the present paper, we not only demonstrate that our proposed logicdevice can function as XNOR, NAND and NOR gate based on the configuration but also show that easy cascadability can be achieved by using minimal number of CMOS devices.
Specifically, we combine two scalable physics, 1) the switching of a ferromagnet through a multiferroic material using the ME effect and 2) the resistance change of an MTJ as a function of the magnetization directions of the constituting ferromagnets, to propose a nonvolatile cascadable MagnetoElectric Spin Logic (MESL). The key highlights of the present work are as follows:
 1
We exploit the inherent coupling of multiferroic materials with the underlying magnetization direction of a ferromagnet to achieve voltage driven low energy switching of nanomagnets. By stacking two such nanomagnets, in contact with respective ME oxides, we form an MTJ stack. We demonstrate that the resulting device can be used as a logicelement that can be used to implement complex Boolean functions.
 2
Using a coupled magnetization dynamics and electron transport simulation, we show that the proposed logicdevice exhibits good scalability, better robustness with respect to the influence of thermal noise and high switching speed as compared to the conventional current driven switching of nanomagnets.
 3
Realizations of two input XNOR, NAND and NOR gates, forming a complete logic family, has been demonstrated. Further, we show that the proposed MESL gates can be easily cascaded using a globalreset operation and dominostyle clocking.
 4
Typically CMOS logic family requires area expensive storage elements (for example, a flipflop circuit), in order to retain the output of the logic gates. Such flipflop circuits become redundant in the proposed MESL gates due to its inherent nonvolatility.
ME Effect and Proposed LogicDevice
ME effect is the physics of generating magnetization from an applied electric field^{12,13}. ME devices usually consists of a single phase or composite multiferroic material (for example BiFeO_{3}^{14}, BaTiO_{3}^{15}) in contact with a nanomagnet. Application of an electric field to the multiferroic material results in an effective magnetic field experienced by the nanomagnet. If the generated magnetic field is strong enough, the magnetization of the nanomagnet can be reversed. This electric field driven switching of the nanomagnets, shows better energy efficiency and speed, as compared to the classic current induced spintransfertorque switching^{16}.
In the case of a single phase ME oxide (like BiFeO_{3}), the switching of the nanomagnet due to applied electric field can be explained as follows^{14}. BiFeO_{3} is a ferroelectric material. Ferroelectricity in BiFeO_{3} arises due to the shift of Bi^{+} cations owing to its hybridization with the surrounding oxygen atoms^{17}. The electric polarization of BiFeO_{3}, which is coupled to the (anti) ferromagnetism of the constituent Fe atoms, can be switched by the application of an electric field. Further, the (anti) ferromagnetism of BiFeO_{3} can be coupled to the ferromagnetism of an underlying nanomagnet. The magnetization of the nanomagnet can be switched in response to the applied electric field across BiFeO_{3}^{14}. The various coupling mechanisms that lead to electricfield driven reversal of magnetization direction in the underlying ferromagnet is currently a topic of intense research^{14,15}.
Nevertheless, the efficiency of the ME effect is usually abstracted by the MEcoefficient denoted as α_{ME}^{14}. α_{ME} is the ratio of magnetic field generated per unit applied electric field. Experimentally, α_{ME} of 1 × 10^{−7} s m^{−1} has been reported in the literature^{14}. Schematically, an ME switched nanomagnet is shown in Fig. 1(a). When an electric field is applied in the +z direction, the nanomagnet switches to the +x direction due to the ME effect. If the direction of the applied electric field is reversed, the nanomagnet switches to −x direction.
Along side the ME switched magnet in Fig. 1(a), we also show a conventional MTJ stack consisting of an oxidespacer separating two nanomagnets. A parallel alignment of the magnetization directions in both the nanomagnets results in a low resistance state, while an antiparallel alignment leads to a high resistance state. The difference in the parallel and antiparallel resistance is usually indicated by a term called Tunnel MagnetoResistance (TMR) ratio. In order to maximize the TMR, MgO is usually used as the oxidespacer in the MTJ stack. The use of MgO as the oxidespacer can be justified from first principles analysis^{18}, which indicates that the coupling of the Bloch states between the nanomagnet and MgO has an important role in deciding the overall resistance of the MTJ stack.
Thus, we are dealing with two oxides  ME oxide for switching the nanomagnet and MgO for reading the state of the MTJ stack. In Fig. 1(b), we show the proposed device structure. It consists of two nanomagnets in contact with respective ME oxides. Due to their multiferroic nature, each of the ME oxides are coupled to the magnetization direction of the underlying nanomagnet. When a positive voltage is applied on the upper or the lower ME oxide, the corresponding nanomagnets switch to +x direction, while for a negative voltage the magnets point in the −x direction. The upper and the lower nanomagnets are separated by an MgO spacer to form an MTJ stack, thus constituting the fourterminal device structure.
The fourterminal nature of the proposed device leads to decoupled read and write paths. Terminals 1 and 2 can be used as the write terminals by applying proper voltage levels to switch the underlying nanomagnets. We assume the ME oxides are thick enough such that the tunneling current flowing through the ME oxides is small enough to be neglected. On the other hand, the state of the device can be read by passing a current (or applying a voltage) between terminals 3 and 4. The proposed logicdevice thus exhibits 1) low energy consumption due to electric field switching 2) decoupled read/write path such that respective oxides (ME oxide and MgO) can be optimized separately for read and write operations. In the next section, we describe how the proposed logicdevice of Fig. 1(b) can be used for constructing nonvolatile XNOR, NAND and NOR gates.
ME Logic Family and Cascadability
The proposed twoinput XNOR gate is shown in Fig. 2(a). The inputs to the XNOR gate are terminals ‘A’ and ‘B’. A positive voltage represents a digital ‘1’ and a negative voltage represents a digital ‘0’. If, both the inputs are the same, the two nanomagnets will either point in +x direction or in −x direction and the MTJ stack would be in the low resistance (parallel) state. The voltage divider consisting of the reference MTJ and the actual MTJ stack, will drive the output of the inverter high, if and only if the MTJ stack is in parallel (low resistance) state. Thus, an XNOR function can be implemented using the configuration shown in Fig. 2(a). It is to be noted that while the resistance of the MTJ is being sensed, the voltage divider effect results in a nonzero voltage on the upper ferromagnet denoted as node ‘T’ in Fig. 2(a). Since the upper ferromagnet constitutes one of the plates of the upper ME capacitor, the voltage at node ‘T’ might switch the direction of the upper ferromagnet. In order to avoid any inadvertent switching of the ferromagnet, the sensing voltage, the resistance of the reference MTJ and the trippoint of the inverter were selected such that the voltage at node ‘T’ is less than the minimum voltage required to switch the ferromagnet.
Next, we propose an ME NAND gate as illustrated in Fig. 2(b). The proposed NAND gate is composed of a series connection of two ME logicdevices. Each of the two series connected ME logicdevice consists of an ME oxide in contact with a nanomagnet and separated by a fixed magnet using MgO spacer. The two MTJ stacks, shown in Fig. 2(b), switch to the high resistance antiparallel state, only if the corresponding inputs are high. The output circuit forms a voltage divider as shown on the right hand part of Fig. 2(b). The ratio of the widths of PMOS and NMOS transistors in the output inverter are chosen such that the inverter output goes low if and only if both the MTJ stacks are in the high resistance (antiparallel) state (or in other words both the inputs ‘A’ and ‘B’ are high). Thus, the circuit in Fig. 2(b) implements a NAND function by proper selection of the transistor widths. Interestingly, the same circuit shown in Fig. 2(b) can also mimic the behavior of a NOR gate. For the NOR gate, the PMOS and NMOS transistors in the inverter are sized such that, output of the inverter goes high if and only if both the MTJ stacks are in low resistance state (or in other words only if both ‘A’ and ‘B’ are low).
Now that we have all the basic gates for computations, we would present the cascadability of our proposed logic gates. As an example, let us consider, cascading two MEXNOR gates. As shown in Fig. 3, the output of the first XNOR gate is connected directly to the input of the next XNOR gate. Initially, we do a reset operation by application of a negative voltage pulse so that all the magnets point in the −x direction. This can be achieved by applying a negative voltage on input terminals ‘A’, ‘B’ and ‘C’, as shown in the timing diagram of Fig. 3. Simultaneously, we pull the ‘G_{1}’ and ‘G_{2}’ terminals of the inverter to negative reset voltages, thus, driving the output of inverters to negative voltages. As such, a global reset can be achieved by simply applying a negative voltage on all the input and the intermediate terminals.
After the reset phase, terminals ‘G_{1}’ and ‘G_{2}’ are kept at zero volts for normal operation. Data inputs can now be applied on terminals ‘A’, ‘B’ and ‘C’. Based on the the inputs at ‘A’, ‘B’ and ‘C’ the input magnets would flip if required, making the MTJ stack either high or low resistance. We then apply, voltage pulses on nodes ‘V_{1}’ and ‘V_{2}’ one after another, in a typical domino style^{19}. When a voltage pulse is applied on node ‘V_{1}’, stage 1 (see Fig. 3) evaluates and the node ‘Out_{1}’ goes high or to zero volts. If ‘Out_{1}’ is high, the next stage magnet corresponding to terminal ‘Out_{1}’ switches to +xdirection. If, however, the inputs are such that node ‘Out_{1}’ remains at zero volts, the corresponding next stage magnet does not switch and stays in the desired −x direction. Once the first stage has evaluated, we apply a pulse on terminal ‘V_{2}’, stage2 evaluates and produces the desired output on ‘Out_{2}’.
Thus, easy cascadability, is achieved by use of the reset scheme and domino style clocking. Though, clocking is necessary for functioning of the proposed gates, it has been used in almost all nonvolatile spin logic to reduce leakage power consumption^{6,11}.
Modeling and Simulation
Next, we describe the modeling and simulation framework that was developed to evaluate the proposed ME gates. The device level simulation framework consisted of coupled magnetization dynamics and electron transport model. The required voltage and the time taken for deterministic switching of the nanomagnet due to the ME effect were obtained from stochasticmagnetization dynamics equations including thermal noise. On the other hand, the resistance of the MTJ stack in parallel and antiparallel state, as a function of the applied voltage was estimated using Nonequilibrium Greens function (NEGF) formalism^{20}.
Under monodomain approximation, the magnetization dynamics were modeled using the wellknow phenomenological equation called the LandauLifshizGilbert (LLG) equation^{21}. LLG equation can be written as ref. 22
where τ is . In (1), α is the Gilbert damping constant, γ is the gyromagnetic ratio, is the unit vector in the direction of the magnetization, t is time and H_{EFF} is the effective magnetic field. H_{EFF} can be written as
where is the demagnetization field due to shape anisotropy. is interfacial perpendicular anisotropy, is the stochastic field due to thermal noise and is the field due to ME effect.
can be written in SI units as ref. 23
where m_{x}, m_{y} and m_{z} are the magnetization moments in x, y and z directions respectively. N_{xx}, N_{yy} and N_{zz} are the demagnetization factors for a rectangular magnet estimated from analytical equations presented in ref. 24. M_{s} is the saturation magnetization. The interfacial anisotropy can be represented as ref. 25
where K_{i} is the effective energy density for interface perpendicular anisotropy and t_{FL} is thickness of the free layer. As mentioned earlier, the ME effect can be abstracted through the parameter α_{ME}^{11}
where, α_{ME} is the coefficient for ME effect, V_{ME} is the voltage applied across the terminals of the ME capacitor and t_{ME} is thickness of the ME oxide, responsible for induction of a magnetic field in response to an applied electric field. was multiplied with suitable constant for unit conversion.
The thermal field was included by the following stochastic equation^{26}
where is a vector with components that are zero mean Gaussian random variables with standard deviation of 1. Vol is the volume of the nanomagnet, T is ambient temperature, dt is simulation time step and k_{B} is Boltzmann’s constant.
Equations (1) and (6) constitute a set of stochastic differential equations. This system of equations was solved numerically by using the Heun’s method^{27}. The solution of the given set of equations, enable us to get the required voltage as well as the switching time for the nanomagnets used in our simulations. The various device dimensions and material parameters used in our simulations are summarized in Table 1. A typical evolution of the magnetization components in response to an voltage pulse is shown in Fig. 4(a).
It is to be noted that, multiferroics and ME effect is currently an active research area^{12,13}. Although many theoretical works have proposed models for describing the origin and the behavior of the ME effect, yet a detailed understanding of the physics of ME effect is still under intense research investigation. Further, experimental demonstration of ME switched ferromagnets can be found in the literature as in ref. 14, yet a global magnetization reversal by ME effect has remained elusive. Due to lack of such experimental results, the ME parameters mentioned in Table 1, are not tied to any particular material system or experiment, they are more like predictive parameters that abstract the details of the ME switching into a simple model which can be used in conjunction with the LLG equation to predict the switching time and energy. A more rigorous benchmarking of ME parameters for future logic devices can be found elsewhere as in ref. 28. Note, the functionality of the present proposal does not depend on the exact values of the ME parameters used for simulation. Therefore, our simple ME model serves the purpose to check the feasibility of our present proposal from device as well as circuits perspective.
The magnetization dynamics equations were coupled with the resistance of the MTJ stack, which was modeled using the nonequilibrium Green’s function (NEGF) formalism. A detailed description of the NEGF model for MTJs can be found in ref. 29. The parallel and antiparallel resistance obtained from our experimentally benchmarked NEGF equations were then abstracted into a behavioral model. The magnetization dynamics equations along with the resistance of the MTJ stack obtained from the NEGF equations, provided a coupled device model that can be used for characterizing the proposed logicdevice.
Device Characteristics
Any emerging logic technology must exhibit desirable characteristics with respect to scalability, speed of operation etc. Using our simulation framework presented in the previous section, we highlight some of the key characteristics of the logicdevice proposed in Fig. 1(b) – a) scalability b) stochastic switching dynamics and c) switching speed.
Scalability
The device in Fig. 1(b) shows good scaling in terms of voltage and energy requirements. This desirable scaling trend can be attributed to the decrease in the ME capacitance with the decrease in ME oxide area. In fact, the switching energy (proportional to CV^{2}, C being the ME oxide capacitance and V the voltage required to switch the nanomagnet) linearly decreases with the scaling in ME oxide area and has a square law dependence with respect to voltage scaling.
Additionally, the proposed device also requires an MTJ stack for a readout operation. Recently, many experimental works^{30,31} have demonstrated scaling of the MTJ structure to as small as 20 nm in diameter. Thus, in terms of areal dimensions of the ME oxide as well as the MTJ stack, the proposed device shows desirable scaling trend.
Stochasticity
The switching dynamics of a nanomagnet is known to be a stochastic process due to the random thermal field as per equation (6). The stochastic switching behavior of the nanomagnets have been exploited for random number generation^{32} and in neuromorphic applications^{33}. However, in the present scenario for Boolean logic gates, we need deterministic switching process. Thermal noise has a lesser detrimental effect on ME switched nanomagnets as opposed to the conventional Spin Transfer Torque (STT) switching mechanism. For STT switching, the switching process is initiated by the thermal field. STT effect dominates only when thermal noise has sufficiently disturbed the initial direction of the magnetization vector from its easy axis^{34}. On the other hand, ME switching does not require such an initiation of the switching process by the thermal field.
In Fig. 5(a), we have shown the distribution of the switching time for constant current in case of STT switching process and for constant voltage for ME switched nanomagnets. It can be observed that, the spread in distribution of switching time is much lesser for ME switching as compared to the STT switching dynamics. For a given time duration (500 ps), the switching probability versus the voltage across the ME oxide is shown in Fig. 5(b). Based on Fig. 5(b), we selected the operating voltage such that the switching probability is ~1.
Switching Speed
ME driven magnetization dynamics can lead to sub1ns switching speed as compared to the STT mechanism which typically requires 5–10 ns of switching time. For STT switching, the STT effect acts as an antidamping torque initially and as a damping torque subsequently thereby switching the nanomagnet. A typical STT switching curve is shown in Fig. 6(a). On the other hand, ME switching follows a much simpler dynamics, similar to the switching process due an external magnetic field, as shown in Fig. 6(b). For the material parameters shown in Table 1, the nanomagnets used in our simulations could switch within 500 ps. Thus, as compared to other nonvolatile logic devices based on injection of spin current and STT switching mechanism^{7}, the proposed logicdevice shows faster switching speed. In practice, the total switching time would be the sum of the switching time of the ferroelectric polarization in the ME oxide and the ferromagnetic switching estimated from our LLG equations. Theoretically, the ferroelectric switching time can be of the order ~70 ps^{35}, which is much less than the ferromagnetic switching time estimated from our simulations. We have therefore, neglected the ferroelectric switching time in our calculations.
Results and Discussions
The energy associated with a single gate, for example the XNOR gate of Fig. 2(a), can be estimated as follows. The total switching energy would consists of the energy to reset the two nanomagnets, the energy to switch the nanomagnets depending on the incoming data and the energy to turn ON the transistor M1 shown in Fig. 2(a).
where C_{ME} is the capacitance of the ME capacitor, C_{G} is the gate capacitance of transistor M1, V_{Reset} is the reset voltage, V_{Data} is the voltage on terminals ‘A’/‘B’ and V_{G} is the gate voltage for transistor M1. Using our simulation E_{Swi Total} was estimated to be 5.5 fJ. Similarly, the readout energy would consists of the energy associated with the voltage divider and the inverter. From our simulations, the readout energy was estimated to be 30 fJ, assuming a time duration of 500 ps.
Conclusions
Nonvolatile logic devices are of particular interest given the current emphasis on low power mobile devices, eventdriven sensing and Internet of Things. In the present work, we have exploited the ability to switch a ferromagnet using the ME effect, to propose a nonvolatile logicdevice. Besides its inherent nonvolatility, the present proposal achieves significant benefits in terms of switching energy of the ferromagnets due to the use of ME effect. Further, the proposed MESL gates can be easily cascaded to implement more complex Boolean functions. From a device perspective, the proposed logicdevice shows good scalability, better robustness to thermal fluctuations and high switching speed. We envisage that the proposed MESL gates could be a promising candidate for beyondCMOS low leakage logic devices.
Additional Information
How to cite this article: Jaiswal, A. and Roy, K. MESL: Proposal for a Nonvolatile Cascadable MagnetoElectric Spin Logic. Sci. Rep. 7, 39793; doi: 10.1038/srep39793 (2017).
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Acknowledgements
The work was supported in part by, Center for Spintronic Materials, Interfaces, and Novel Architectures (CSPIN), a MARCO and DARPA sponsored StarNet center, by the Semiconductor Research Corporation, the National Science Foundation, Intel Corporation and by the DoD Vannevar Bush Fellowship.
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A. Jaiswal wrote the paper and performed the simulations. K. Roy assisted in developing the concepts. All authors helped with the writing of the paper and discussing the results.
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Jaiswal, A., Roy, K. MESL: Proposal for a Nonvolatile Cascadable MagnetoElectric Spin Logic. Sci Rep 7, 39793 (2017). https://doi.org/10.1038/srep39793
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