Equivalent Circuit for Magnetoelectric Read and Write Operations

We describe an equivalent circuit model applicable to a wide variety of magnetoelectric phenomena and use spice simulations to benchmark this model against experimental data. We use this model to suggest a different mode of operation where the 1 and 0 states are represented not by states with net magnetization (like $m_x$, $m_y$, or $m_z$) but by different easy axes, quantitatively described by ($m_x^2-m_y^2$), 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 ($m_x^2-m_y^2$) to represent a bit is a radical departure from the standard convention of using the magnetization ($m$) to represent information. We then show how the equivalent circuit can be used to build a device exhibiting tunable randomness and suggest possibilities for extending it to nonvolatile memory with read and write capabilities, without the use of external magnetic fields or magnetic tunnel junctions.

Figure 1

Equivalent circuit for ME read and write operations. (a) The charge on the piezoelectric (PE) capacitor changes the easy axis of the ferromagnet (FM), which causes a change in the output voltage $V_L$ through the inverse effect. (b) Equivalent circuit model obtained from Eq. (1). The write operation is through the effective field ${\overrightarrow{H}}_{\mathrm{ME}}=-{\mathbf{\nabla }}_{\mathbf{m}}{E}_m/(M_s\mathcal{V})$ that enters the SLLG equation. Read operation is through the dependent voltage source $V$ that is proportional to $\partial E_m/\partial Q$, where $E_m$ is the magnetic energy.

Figure 2

Experiment vs circuit model. (a) The results of the self-consistent circuit model for the structure in (b) are in good agreement with the experimental results in Ref. [25]. $V_{\mathrm{ME}}$ is the mathematical difference of two measurements of $V_R$ with and without the external magnetic field, $V_{\mathrm{ME}}=V_R(H\ne 0)-V_R(H=0)$. (b) The experimental structure reported in Ref. [25] where the piezeoelectric (PE) is a $⟨011⟩$-cut PMN-PT substrate and the ferromagnet (FM) is $N$ layers of ${\mathrm{TbCo}}_2/\mathrm{FeCo}$. The back voltage is $V=v_M\mu$, where $\mu =m_x^2-m_y^2$ and the magnetic energy is $E_m=Q_{\mathrm{PE}}{v}_M\mu$, where $Q_{\mathrm{PE}}$ is the charge on the capacitor $C_{\mathrm{PE}}$. The following parameters are used: coercivity for FM ($H_K=200\mathrm{Oe}$); saturation magnetization, $M_s=1100\mathrm{emu}/{\mathrm{cm}}^3$; FM thickness, $t_{\mathrm{FM}}=200\mathrm{nm}$; PE thickness, $t_{\mathrm{PE}}=30\mu \mathrm{m}$; $\mathrm{area}=520\times 520{\mathrm{nm}}^2$; magnetoelastic constant, $B=-7\mathrm{MPa}$; a net PE constant, $d=d_{31}-d_{32}=2500\mathrm{pC}/\mathrm{N}$; permittivity, $\epsilon =4033{\epsilon }_0$; resistance, $R=2\mathrm{M}\mathrm{\Omega }$; back voltage, $v_M=Bd{t}_{\mathrm{FM}}/2\epsilon$. In the experiment, the magneto-optic Kerr effect (MOKE) is used to show the variation of magnetization, which is compared to the pseudomagnetization in our simulation. Experimental panel is reproduced with permission of AIP Publishing, from Ref. [25].

Figure 3

Tunable randomness. Results for the structure in Fig. 1 using the circuit model in Fig. 1 with a circular magnet ($H_K\to 0$, $E_A\to 0$). $C=C_L=50\mathrm{aF}$ and $v_M=10\mathrm{mV}$, such that $C_{\mathrm{eff}}{v}_M^2/kT<1$. (a) Three results are shown for the magnetization, $\mu$: transient spice simulations (solid blue) where the input voltage ($V_{\mathrm{in}}$) is swept from $-50$ to $+50\mathrm{mV}$ in $1\mu \mathrm{s}$ and pseudomagnetization, and $\mu$ is plotted against $V_{\mathrm{in}}$. Separate spice simulations are shown for each solid square where an average magnetization is obtained over 100 ns. The exact Boltzmann integral is obtained from Eq. (6). (b) The same results for the differential load voltage, $\mathrm{\Delta }{V}_L=V_L-V_L(v_M=0)$, in this case, $V_L(v_M=0)=V_{\mathrm{in}}/2$. The actual load voltage has a linear $V_{\mathrm{in}}$ dependence superimposed on $\mathrm{\Delta }{V}_L$, similar to Fig. 4. The differential load voltage is shown here for clarity.

Figure 4

Nonvolatility. When $C_{\mathrm{eff}}{v}_M^2/kT$ exceeds 1, the pseudomagnetization ($\mu$) for the circular magnet ($H_K\to 0$, $E_A\to 0$) investigated in Fig. 3 becomes stable. (a) The input voltage ($V_{\mathrm{in}}$) doing a negative-positive-negative sweep as a function of time. (b) Load voltage ($C=C_L=100\mathrm{aF}$) as a function of $V_{\mathrm{in}}$. (c) $\mu$ as a function of $V_{\mathrm{in}}$, exhibiting hysteresis.