Ultra-low-energy straintronics using multiferroic composites
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un ULTRA-LOW-ENERGY STRAINTRONICS USINGMULTIFERROIC COMPOSITES
KUNTAL ROY * Electrical and computer Engineering DepartmentVirginia Commonwealth University, RichmondVA 23284, [email protected]
This paper reviews the recent developments on building nanoelectronics for our future informa-tion processing paradigm using multiferroic composites. With appropriate choice of materials,when a tiny voltage of few tens of millivolts is applied across a multiferroic composite, i.e. a piezo-electric layer stain-coupled with a magnetostrictive layer, the piezoelectric layer gets strainedand the generated stress in the magnetostrictive layer switches the magnetization direction be-tween its two stable states. We particularly review the switching dynamics of magnetizationand calculation of associated metrics like switching delay and energy dissipation. Such voltage-induced magnetization switching mechanism dissipates a minuscule amount of energy of only ∼ Keywords : Nanoelectronics; energy-efficient design; spintronics; straintronics; multiferroics.
1. Introduction
The conventional charge-based electronics for morethan past fifty years has a history of great success1.However, the proven concept of enhancing the per-formance metrics by miniaturization2 of devices isapproaching its fundamental limits3; while there areissues due to process variation, basically the exces-sive energy dissipation in the devices limits the fur-ther improvement of transistor-based electronics4.The Nanoelectronics Research Initiative (NRI)5 atUnited States says “Future generations of electron-ics will be based on new devices and circuit architec-tures, operating on physical principles that cannotbe exploited by conventional transistors. NRI seeksthe next device that will propel computing beyondthe limitations of current technology.” The chal- lenge is to invent switching devices that dissipateminiscule amount of energy, e.g. ∼ ∗ Current affiliation: School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA.1
Kuntal Roy switching barrier between the states. It is widelybelieved that using “spin” as state variable is ad-vantageous over the charge-based counterpart. Un-fortunately, however, this advantage will be squan-dered if the method adopted to switch the spin is soenergy-inefficient that the energy dissipated in theswitching circuit far exceeds the energy dissipatedinside the switch. Regrettably, this is often the case,e.g. switching spins with a magnetic field6 ; ∼
100 nm) because of the competition be-tween the magnetostatic energy and the quantum-mechanical exchange energy, causing nanomagnetsto behave like single giant spins. These “giant”spins can beat superparamagnetic limit at room-temperature11, which is crucial for general-purposeinformation processing. The minimum energy dissi-pated to switch such a single-domain nanomagnet(a collection of M spins) can be only ∼ kT ln (1 /p ),where T is temperature and p is error probability,since the exchange interaction between spins makes M spins rotate together in unison like a giant clas-sical spin12 ;
13. On the contrary, the minimumenergy dissipated to switch a charge-based devicelike a transistor would be ∼ N kT ln (1 /p ), where N isthe number of information carriers. This gives nano-magnets an inherent advantage over the traditionaltransistors with regards to energy dissipation.In multiferroics , different ferroic orders suchas ferroelectric, ferromagnetic/ferrimagnetic, fer-roelastic etc. coexist. For our discussion, we willassume the coexistence of ferroelectric and ferro-magnetic orders to mean mutiferroism. Single-phasemultiferroic materials are rare and moreover themagnetoelectric responses of those is very weak oroccur only at low temperatures so their technologi-cal applications is not yet very promising14. On thecontrary, 2-phase multiferroic composites consist-ing of magnetostrictive layers strain-coupled withpiezoelectric layers14 ; ; ; ;
18 do not have thebottlenecks as of their single-phase counterparts.Thus multiferroic composites are more promising for technological applications.
Fig. 1. A 2-phase multiferroic nanomagnet in the shape ofan elliptical cylinder is stressed with an applied voltage viathe d coupling in the piezoelectric. The multiferroic is pre-vented from expanding or contracting along the in-plane hardaxis ( y -axis), so that a uniaxial stress is generated along theeasy axis ( z -axis). (Reprinted with permission from Ref. 19.Copyright 2012, AIP Publishing LLC.) The magnetization of the shape-anisotropicsingle-domain magnetostrictive nanomagnet in mul-tiferroic composites can be switched in lessthan 1 nanosecond while dissipating only ∼ ;
19. Hence such devices have emergedas potential candidates as storage and switchingelements for our future non-volatile memory andlogic systems. Particularly due to high switchingspeed while simultaneously being highly energy ef-ficient, this has lead to logic proposals incorpo-rating such systems20 ; ; ;
23. The magnetiza-tion of the nanomagnet has two stable states (mu-tually anti-parallel) along the easy axis encodingthe binary bits 0 and 1. The magnetization can beswitched from one stable state to the other when atiny voltage of few tens of millivolts is applied acrossthe piezoelectric layer while constraining it from ex-panding or contracting along its in-plane hard-axis(see Fig. 1). The applied voltage produces a strainin the piezoelectric layer, which is then transferredto the magnetostrictive layer. This in turn generatesa uniaxial stress in the magnetostrictive nanomag-net along its easy-axis and rotates the magnetiza-tion towards the in-plane hard axis as long as theproduct of the stress and the magnetostrictive coef-ficient is negative . It is assumed by convention thata tensile stress is positive and a compressive stressis negative. There have been experimental efforts todemonstrate such electric-field induced magnetiza-tion rotation24 ; ; ; ; ltra-low-energy straintronics using multiferroic composites netization dynamics and to calculate the associatedperformance metrics like switching delay and energydissipation. Also, we present the model for deter-mining magnetization dynamics in a circuit madeof multiple multiferroic devices. Section 3 presentsthe simulation results for both a single memory de-vice and a circuit of multiple multiferroic devices.Finally, Section 4 summarizes this review and pro-vides the outlook of the multiferroic straintronic de-vices on building nanoelectronics for our future in-formation processing paradigm.
2. Model
In this Section, we will first review the model of asingle multiferroic device. The emphasis would beon magnetization dynamics of the magnetostrictivenanomagnet in a multiferroic composite by solv-ing Landau-Lifshitz-Gilbert (LLG) equation29 ; stochastic LLG equation31 ;
19. Then we describe how thesame model can be used for unidirectional signalpropagation in a chain of multiferroic devices21 ; Single Multiferroic Device
We consider a single isolated nanomagnet in theshape of an elliptical cylinder with its elliptical crosssection lying in the y - z plane; the major axis isaligned along the z -direction and minor axis alongthe y -direction (see Fig. 1). The dimensions of themajor axis, the minor axis, and the thickness are a , b , and l , respectively. So the magnet’s volume isΩ = ( π/ abl . The z -axis is the easy axis, the y -axisis the in-plane hard axis and the x -axis is the out-of-plane hard axis. Since l ≪ b , the out-of-plane hardaxis is much harder than the in-plane hard axis. Let θ ( t ) be the polar angle and φ ( t ) the azimuthal angleof the magnetization vector in standard sphericalcoordinate system. Note that when φ = ± ◦ , themagnetization vector lies in the plane of the nano-magnet. Any deviation from φ = ± ◦ correspondsto out-of-plane excursion.We can write the total energy of the magne-tostrictive single-domain nanomagnet when it issubjected to uniaxial stress along the easy axis (ma-jor axis of the ellipse) as the sum of the uniax-ial shape anisotropy energy and the uniaxial stressanisotropy energy32. We assume that the magne- tostrictive layer is polycrystalline, so that we ignorethe magnetocrystalline energy. The uniaxial shapeanisotropy energy at an instant of time t is givenby32 E SHA ( t ) = E SHA ( θ ( t ) , φ ( t ))= ( µ / M s Ω N d ( θ ( t ) , φ ( t )) (1)where M s is the saturation magnetization and thedemagnetization factor N d ( t ) is expressed as32 N d ( t ) = N d ( θ ( t ) , φ ( t )) = N d − xx sin θ ( t ) cos φ ( t )+ N d − yy sin θ ( t ) sin φ ( t ) + N d − zz cos θ ( t ) (2)with N d − mm being the m th (m=x,y,z) componentof the demagnetization factor33. Note that thesefactors depend on the dimensions of the nanomag-net and not on the material properties. The dimen-sions of the nanomagnet is chosen as a = 100 nm, b = 90 nm and l = 6 nm, which ensures that thenanomagnet has a single ferromagnetic domain13.These dimensions alongwith the material parametersaturation magnetization M s determine the shapeanisotropy energy barrier, which separates the twostable states of the nanomagnet. The in-plane en-ergy barrier E b ( φ = ± ◦ , see Fig. 1), which is the lowest difference between the shape anisotropy en-ergies when θ = 90 ◦ and θ = 0 ◦ , ◦ determinesthe static error probability of spontaneous magne-tization reversal due to thermal fluctuations. Ac-cording to Boltzmann distribution, this probabil-ity is exp [ − E b /kT ]. This probability should be lowenough for technological application purposes. Withthe dimensions and material chosen, E b = 44 kT atroom temperature, so that the static error proba-bility at room temperature is e − . Note that the dynamic error probability on the other hand signi-fies the switching error probability when magneti-zation fails to flip from one state to another duringswitching events. Usually the dynamic error prob-ability can be much higher than that of static er-ror probability. So we must put emphasis on this dynamic error probability to meet the demand oftechnological viability.The stress anisotropy energy is given by34 ; E ST A ( t ) = E ST A ( θ ( t ) , σ ( t ))= − (3 / λ s σ ( t )Ω cos θ ( t ) (3)where (3 / λ s is the magnetostriction coefficient ofthe magnetostrictive nanomagnet and σ ( t ) is thestress generated in it by an external voltage. A pos-itive λ s σ ( t ) product will favor alignment of the mag-netization along the major axis ( z -axis), while a Kuntal Roy negative λ s σ ( t ) product will favor alignment alongthe minor axis ( y -axis), because that will minimize E ST A ( t ). We will use the following convention thata compressive stress is negative and tensile stressis positive. Therefore, in a material like Terfenol-Dthat has positive λ s , a compressive stress will favoralignment along the minor axis, and tensile alongthe major axis. The situation will be exactly oppo-site with nickel and cobalt that have negative λ s .At any instant of time t , the total energy of thenanomagnet can be expressed as19 E ( t ) = E ( θ ( t ) , φ ( t ) , σ ( t ))= B ( φ ( t ) , σ ( t )) sin θ ( t ) + C ( t ) (4)where B ( t ) = B ( φ ( t ) , σ ( t )) = B ( φ ( t )) + B stress ( σ ( t ))(5a) B ( t ) = B ( φ ( t )) = ( µ / M s Ω[ N d − xx cos φ ( t )+ N d − yy sin φ ( t ) − N d − zz ] (5b) B stress ( t ) = B stress ( σ ( t )) = (3 / λ s σ ( t )Ω (5c) C ( t ) = C ( σ ( t )) = ( µ / M s Ω N d − zz − (3 / λ s σ ( t )Ω . (5d)The magnetization M (t) of the nanomagnethas a constant magnitude at any given temperaturebut a variable direction, so that we can represent itby the vector of unit norm n m ( t ) = M ( t ) / | M | = ˆe r where ˆe r is the unit vector in the radial direction inspherical coordinate system represented by ( r , θ , φ ).The other two unit vectors in the spherical coor-dinate system are denoted by ˆe θ and ˆe φ for θ and φ rotations, respectively. The torque acting on themagnetization per unit volume due to shape andstress anisotropy is19 T E ( t ) = − n m ( t ) × ∇ E ( θ ( t ) , φ ( t ) , σ ( t ))= − B ( φ ( t ) , σ ( t )) sinθ ( t ) cosθ ( t ) ˆe φ − B e ( φ ( t )) sinθ ( t ) ˆe θ , (6)where B e ( t ) = B e ( φ ( t ))= ( µ / M s Ω( N d − xx − N d − yy ) sin (2 φ ( t )) . (7)The effect of room-temperature random ther-mal fluctuations is incorporated via a random mag-netic field h ( t ), which is expressed as h ( t ) = h x ( t ) ˆe x + h y ( t ) ˆe y + h z ( t ) ˆe z (8)where h i ( t ) (i=x,y,z) are the three components ofthe random thermal field in Cartesian coordinates. We assume the properties of the random field h ( t )as described in Ref. 31. The random thermal fieldcan be written as19 h i ( t ) = s αkT | γ | M V ∆ t G (0 , ( t ) ( i = x, y, z ) (9)where α is the dimensionless phenomenologicalGilbert damping parameter, γ = 2 µ B µ / ~ is thegyromagnetic ratio for electrons and is equal to2 . × (rad.m).(A.s) − , µ B is the Bohr mag-neton, M V = µ M s Ω, and 1 / ∆ t is proportionalto the attempt frequency of the thermal field, ∆ t is the simulation time-step used, and the quantity G (0 , ( t ) is a Gaussian distribution with zero meanand unit variance35.The thermal torque can be written as19 T TH ( t ) = M V n m ( t ) × h ( t ) = P θ ( t ) ˆe φ − P φ ( t ) ˆe θ (10)where P θ ( t ) = M V [ h x ( t ) cosθ ( t ) cosφ ( t )+ h y ( t ) cosθ ( t ) sinφ ( t ) − h z ( t ) sinθ ( t )] , (11) P φ ( t ) = M V [ h y ( t ) cosφ ( t ) − h x ( t ) sinφ ( t )] . (12)The magnetization dynamics under the actionof the torques T E ( t ) and T TH ( t ) is described by thestochastic Landau-Lifshitz-Gilbert (LLG) equationas follows. d n m ( t ) dt − α (cid:18) n m ( t ) × d n m ( t ) dt (cid:19) = − | γ | M V [ T E ( t ) + T TH ( t )] . (13)After solving the above equation analytically,we get the following coupled equations of magneti-zation dynamics for θ ( t ) and φ ( t )19. (cid:0) α (cid:1) dθ ( t ) dt = | γ | M V [ B e ( φ ( t )) sinθ ( t ) − αB ( φ ( t ) , σ ( t )) sinθ ( t ) cosθ ( t )+ ( αP θ ( t ) + P φ ( t ))] , (14) (cid:0) α (cid:1) dφ ( t ) dt = | γ | M V [ αB e ( φ ( t ))+ 2 B ( φ ( t ) , σ ( t )) cosθ ( t ) − [ sinθ ( t )] − ( P θ ( t ) − αP φ ( t ))]( sinθ = 0) . (15) ltra-low-energy straintronics using multiferroic composites We need to solve the above two coupled equationsnumerically to track the trajectory of magnetizationover time, in the presence of thermal fluctuations.When sin θ = 0 ( θ = 0 ◦ or θ = 180 ◦ ), i.e. whenthe magnetization direction is exactly along the easyaxis, the torque on the magnetization vector givenby Eq. (6) becomes zero. That is why only ther-mal fluctuations can budge the magnetization vec-tor from the easy axis. Consider the situation when θ = 180 ◦ . From Eqs. (14) and (15), we get19 φ ( t ) = tan − (cid:18) αh y ( t ) + h x ( t ) h y ( t ) − αh x ( t ) (cid:19) , (16) dθ ( t ) dt = −| γ | ( h x ( t ) + h y ( t )) p ( h y ( t ) − αh x ( t )) + ( αh y ( t ) + h x ( t )) . (17)We can see from the Eq. (17) clearly that thermaltorque can deflect the magnetization from the easyaxis since the time rate of change of θ ( t ) wouldbe non-zero in the presence of the thermal fluctu-ations. Note that dθ ( t ) /dt does not depend on thecomponent of the random thermal field along the z -axis, i.e. h z ( t ), which is a consequence of having z -axis as the easy axis of the nanomagnet. However,once the magnetization direction is even slightly de-flected from the easy axis, all three components ofthe random thermal field along the x -, y -, and z -direction would come into play.When no stress is applied on the magnetostric-tive nanomagnet, magnetization would just fluctu-ate around an easy axis provided that the shapeanisotropy energy barrier is enough high to pre-vent spontaneous reversal of magnetization fromone state to another in a short period of time. Wecan solve the Eqs. (14) and (15) while setting B stress = 0 to track the dynamics of magnetization due tothermal fluctuations. So this will yield the distri-bution of the magnetization vector’s initial orienta-tion when stress is turned on. Since the most prob-ably value of magnetization is along easy axis, the θ -distribution is Boltzmann peaked at θ = 0 ◦ or180 ◦ , while the φ -distribution is Gaussian peakedat φ = ± ◦ because these positions are minimumenergy positions36.Stress is ineffective when θ is around 0 ◦ or180 ◦ , i.e. when magnetization is around the easyaxis (mathematically, note that the expression inEq. (6) is proportional to sin θ , which is zero when θ is equal to 0 ◦ or 180 ◦ ). Hence, we would get a longtail in the switching delay distribution should mag-netization starts very near from easy axis. When we start out from θ = 0 ◦ , ◦ , we have to wait a while;thermal fluctuations may help here while gettingstarted but random thermal kicks may also causemagnetization to traverse towards the opposite di-rection than the intended dirction of switching.Thus, switching trajectories initiating from neareasy axis may be very slow. We do need to worryabout the switching delay tail more than mean switching delay since the extent of tail will set therequirement of pulse width of stress for switchingto take place with sufficiently high probability.In order to eliminate the long tail in the switch-ing delay distribution, we can apply a static biasfield that will shift the peak of θ initial distributionaway from the easy axis, so that the most proba-ble starting orientation will no longer be the easyaxis19. This field is applied along the out-of-planehard axis (+ x -direction) and the potential energydue to the applied magnetic field can be expressedas E mag ( θ ( t ) , φ ( t )) = − M V H sinθ ( t ) cosφ ( t ) , (18)where H is the magnitude of magnetic field. Thetorque generated due to this field is T M ( t ) = − n m ( t ) × ∇ E mag ( θ ( t ) , φ ( t )) . (19)We assume that a permanent magnetic sheet willbe employed to produce the bias field and thus willnot require any additional energy dissipation to begenerated. The presence of this field will modifyEqs. (14) and (15) for magnetization dynamics to19 (cid:0) α (cid:1) dθ ( t ) dt = | γ | M V × [ B e ( φ ( t )) sinθ ( t ) − αB ( φ ( t ) , σ ( t )) sinθ ( t ) cosθ ( t )+ αM V H cosθ ( t ) cosφ ( t ) − M V H sinφ ( t )+ ( αP θ ( t ) + P φ ( t ))] , (20) (cid:0) α (cid:1) dφ ( t ) dt = | γ | M V × [ αB e ( φ ( t )) + 2 B ( φ ( t ) , σ ( t )) cosθ ( t ) − [ sinθ ( t )] − ( M V H cosθ ( t ) cosφ ( t ) + αM V H sinφ ( t )) − [ sinθ ( t )] − ( P θ ( t ) − αP φ ( t ))] ( sinθ = 0) . (21)Note that the bias field makes the potential en-ergy profile of the nanomagnet asymmetric in φ -space and the energy minimum gets shifted from φ min = ± ◦ (the plane of the nanomagnet) to φ min = cos − (cid:20) HM s ( N d − xx − N d − yy ) (cid:21) . (22) Kuntal Roy
However, the potential profile will remain symmet-ric in θ -space, with θ = 0 ◦ and θ = 180 ◦ remainingas the minimum energy locations. A bias magneticfield of flux density 40 mT applied perpendicular tothe plane of the magnet would make φ min ≃ ± ◦ deflecting the magnetization vector ∼ ◦ from themagnet’s plane. Application of the bias magneticfield will also affect the in-plane shape anisotropyenergy barrier E b as it gets reduced from 44 kT to36 kT at room temperature.We consider both the energy dissipated inter-nally in the nanomagnet due to Gilbert damping(termed as E d ) and the energy dissipated in theswitching circuit while applying voltage across themultiferroic structure generating stress on the nano-magnet (termed as ‘ CV ’ dissipation, where C and V denote the capacitance of the piezoelectric layerand the applied voltage, respectively). If the voltageis turned on or off abruptly then the energy dissi-pated during either turn on or turn off is (1 / CV ,however, if the ramp rate is finite, the energy dissi-pated can be significantly reduced37. The internalenergy dissipation E d , is given by the expression R τ P d ( t ) dt , where τ is the switching delay and P d ( t )is the power dissipated during switching given as19 P d ( t ) = α | γ | (1 + α ) M V | T E ( t ) + T M ( t ) | . (23)We sum up the power P d ( t ) dissipated during theentire switching period to get the corresponding en-ergy dissipation E d and add that to the ‘ CV ’ dis-sipation in the switching circuit to find the totaldissipation E total . There is no net dissipation dueto random thermal torque, however, it affects E d since it raises the critical stress needed to switchwith ∼ Array of Multiferroic Devies
Here we will use the same model as derived fora single multiferroic device and see how unidirec-tional flow of signal is possible in a horizontal chainof multiferroic devices20 ; ;
22 using dipole cou-pling between nanomagnets and Bennett clockingmechanism38.The dipole coupling between two magnetic mo-ments M1 and M2 separated by a distance vector R can be expressed as39: E dipole = 14 πµ R (cid:20) ( M . M ) − R ( M . R )( M . R ) (cid:21) . (24) Fig. 2. Dipole coupling between two magnetic moments.(Reprinted with permission from Ref. 21. Copyright 2012,Kuntal Roy.)
In standard spherical coordinate system (seeFig. 2), the expression of dipole coupling can beformulated as E dipole = µ πR M s Ω [ cosθ cosθ + sinθ sinθ ( cosφ cosφ − sinφ sinφ )](25)where | M | = | M | = µ M s Ω, Ω is the volume ofthe nanomagnets, M s is the saturation magnetiza-tion, and R = R ˆe y .Note that dipole coupling is bi-directional, i.e. E dipole = E dipole, = E dipole, . Because of the dipolecoupling between the magnetizations of the nano-magnets, the potential profiles of both the nano-magnets are tilted and the ground state of the mag-netizations are antiferromagnetically coupled as de-picted in the Fig. 2.If we somehow change the magnetization direc-tion of one nanomagnet, the magnetization of theother nanomagnet would not automatically changeits direction to assume an antiferromagnetic order.It is because of the reason that there is a barrierseparating two magnetization states. It’s true thatantiferromagnetic order is the ground state, how-ever, during operation of devices, we must removethe barrier and then again restore it to make surethat antiferromagnetic order is maintained. Magne-tization may come to antiferromagnetic order aftera very long time depending on the barrier heightbut the operation of devices cannot be dependenton that.In general, we need to propagate a logic bit uni-directionally along a chain of nanomagnets. It re-quires a clock signal to periodically reset the mag-netization direction of each nanomagnet. If a globalmagnetic field is utilized for such a purpose, it wouldnot allow pipelining of data, and magnetization ofevery nanomagnet must be maintained along hardaxis until a bit propagates. It needs an energy min-ima along hard axis, which can be introduced by bi- ltra-low-energy straintronics using multiferroic composites axial anisotropy40, but thermal fluctuations wouldproduce a large bit error probability41. Using a lo-cal magnetic field eliminates the problems of us-ing global magnetic field, but it is difficult to main-tain a magnetic field locally within a dimension of ∼
100 nm. Furthermore, generating magnetic fieldis highly energy consuming. We can use electric-field operated (since electric-field can be maintainedlocally) multiferroic devices to propagate signalsin a chain of nanomagnets20 using so-called Ben-nett clocking mechanism, termed in the name ofBennett38. Ref. 20 performed the steady-state anal-ysis, while Refs. 21, 22 solved the magnetization dy-namics using the same model as for a single multi-ferroic device to show that the switching may takeplace in sub-nanosecond9 ; ;
19, which is crucialfor building nanomagnetic logic23.Fig. 3 depicts the issue (and also solution) be-hind Bennett clocking in a chain of nanomagnet.First of all, it needs to be emphasized that dipolecoupling is bi-directional. So the 2nd nanomagnet experiences dipole coupling effect from both of itsneighbors, i.e. 1st and 3rd nanomagnets. Note thatwe are considering only nearest neighbor interac-tion, since dipole coupling reduces drastically withdistance [see Eq. (24)]. Thus, if the 1st nanomag-net is switched, the 2nd nanomagnet finds itself ina locked condition as the 1st nanomagnet is tellingit to go up , while the 3rd nanomagnet is telling itto go down . Therefore, it remains on its previousposition and thus the change in information on the1st nanomagnet cannot be propagated through thechain of nanomagnets.To prevent this lockjam, we need to impose the unidirectionality in time as shown in the Fig. 3.Both the 2nd and 3rd nanomagnets are stressed toget them aligned to their hard axes (note the thirdrow in Fig. 3) and then stress is removed/reversedon the 2nd nanomagnet (note the fourth row inFig. 3) to relax its magnetization towards the de-sired state. In this way, subsequently applying stresson the nanomagnets and then releasing/reversingthe stress, we can propagate a logic bit unidirec- Kuntal Roy tionally along a chain of nanomagnets. The slightdeflection in the magnetization of the 4th nanomag-net in the third row of Fig. 3 is due to dipole cou-pling, while in the fourth row, the magnetization of4th nanomagnet is aligned along its hard axis be-cause of applied stress on it. A 3-phase clock wouldbe sufficient to propagate a signal along the chain ofnanomagnet. Note that we are explaining the oper-ation with two-dimensional in-plane potential land-scapes of the nanomagnets (assuming azimuthalangle φ = ± ◦ ), but solution of the full three-dimensional dynamics is necessary since the out-of-plane excursion of magnetization has immense influ-ence in shaping the magnetization dynamics36 andreducing the switching delay by a couple of ordersin magnitude to sub-nanosecond9 ; ; E dipole, = µ πR M s Ω [ cosθ cosθ + cosθ cosθ + sinθ sinθ ( cosφ cosφ − sinφ sinφ )+ sinθ sinθ ( cosφ cosφ − sinφ sinφ )] . (26)The torque acting on the 2nd nanomagnet due todipole coupling21 T dipole , ( t ) = − n m ( t ) × ∇ E dipole, = − ∂E dipole, ∂θ ˆe φ + 1 sinθ ∂E dipole, ∂φ ˆe θ = − T dipole,φ ˆe φ + T dipole,θ ˆe θ , (27)where T dipole,φ = ∂E dipole, ∂θ = µ πR M s Ω [ − sinθ cosθ − sinθ cosθ + sinθ cosθ ( cosφ cosφ − sinφ sinφ )+ sinθ cosθ ( cosφ cosφ − sinφ sinφ )] , (28) and T dipole,θ = 1 sinθ ∂E dipole, ∂φ = − µ πR M s Ω × [ sinθ ( cosφ sinφ + 2 sinφ cosφ )+ sinθ ( cosφ sinφ + 2 sinφ cosφ )] . (29)The torque acting on the 2nd nanomagnet due toshape and stress anisotropy can be derived similarlyfollowing the Eq. (6) as21 T E , ( t ) = − B ( φ ( t )) sinθ ( t ) cosθ ( t ) ˆe φ − B e ( φ ( t )) sinθ ( t ) ˆe θ , (30)where B ( φ ( t )) = µ M s Ω[ N d − xx cos φ ( t )+ N d − yy sin φ ( t ) − N d − zz ]+(3 / λ s σ Ω , (31a) B e ( φ ( t )) = µ M s Ω( N d − xx − N d − yy ) sin (2 φ ( t )) . (31b)After solving the Landau-Lifshitz-Gilbert(LLG) equation considering the dipole couplingterm in a very similar way as done for a singlemultiferroic device, we get the coupled dynamicsbetween the polar angle θ and azimuthal angle φ for the 2nd nanomagnet as21 (cid:0) α (cid:1) dθ ( t ) dt = | γ | M V [ B e ( φ ( t )) sinθ ( t ) − αB ( φ ( t )) sinθ ( t ) cosθ ( t ) − T dipole,θ − αT dipole,φ ] , (32) (cid:0) α (cid:1) dφ ( t ) dt = | γ | M V sinθ ( t )[ αB e ( φ ( t )) sinθ ( t )+ 2 B ( φ ( t )) sinθ ( t ) cosθ ( t )+ αT dipole,θ + T dipole,φ ]( sinθ = 0) . (33)Note that in a very similar way the equationsof dynamics for the other three nanomagnets canbe derived.On energy dissipation, we have one more com-ponent contributing to the total energy apart fromthe shape anisotropy and stress anisotropy energy,which is the energy due to dipole coupling. While ltra-low-energy straintronics using multiferroic composites calculating internal energy dissipation, the sum ofthe energy dissipations in all the four nanomagnetsare considered but note that the dissipations in 1stand 4th nanomagnets are quite negligible since theydon’t quite switch and dissipation in 2nd nanomag-net is around twice that of in 3rd nanomagnet since2nd nanomagnet switches a complete 180 ◦ , whilethe 3rd nanomagnet switches only about 90 ◦ . Theinstantaneous power dissipation for the 2nd nano-magnet can be determined as21 P d, ( t ) = α | γ | (1 + α ) M V | T E , ( t ) + T dipole , ( t ) | . (34) Fig. 4. Schematic of universal logic gates employing Mag-netic Quantum Cellular Automata (MQCA) based architec-ture: (a) NAND gate, and (b) NOR gate. Note that a weak bias field in the specified direction is required to break the tiewhen the input bits are different. The bias field must be weakenough so that it does not interfere in the operation whenthe input bits are 0s for NAND gate and 1s for NOR gate.(Reprinted with permission from Ref. 21. Copyright 2012,Kuntal Roy.)
Note that we have not considered thermal flu-cutations and also have not applied any out-of-planebias field so the term T M ( t ) term as in Eq. (23) isabsent here. The power dissipations are integratedthroughout the switching period to get the energydissipation due to Gilbert damping. We have alsoconsidered ‘ CV ’ energy dissipation, which can besignificantly brought down by decreasing the stresssince, stress is proportional to voltage applied, whilesacrificing switching delay a bit. Since we have con-sidered instantaneous ramp and stress is reversedduring ramp-down phase, the ‘ CV ’ energy dissipa-tion is simply 3 CV for the 2nd nanomagnet.We have considered Bennett clocking in anantiferromagnetically coupled horizontal wire fordemonstration of magnetization dynamics in an ar-ray of multiferroic devices, however, a similar anal-ysis is possible in the context of a ferromagneticallycoupled vertical wire. We have not incorporatedramp rate effect or thermal fluctuations, which oneneeds to consider and analyze further. Universallogic gates (e.g. NAND and NOR gates) can alsobe constructed and analyzed using the very samemodel that includes dipole coupling. Fig. 4 depictssuch possibilities. In general, magnetizations of anarray of nanomagnets can be manipulated to im-plement computing in MQCA (Magnetic QuantumCellular Automata) based architecture42. Althoughsuch architecture has complexity of clocking eachnanomagnet in the array, this is a regular structureand circuits based on such structure can be designedsystematically. Anyway, unconventional design oflogic gates and building blocks for large-scale cir-cuits using multiferroic composites can possibly beworked out too. Such designs may incur less com-plexity and possess better performance metrics thanthat of Bennett clocking mechanism.
3. Simulation Results andDiscussions
In this Section, we review the simulation results forboth single multiferroic devices9 ;
19 and an array ofmultiferroic devices21 ;
22. The performance metricsswitching delay and energy dissipation are deter-mined and trade-off between them is presented, i.e.if we want to make the switching faster, it wouldcost higher energy dissipation. Also, we determinethe distributions of switching delay and energy dis-sipation, and number of successful switching eventsin the presence of room-temperature thermal fluc-tuations for a single multiferroic device. Kuntal Roy
Single Multiferroic Device
We consider the magnetostrictive layer to be madeof polycrystalline Terfenol-D, nickel, or cobalt9.Terfenol-D has 30 times higher magnetostriction co-efficient in magnitude and it has the following mate-rial properties – Young’s modulus (Y): 8 × Pa,magnetostrictive coefficient ((3 / λ s ): +90 × − ,saturation magnetization ( M s ): 8 × A/m, andGilbert’s damping constant ( α ): 0.1 (Refs. 43, 44,45, 46). For the piezoelectric layer, we have con-sidered lead-zirconate-titanate (PZT) having a di-electric constant of 1000. The maximum strain thatcan be generated in the PZT layer is 500 ppm47 ; d =1.8 × − m/V for PZT49. The PZT layer isassumed to be four times thicker than the mag-netostrictive layer so that any strain generated init is transferred almost completely to the magne-tostrictive layer20 ;
9. So the corresponding stressin Terfenol-D is the product of the generated strain(500 × − ) and the Young’s modulus (8 × Pa).Hence, 40 MPa is the maximum stress that canbe generated in the Terfenol-D nanomagnet. Thestrain-voltage relationship in PZT is actually su-perlinear since d increases with electric field48.Hence, the voltage needed to produce 500 ppmstrain in the Terfenol-D layer will be less than 66.7mV and the energy dissipation would be a bit over-estimated too.We first review the results for low stress lev-els leading to slow switching speed (10-100 ns) andlow energy dissipation9. Fig. 5 (taken from Ref. 9)shows the energy dissipated in the switching cir-cuit ( CV ) and the total energy dissipated ( E total )as functions of delay for three different materials(Terfenol-D, nickel, and cobalt) used as the mag-netostrictive layer in the multiferroic nanomagnet.We solve magnetization dynamics to calculate theswitching delay τ and also energy dissipation (‘ CV ’dissipation and internal one E d ) for a given stress σ ,and then we plot the switching delays and energydissipations for different stress values. Terfenol-Dincurs much less energy dissipation than the othertwo materials because it has much higher magne-tostriction coefficient requiring a less stress level togenerate a certain stress anisotropy. For Terfenol-D, the stress required to switch in 100 ns is 1.92MPa and that required to switch in 10 ns is 2.7MPa. Note that for a stress of 1.92 MPa, the stressanisotropy energy B stress is 32.7 kT while for 2.7MPa, it is 46.2 kT . Since the stress level is assumed to be applied instantaneously, as expected, the en-ergy dissipation numbers are larger than the shapeanisotropy barrier of ∼ kT . A larger excess en-ergy is needed to switch faster signifying the delay-energy trade-off. The energy dissipated and lost asheat in the switching circuit ( CV ) is only 12 kT fora delay of 100 ns, while that is 23.7 kT for a delayof 10 ns. The total energy dissipated is 45 kT forswitching delay of 100 ns and 70 kT for switchingdelay of 10 ns. Fig. 5. Energy dissipated in the switching circuit ( CV ) andthe total energy dissipated ( E total ) as functions of delay forthree different materials used as the magnetostrictive layerin the multiferroic nanomagnet. (Reprinted with permissionfrom Ref. 9. Copyright 2011, AIP Publishing LLC.) With a nanomagnet density of 10 cm − in amemory or logic chip, and if we consider 10% of thenanomagnets switch at any given time (10% activitylevel), the dissipated power density would have beenonly 2 mW/cm to switch in 100 ns and 30 mW/cm to switch in 10 ns. Such extremely low power andyet high density magnetic logic and memory sys-tems can be powered by existing energy harvestingsystems50 ; ; ;
53 that harvest energy from theenvironment without the need for an external bat-tery. These processors are deemed to be suitable forimplantable medical devices, e.g. those implanted ina patient’s brain that monitor brain signals to warnof impending epileptic seizures. They can run onenergy harvested from the patient’s body motion.For such applications, 10-100 ns switching delay isadequate.We now review multiferroic devices for higherstress levels and fast sub-nanosecond switchingspeed, which is particularly important for logic and ltra-low-energy straintronics using multiferroic composites computing purposes19. We performed simulationsin the presence of room-temperature thermal fluc-tuations and we consider only Terfenol-D as magne-tostrictive material since it has much higher magne-tostrictive coefficient (than nickel and cobalt) thusbeing fruitful in resisting the adverse effects of ther-mal fluctuations.Fig. 6 plots the distributions of initial angles θ initial and φ initial in the presence of thermal fluc-tuations and a bias magnetic field applied along theout-of-plane direction (+ x -axis). The bias field hasshifted the peak of θ initial exactly from the easy axis( θ = 180 ◦ ) as shown in Fig. 6(a). The φ initial dis-tribution (see Fig. 6(b)) has two peaks and residesmostly within the interval [-90 ◦ ,+90 ◦ ] since the biasmagnetic field is applied in the + x -direction. Be-cause the magnetization vector starts out from nearthe south pole ( θ ≃ ◦ ) when stress is turned on,the effective torque on the magnetization [ ∼ M × H ,where M is the magnetization and H is the effec-tive field] due to the + x -directed magnetic field issuch that the magnetization prefers the φ -quadrant(0 ◦ ,90 ◦ ) over the φ -quadrant (270 ◦ ,360 ◦ ), which isthe reason for the asymmetry in the two distribu-tions of φ initial . Consequently, when the magneti-zation vector starts out from θ ≃ ◦ , the initialazimuthal angle φ initial is more likely to be in thequadrant (0 ◦ ,90 ◦ ) than in the quadrant (270 ◦ ,360 ◦ ).We assume that when a compressive stressis applied to initiate switching (since Terfenol-Dhas positive magnetostrictive coefficient), the mag-netization vector starts out from near the southpole ( θ ≃ ◦ ) with a certain ( θ initial , φ initial )picked from the initial angle distributions at room-temperature. Stress is ramped up linearly and keptconstant until the magnetization reaches x - y plane( θ = 90 ◦ ). Then stress is ramped down at the samerate at which it was ramped up, and reversed inmagnitude to aid switching. The magnetization dy-namics ensures that θ continues to rotate towards0 ◦ . When θ becomes ≤ ◦ , switching is deemedto have completed. A moderately large number(10,000) of simulations, with their corresponding( θ initial , φ initial ) picked from the initial angle distri-butions, are performed for each value of stress andramp duration to generate the simulation results inthe presence of thermal fluctuations.Fig. 7 shows the switching probability as a func-tion of stress levels (10-30 MPa) and as well as volt-age applied across the piezoelectric layer for differ-ent ramp durations (60 ps, 90 ps, 120 ps)37 ;
54 at room temperature (300 K). The minimum stressneeded to switch the magnetization with ∼ ∼
14 MPa for 60 ps rampduration and ∼
17 MPa for 90 ps ramp duration.For higher ramp duration of 120 ps, the curve isnon-monotonic, which we will discuss later.
Fig. 6. Distribution of polar angle θ initial and azimuthalangle φ initial due to thermal fluctuations at room tempera-ture (300 K) when a magnetic field of flux density 40 mT isapplied along the out-of-plane hard axis (+ x -direction). (a)Distribution of polar angle θ initial at room temperature (300K). The mean of the distribution is 173 . ◦ , and the mostlikely value is 175.8 ◦ . (b) Distribution of the azimuthal an-gle φ initial due to thermal fluctuations at room temperature(300 K). There are two distributions with peaks centered at ∼ ◦ and ∼ ◦ . (Reprinted with permission from Ref. 19.Copyright 2012, AIP Publishing LLC.) At low stress levels (10-20 MPa), the switchingprobability increases with stress, regardless of theramp rate. This happens because a higher stress can Kuntal Roy more effectively counter the adverse effects of ther-mal fluctuations to facilitate switching, and henceincreases the success rate of switching. This featureis independent of the ramp rate for lower stress lev-els. However, for higher stress levels accompaniedby a higher ramp duration, the switching dynamicsis complex, which we will explain onwards.
Fig. 7. Percentage of successful switching events among thesimulated switching trajectories (or the switching probabil-ity) at room temperature in a Terfenol-D/PZT multiferroicnanomagnet versus (lower axis) stress (10-30 MPa) and (up-per axis) voltage applied across the piezoelectric layer, fordifferent ramp durations (60 ps, 90 ps, 120 ps). The stressat which switching becomes ∼ ∼ We will now describe the significance of ramp-rate on success probability. When stress ( compres-sive stress for Terfenol-D) is made active, mag-netization traverses from θ ≃ ◦ towards x - y plane ( θ = 90 ◦ ). Once the magnetization vectorcrosses the x - y plane, the stress needs to be with-drawn as soon as possible. This is because the stressforces the energy minimum to remain at θ = 90 ◦ ,which will make the magnetization linger around θ = 90 ◦ instead of rotating towards the desireddirection θ ≃ ◦ . This is why stress must be re-moved or reversed immediately upon crossing the x - y plane so that the energy minimum moves to θ = 0 ◦ , ◦ . Then the magnetization vector rotatestowards θ = 0 ◦ rather than θ = 180 ◦ due to in-herent switching dynamics36. If the removal rate isfast, then the success probability is high since theharmful stress does not stay active long enough tocause significant backtracking of the magnetization vector towards θ = 90 ◦ . However, if the ramp rate istoo slow, then significant backtracking can possiblyoccur whereupon the magnetization vector may re-turn to the x - y plane. Then thermal torque decidesthe fate whether magnetization backtracks towards θ ≃ ◦ , causing switching failure or switchessuccessfully towards θ ≃ ◦ ; in general there is50% switching probability then. Hence the switch-ing probability drops with decreasing ramp rate. Fig. 8. The thermal mean of the switching delay (at 300 K)versus (lower axis) stress (10-30 MPa) and (upper axis) volt-age applied across the piezoelectric layer, for different rampdurations (60 ps, 90 ps, 120 ps). Switching may fail at lowstress levels and also at high stress levels for long ramp du-rations. Failed attempts are excluded when computing themean. (Reprinted with permission from Ref. 19. Copyright2012, AIP Publishing LLC.)
A similar explanation is also applicable for thenon-monotonic stress dependence of the switchingprobability when the ramp rate is slow (ramp du-ration of 120 ps). When θ is in the quadrant [180 ◦ ,90 ◦ ], a higher stress is helpful since it provides alarger torque to move towards the x - y plane, butwhen θ is in the quadrant [90 ◦ , 0 ◦ ], a higher stressis harmful since it increases the chance of backtrack-ing, particularly when the ramp-down rate is slow.These two counteracting effects are the reason forthe non-monotonic dependence of the success prob-ability on stress in the case of the slowest ramprate. At higher stress levels accompanied by a slowramp rate, it causes a significant amount of back-tracking causing the switching probability to dropfast. For slow ramp rate (ramp duration of 120 ps),we have not observed 100% switching probability atany stress for the 10,000 simulations performed.Fig. 8 plots the thermal mean (averaged from10,000 simulations) switching delay versus stress for ltra-low-energy straintronics using multiferroic composites different ramp durations. Specifically, only success-ful switching events are considered here since we donot have a value of switching delay for an unsuc-cessful switching event. For ramp durations of 60ps and 90 ps, the switching delay decreases withincreasing stress since the torque, which rotatesthe magnetization, increases when stress increases.However, for 120 ps ramp duration, the dependenceis non-monotonic, due to same reason causing thenon-monotonicity in Fig. 7. For a certain stress,decreasing the ramp duration (or increasing theramp rate) decreases the switching delay becausethe stress reaches its maximum value quicker andhence switches the magnetization faster. Fig. 9. The standard deviations in switching delay versus(lower axis) stress (10-30 MPa) and (upper axis) voltage ap-plied across the piezoelectric layer for 60 ps ramp duration at300 K. We consider only the successful switching events in de-termining the standard deviations. The standard deviationsin switching delay for other ramp durations are of similarmagnitudes and show similar trends. (Reprinted with per-mission from Ref. 19. Copyright 2012, AIP Publishing LLC.)
Fig. 9 plots the standard deviation in switchingdelay versus stress for 60 ps ramp duration. The re-sults for other ramp durations are similar and henceare not shown for brevity. At higher values of stress,the torque due to stress is stronger and dominatesover the random thermal torque, which causes thespread in the switching delay. This decreases thestandard deviation in switching delay with increas-ing stress and thus the switching delay distributiongets more peaked as we increase the stress.Fig. 10 plots the thermal mean of the totalenergy dissipated to switch the magnetization asa function of stress and voltage across the piezo-electric layer for different ramp durations. To un-derstand these curves, we need to consider first the trend of average power dissipation ( E total /τ ),which increases with stress for a given ramp dura-tion and decreases with increasing ramp durationfor a given stress. More stress corresponds to more‘ CV ’ dissipation and also more internal power dis-sipation because it results in a higher torque. Slowerswitching decreases the power dissipation since itmakes the switching more adiabatic. Note that theswitching delay curves show the opposite trend (seeFig. 8) than that of average power dissipation. Ata lower ramp rate (higher ramp duration), the av-erage power dissipation E total /τ is always smallerthan that of a higher ramp rate, but the switchingdelay does not decrease as fast as with higher val-ues of stress (in fact switching delay may increasefor higher ramp duration), which is why the energydissipation curves in Fig. 10 exhibit the cross-overs. Fig. 10. Thermal mean of the total energy dissipation ver-sus (lower axis) stress (10-30 MPa) and (upper axis) voltageacross the piezoelectric layer for different ramp durations (60ps, 90 ps, 120 ps). Once again, failed switching attempts areexcluded when computing the mean. (Reprinted with permis-sion from Ref. 19. Copyright 2012, AIP Publishing LLC.)
Fig. 11 plots the ‘ CV ’ energy dissipation inthe switching circuitry versus stress and the volt-age applied across the PZT layer. Increasing stressrequires increasing the voltage V , which is whythe ‘ CV ’ energy dissipation increases rapidly withstress. This dissipation however is a small fractionof the total energy dissipation ( < CV ’ dissipation decreases when the rampduration increases because then the switching be-comes more ‘adiabatic’ and hence less dissipative.This component of the energy dissipation wouldhave been several orders of magnitude higher had Kuntal Roy we switched the magnetization with an externalmagnetic field6 ; Fig. 11. The ‘ CV ’ energy dissipation in the external circuitas a function of (lower axis) stress and (upper axis) voltageapplied across the PZT layer for different ramp durations.The dependence on voltage is not exactly quadratic since thevoltage is not applied abruptly, but instead ramped up grad-ually and linearly in time. (Reprinted with permission fromRef. 19. Copyright 2012, AIP Publishing LLC.) Fig. 12 plots the switching delay and energydistributions in the presence of room-temperaturethermal fluctuations for 15 MPa stress and 60 psramp duration. The high-delay tail in Fig. 12(a) isin general associated with those switching trajecto-ries that start close to θ = 180 ◦ . In such trajectories,the starting torque is vanishingly small as explainedearlier, which makes the switching sluggish at thebeginning. During this time, switching also becomessusceptible to backtracking because of thermal fluc-tuations, which may also increase the delay further.However, it may well happen that random thermaltorque quite facilitates a switching trajectory evenit gets started very close to the easy axis making the net switching fast. Note that there was not a singleevent where the delay exceeded 1 ns out of 10,000simulations of switching trajectories, showing thatthe probability of that happening is less than 0.01%and probably far less than 0.01%. We have analyzedthe cause of such high-delay tail and have been ableto reduce it by application of an out-of-plane biasfield. The magnitude of bias field can be calibratedto reduce the extent of such tail further. Since theenergy dissipation is the product of the power dissi-pation and the switching delay, similar behavior is found in Fig. 12(b). Fig. 12. Delay and energy distributions for 15 MPa appliedstress and 60 ps ramp duration at room temperature (300K). (a) Distribution of the switching delay. The mean andstandard deviation of the distribution are 0.44 ns and 83ps, respectively. (b) Distribution of energy dissipation. Themean and standard deviation of the distribution are 184 kT and 15.5 kT at room temperature, respectively. (Reprintedwith permission from Ref. 19. Copyright 2012, AIP Publish-ing LLC.) Array of Multiferroic Devices
Here we review the simulation results of unidi-rectional information propagation through an ar-ray of multiferroic devices using Bennett clockingmechanism21 ;
22. We do not consider thermal fluc-tuations, nonetheless in all the simulations, the ini-tial orientation of the magnetization vector is as-sumed as θ = 175 ◦ and φ = 90 ◦ . Stress is appliedinstantaneously and we solve Eqs. (32) and (33) forthe 2nd nanomagnet (and similar equations for theother three nanomagnets) at each time step. Once θ becomes 90 ◦ , stress is reversed instantaneously and ltra-low-energy straintronics using multiferroic composites we follow the magnetization vector in time until θ becomes ≤ ◦ . At that point, switching is deemedto have completed.Fig. 13 shows the magnetization dynamics forall the nanomagnets when propagating a signal uni-directionally in a chain of four Terfenol-D/PZTmultiferroic nanomagnets using Bennett clockingmechanism for the case of 5.2 MPa stress. When5.2 MPa compressive stress is applied on the 2ndand 3rd nanomagnets and after their magnetiza-tions come to their hard axes, stress is reversedon the 2nd nanomagnet to relax its magnetizationtowards its desired state. In the Fig. 13(b), notethat the applied stress has deflected the azimuthalangle of magnetization φ of the 2nd nanomagnetin the quadrant (90 ◦ , ◦ ) while that for the 3rdnanomagnet is deflected in the quadrant (0 ◦ , ◦ ).These out-of-plane excursions aid the switching tobe very fast in sub-nanosecond; the same physicsas for a single multiferroic device9 ;
19 applies heretoo. Had we not considered such out-of-plane ex-cursion, the switching delay would have been acouple of magnitudes larger, which signifies thatsteady-state analysis without considering out-of-plane motion20 is neither qualitatively nor quan-titatively accurate21 ; ◦ during 90 ◦ switching,i.e. when the magnetizations come near to theirhard axes. Upon reversing the stress on the 2ndnanomagnet, its magnetization rotates out-of-planemore, but this time at the very end, it has comeback to its plane because of the dipole coupling withthe 3rd nanomagnet, which tries to align the 2ndnanomagnet’s magnetization with its own magneti-zation. Finally, note that the magnetizations of 1stand 4th nanomagnets remain quite unchanged be-cause no stress is applied on these two nanomagnets,the slight changes in the directions of the magneti-zations occurred because of dipole coupling effectwith the neighboring nanomagnets.Fig. 14 shows magnetization dynamics for Ben-nett clocking with 30 MPa stress, rather than 5.2MPa stress as we have presented earlier. With highstress, magnetizations have deflected out-of-planemore ( ∼ ◦ ) than that for the lower stress whilereaching θ = 90 ◦ and the magnetization of 2ndnanomagnet has executed a precessional motion be-fore completing switching. Apparently, almost halfof the time taken during switching, magnetizationexperiences such unfruitful motion. To reduce such unfruitful motion, one can just use a lower stresslevel. If we apply 10 MPa stress, rather than thehigh stress 30 MPa, and magnetization does notexecute any precessional motion and the switchingdelay increases just a bit from 0.32 ns to 0.34 ns,while energy dissipation decreases 10 times due to alower stress level. Thus it is possible to engineer thestress levels to achieve good performance metricsfor application purposes. Fig. 13. Magnetization dynamics for Bennett clocking in achain of four Terfenol-D/PZT multiferroic nanomagnets withstress 5.2 MPa and assuming instantaneous ramp: (a) polarangle θ versus time, and (b) azimuthal angle φ over time whileswitching occurs, i.e. during the time θ changes from 175 ◦ to 5 ◦ . (Reprinted with permission from Ref. 21. Copyright2012, Kuntal Roy.) Similarly, we can analyze and simulate logicgates (see Fig. 4) and bigger circuits using multifer-roic composites. Since the 3-phase clocking circuitryfor Bennett clocking may be complex for applicationpurposes, any unconventional design of logic gatesand building blocks can be worked out and simu- Kuntal Roy lated too, however, based on the same techniquepresented here.
Fig. 14. Magnetization dynamics for Bennett clocking in achain of four Terfenol-D/PZT multiferroic nanomagnets withstress 30 MPa and assuming instantaneous ramp: (a) polarangle θ versus time, and (b) azimuthal angle φ over time whileswitching occurs, i.e. during the time θ changes from 175 ◦ to 5 ◦ . (Reprinted with permission from Ref. 21. Copyright2012, Kuntal Roy.)
4. Summary and Outlook
We have theoretically investigated electric-field in-duced switching of magnetization in a magnetostric-tive nanomagnet strain-coupled with a piezoelectriclayer in a multiferroic composite structure. The per-formance metrics of switching like sub-nanosecondswitching delay and ∼ Acknowledgments
I acknowledge discussions with Jayasimha Atu-lasimha, Supriyo Bandyopadhyay, Supriyo Datta,and Avik Ghosh.
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