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Equivalent Circuit for Magnetoelectric Read and Write Operations

Kerem Y. Camsari,1 Rafatul Faria,1 Orchi Hassan,1 Brian M. Sutton,1 and Supriyo Datta1

1School of Electrical and Computer Engineering, Purdue University, IN, 47907(Dated: April 18, 2018)

We describe an equivalent circuit model applicable to a wide variety of magnetoelectric phenomenaand use SPICE simulations to benchmark this model against experimental data. We use this modelto suggest a different mode of operation where the “1” and “0” states are not represented by stateswith net magnetization (like mx, my or mz) but by different easy axes, quantitatively describedby (m2

x − m2y) which switches from “0” to “1” through the write voltage. This change is directly

detected as a read signal through the inverse effect. The use of (m2x − m2

y) to represent a bitis a radical departure from the standard convention of using the magnetization (m) to representinformation. We then show how the equivalent circuit can be used to build a device exhibitingtunable randomness and suggest possibilities for extending it to non-volatile memory with read andwrite capabilities, without the use of external magnetic fields or magnetic tunnel junctions.

I. INTRODUCTION

In magnetic random access memory (MRAM) tech-nology write units are typically based on spin-torque orspin-orbit torque, while read operations are based on themagnetoresistance of magnetic tunnel junctions (MTJ).But there is increasing interest in voltage-driven unitsdue to the potential for low power operation, both activeand stand-by based on different types of magnetoelectricphenomena [1–20].The central result of this paper is an equivalent circuit

model (Fig. 1) applicable to a range of magnetoelectric(ME) phenomena including both write and read opera-tions. It consists of a capacitor circuit which incorporatesthe back voltage from the magnetoelectric coupling de-scribed by (1):

VIN =Q

CL

+Q

C+

∂Em

∂Q(1)

where Em is the magnetic energy including the partcontrolled by the charge Q on an adjacent capacitorC, through the ME effect. Equation (1) is solved self-consistently with the stochastic Landau-Lifshitz-Gilbert(s-LLG) equation which feels an effective field ( ~Hme =−∇m Em/MsVol.), ∇m represents the gradient oper-ator with respect to magnetization directions mi, Ms isthe saturation magnetization and Vol. is the volume ofthe magnet. The s-LLG treatment for all simulations inthis paper is similar to what is described in [21–23] andis not repeated here. We first benchmark this equivalentcircuit against the recently demonstrated MagnetoELec-tric Random Access Memory (MELRAM) device [24, 25]which uses the magnetoelectric effect (ME) and its in-verse (IME) for write and read operations, using a struc-ture whose energy Em is given by Eq. 2. We then arguethat, unlike MELRAM, the “1” and the “0” states neednot be represented by states with a net magnetization.For example, using a structure whose energy Em is givenby Eq. 4, one could instead switch the easy axis with awrite voltage, and this change in the easy axis can be

LLG

(a) (b)

Fig. 1. Equivalent circuit for magnetoelectric (ME)read and write operations (a) The charge on the piezo-electric (PE) capacitor changes the easy-axis of the ferromag-net (FM) and this causes a change in the output voltage VL

through the inverse effect. (b) Equivalent circuit model ob-tained from (1). Write operation is through the effective

field ~Hme = −∇m Em/(MsVol.) that enters the stochasticLandau-Lifshitz-Gilbert (s-LLG) equation. Read operation isthrough the dependent voltage source V that is proportionalto ∂Em/∂Q, where Em is the magnetic energy.

read as a change in the voltage across a series capacitorthrough the inverse effect, allowing a “field-free” opera-tion without any symmetry breaking magnetic field.

II. EXPERIMENTAL BENCHMARK

We start with the MELRAM device (Fig. 2b) reportedrecently in [25] where the magnetic energy has the form

Em = −EAmxmy

+EH/√2(mx −my) + vMQ (m2

x −m2y) (2)

We note that this energy expression is essentially thesame as what was reported in Ref. [25] expressed usingmagnetization components, mx,my,mz. For example,the anisotropy energy is written in [25] as −EA sin2 φ,

2

LLG

(a) (b)

Fig. 2. Experiment vs circuit model: (a) The results of the self-consistent circuit model for the structure in (b) arein good agreement with the experimental results in [25]. VME is the mathematical difference of two measurements of VR

with and without the external magnetic field, VME = VR(H 6= 0) − VR(H = 0). (b) Experimental structure reported in [25]where the piezeoelectric (PE) is 〈011〉-cut PMN-PT and the ferromagnet (FM) is N layers of TbCo2/FeCo. The back-voltageis V=vMµ where µ = m2

x − m2y and the magnetic energy is Em = QPEvMµ where QPE is the charge on the capacitor

CPE. The following parameters are used: Coercivity for FM (HK=200 Oe), saturation magnetization Ms=1100 emu/cc, FMthickness, tFM=200 nm, PE thickness tPE=30 µm, Area=520 × 520 nm2, Magnetoelastic constant B = −7 MPa, a net PEconstant, d = d31 − d32 = 2500 pC/N, permittivity ǫ = 4033 ǫ0, resistance R = 2 MΩ, back voltage vM = BdtFM/2ǫ. Inthe experiment, magneto-optic Kerr effect (M.O.K.E) is used to show the variation of magnetization, which is compared tothe pseudo-magnetization in our simulation. Experimental panel is reproduced with permission of AIP Publishing LLC, fromReference [25].

with φ measured from the magnetic field ~Hext such thatmx = cos(3π/4 − φ),my = sin(π/4 − φ) and mxmy =sin2 φ, ignoring an unimportant constant. Similarly theZeeman term is written in [25] as−EH cosφ which equalsEH(mx − my)/

√2. In [25], the uniaxial anisotropy

energy term and the external magnetic field were inge-niously balanced (by choosing EH = EA

√2) to provide

two unique low energy states that represent “0” and “1”at φ = π/2 and φ = π.Finally, the last term represents the ME effect where

an applied voltage generates a chargeQ, controlled by theinput voltage VIN, which changes the anisotropy energysuch that a positive (or negative) Q causes the magneticenergy to favor the y-axis (or the x-axis) for a positivevM . This is due to the anisotropic piezoelectric coeffi-cients d31 and d32 having different signs, a special prop-erty of the 〈011〉-cut (PMN-PT) that was chosen in theexperiment.The equivalent circuit incorporates the back voltage

from the ME coupling using (1), with the load capacitor

CL replaced by a resistor R:

VIN = RdQ

dt+

Q

C+

∂Em

∂Q

= RdQ

dt+

Q

C+ vM (m2

x −m2y) (3)

It is possible to write the ME energy as qMV in termsof an applied voltage V rather than charge Q, but thischoice would lead to a back charge ∂Em/∂V instead of aback voltage ∂Em/∂Q, giving a different but equivalentlooking circuit model.Fig. 2a shows the write and read signals for the exper-

imental structure in Fig. 2b calculated using a SPICEmodel, that are in good agreement with the experimen-tal results presented in [25]. The reason for the verydifferent time scales of the experiment and the circuitmodel is that the circuit model solves the real-time dy-namics of the nanomagnet with time steps of the order ofa fraction of the inverse FMR frequency of the nanomag-net (1/f ∼ 2π/γ/

√

[HK(HK + 4πMs)] ∼ 0.2 ns for the

3

chosen parameters) to avoid large numerical integrationswhile the experimental measurement is performed withquasi-static pulses. Therefore the RC time constants inboth cases are very different, however the maxima andminima of each signal closely match based on the chosenparameters.

III. FIELD-FREE OPERATION

It is evident from Fig. 2a that our equivalent circuitdescribes the switching process accurately in the experi-ment described in Ref. [25]. Using the same circuit modelwe would like to suggest the possibility of field-free op-eration where “0” and “1” are represented by two dif-ferent easy axes rather than two different magnetizationdirections. For this illustration, we consider a ferro-magnet whose easy axis does not lie along mx = ±my asin Ref. [25], but rather along the y-axis (mx = 0), corre-sponding to an anisotropy energy given by EA(m

2x−m2

y).Also, there is no external magnetic field so that EH = 0giving an overall energy expression of the form

Em = (EA + vMQ)(m2x −m2

y) (4)

instead of Eq. 2.A positive induced chargeQmakes y-direction the easy

axis so that 〈m2x −m2

y〉 = −1, while a negative Q makesx-direction the easy axis so that 〈m2

x − m2y〉 = +1, and

this constitutes the writing operation. The inverse of thesame effect gives rise to a back voltage that allows oneto read the information. Using (4) we obtain from (1):

VIN =Q

CL

+Q

C+ vM (m2

x −m2y) (5)

Use of this “pseudo-magnetization” µ ≡ m2x −m2

y is aradical departure from the standard convention of usingthe magnetization (mx,my or mz) to represent a bit [26],opening up new possibilities for writing and reading.Even though we have limited our discussion to the com-

posite PE / FM structures that give rise to a magneto-electric effect due to a coupling of the strain from a PEmaterial to a magnetostrictive FM material, we believethe circuit description described in Fig. 1 could be of gen-eral use. Indeed, it should be possible to use other quan-tities represented by a function f(mx,my,mz) to repre-sent a bit. Any mediating term due to strain, charge,orbital or other microscopic mechanisms giving rise to aterm of the form (Q × f) in the energy expression thatcan be used to write such a bit, should also give rise toan inverse effect for read out.In the next two sections, we show two example uses of

the external magnetic field-free operation of the equiva-lent circuit.

(a)

(b)

Fig. 3. Tunable randomness: Results for the structurein Fig. 1a using the circuit model in Fig. 1b with a circularmagnet (HK → 0, EA → 0). C = CL = 50 aF and vM = 10mV such that Ceffv

2

M/kT < 1 (a) Three results are shown formagnetization, µ : Transient SPICE simulations (Solid blue)where the input voltage (VIN) is swept from −50 mV to +50mV in 1 µs and pseudo-magnetization, µ is plotted againstVIN. Separate SPICE simulations for each solid square wherean average magnetization is obtained over 100 ns. ExactBoltzmann integral obtained from Eq. 6. (b) Same resultsfor the differential load voltage, ∆VL = VL −VL(vM = 0), inthis case VL(vM = 0) = VIN/2. The actual load voltage hasa linear VIN dependence superimposed on ∆VL, similar toFig. 4. The differential load voltage is shown here for clarity.

IV. EXAMPLE #1: TUNABLE RANDOMNESS

The first example we illustrate using the equivalent cir-cuit of Fig. 1 is obtained by coupling the circuit shown inFig. 1 with a low-barrier circular nanomagnet that doesnot have an easy axis (HK → 0) and no energy barrier(EA = 0) that favors a magnetization axis [27, 28]. Themagnetization of such a magnet fluctuates randomly inthe plane, in the presence of thermal noise. The read andwrite mechanisms of the ME effect convert the fluctua-tions in the pseudo-magnetization µ to a voltage.Fig. 3 shows the differential load voltage ∆VL vs VIN

assuming C = CL =50 aF making Ceff = CCL/(C +CL) = 25 aF and vM = 10 mV, consistent with the ma-terial parameters for the experimental system in Fig. 2b,though the coupling coefficient vM is chosen somewhatsmaller, (such that Ceffv

2M/kT < 1, as we explain in the

next section) in order to avoid any hysteresis or mem-ory effects. Alternatively one could use a smaller load

4

capacitance, reducing Ceff .With this choice of parameters, the magnetizations and

hence the voltages fluctuate with time, and the averagedvalues over a time interval of ≈100 ns match the averageresults obtained from the Boltzmann probability:

〈µ〉 =

∫ Q=+∞

Q=−∞

∫ φ=+π

φ=−π

dφ dQ

µ=m2

x−m2

y

︷ ︸︸ ︷

cos(2φ) ρ(Q,φ)

∫ Q=+∞

Q=−∞

∫ φ=+π

φ=−π

dφ dQ ρ(Q,φ)

(6)

where ρ(Q,φ) = 1/Z exp[−E(φ,Q)/kT ] and E =QvMµ+Q2/(2Ceff)−QVIN represents the total energy.Similar to our previous discussion, we assume that themagnetization for the circular in-plane magnet is confinedto the plane of the magnet due to the strong demagne-tization field. Therefore, the magnetization integral canbe taken in the plane (φ → ±π) and this seems to bein good agreement with the numerical s-LLG results asshown in Fig. 3. The average load voltage is obtainedusing Eq. 6, but replacing cos(2φ) with Q/CL.Eq. 6 does not seem to reduce to a compact closed

form, but assuming Q = Ceff(VIN ± vM |µ|) ≈ CeffVIN forsmall vM , allows a direct evaluation:

〈µ〉 ≈ −I1(x)

I0(x)(7)

where In is the modified Bessel function of the first kind[29], and x = QvM/kT . This approximation (not shown)seems to be in good agreement with an exact numericalevaluation of Eq. 6 that is shown in Fig. 3 and could beuseful as an analytical guide.Note that the SPICE simulation solves the magne-

tization and the load voltage self-consistently followingthe equivalent circuit (Fig. 1a) while the Boltzmann lawtakes these self-consistencies into account exactly. Theagreement between the two constitutes another impor-tant benchmark for our equivalent circuit.With additional gain and isolation that can be incor-

porated by CMOS components this field-free voltage-tunable randomness can become a potential voltage con-trollable “p-bit” (probabilistic bit) that can be used as abuilding block for a new type of probabilistic logic [30–35] or other neuromorphic approaches that make use ofstochastic units [36–41], but this is not discussed further.

V. EXAMPLE #2: NON-VOLATILEOPERATION

It is easy to see by integrating Eq. 6 that even whenone uses a stable magnet (EA > 40 kT ) in Eq. 4, thepseudo-magnetization µ does not show “hysteretic” be-havior as function of VIN, but simply shifts the sigmoidresponse of Fig. 3 to the left or right depending on the

Fig. 4. Non-volatility: When Ceffv2

M/kT exceeds 1, thepseudo-magnetization (µ) for the circular magnet (HK →0, EA → 0) investigated in Fig. 3 becomes stable. (a) Showsthe input voltage (VIN) doing a negative-positive-negativesweep as a function of time. (b) Load voltage (C = CL = 100aF as a function of VIN. (c) µ as a function of VIN, exhibitinghysteresis.

sign of EA. The average sigmoidal behavior of Fig. 3 isnot just a consequence of using circular magnets, even a40 kT magnet would show non-hysteretic behavior, butwith suppressed fluctuations in µ and a shift along theVIN axis. To obtain hysteretic behavior for the pseudo-magnetization µ we need an energy term that is quadratic(∼ µ2) rather than linear (= EA µ) as in Eq. 4. but wewill not discuss the possibility further in this paper. Nextwe show that the energy expression we have used couldlead to hysteretic behavior if the ME coefficient vM werelarge enough. Such a quadratic term could arise natu-rally from the physics which we hope motivates futureinvestigation.Fig. 4 shows the results of a transient simulation of

the equivalent circuit with a circular magnet, similarto Fig. 3 with the only difference that in this examplethe back-voltage (vM ) is increased to 100 mV such thatCeffv

2M/kT ≫ 1. An input voltage is slowly swept from

−200 mV to +200 mV and back to −200 mV within 1

5

µs, where pseudo-magnetization (µ) and the load volt-age (VL) show hysteresis, similar to the magnetizationof an ordinary magnet. One way to understand the hys-teretic behavior is to note that the total energy for thefull circuit in Fig. 1 can be written as:

Etotal =Q2

2Ceff

+QvMµ−QVIN (8)

where C−1

eff= C−1 + C−1

L .Expanding Eq. 7 for small vM , we can approximate

the pseudo-magnetization by µ ≈ −QvM/(2kT ) and wehave:

Etotal ≈Q2

2Ceff

− Q2v2M2kT

−QVIN (9)

suggesting that the ME effect provides a negative capac-

itance −kT/v2M in series with Ceff leading to hystereticbehavior when Ceffv

2M > kT reminiscent of similar be-

havior based on ferroelectrics [42, 43].Numerical simulations of the equilibrium fluctuations

of this magnet also show that the thermal stability ofthe µ is ≈ Ceffv

2M/kT which can be 60 or greater, for

reasonable values of vM and Ceff providing the possibilityof non-volatile memory applications based on the pseudo-magnetization µ.

VI. CONCLUSION

In summary, we have presented an equivalent circuitfor magneotoelectric read and write and showed that itdescribes recent experiments on the MELRAM devicequite accurately. We then used this circuit model toillustrate the possibility of representing “1” with differ-ent easy axes, encoded by the pseudo-magnetization µ,rather than with different magnetizations, allowing a nat-ural field-free operation that can be useful for a numberof applications in stochastic neuromorphic computing.Lastly, we showed the possibility of using the pseudo-magnetization for non-volatile memory applications.

ACKNOWLEDGMENT

The authors acknowledge insightful discussions withV. Ostwal, P. Debashis, Z. Chen, J. Appenzeller, S. Ma-jetich and J. T. Heron. This work was supported in partby the National Science Foundation through the NCN-NEEDS program, contract 1227020-EEC, the Nanoelec-tronics Research Initiative through the Institute for Na-noelectronics Discovery and Exploration (INDEX).

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