The Dynamics of a Soliton Ratchet (repository)

The smallest scales are currently being reached for the fabrication of electronic devices. Another way of increasing the density of these devices can be to extend them into the third dimension. One of the devices that can already be made in a three-dimensional configuration is called a shift register. It transfers data along a chain, and it can be constructed out of transistors. However, this would drastically increase the complexity of the process flow required for the production of the device. Another way of constructing a shift register is by making use of magnetic kink solitons, which are vertically transferred throughout a stack of sub-nanometer thick layers of ferromagnetic material [1].

We start with a stack of magnetic layers with an out-of-plane uniaxial anisotropy. These layers are antiferromagnetically coupled by an RKKY coupling. This means that all layers are aligned antiparallel in the ground state. We now introduce a frustration in the stack, where two spins are aligned parallel. This frustration is called a magnetic kink soliton.

By carefully tuning properties such as the thicknesses of the layers and the coupling strengths, a scheme can be devised to move the soliton vertically throughout the stack. This is done by applying an external magnetic field.

The specifics behind the ratchet mechanism can be worked out analytically by making use of the Ising model [1]. This model assumes that each layer has a single-domain macrospin magnetization, and that the magnetization always points along the easy axis. The switching field of each layer can be computed analytically, and the Ising model assumes that a layer switches when an external field is applied that is larger than the switching field. This simple model gives a good estimation for what kind of fields are needed to switch layers during the ratchet operation. Requirements can be derived which the ratchet parameters have to fulfill in order for the device to function. However, the Ising model is missing time dependence. This means that it can't give predictions about the speed and stability of the ratchet.

The analysis of the soliton ratchet done in previous work [1] can be expanded by making use of a different model for the evolution of the magnetizations of ferromagnetic materials, which is the LLG equation. This model does take into account the time evolution of each magnetization in each layer of the stack, but it still assumes macrospins. The LLG equation is given by: \begin{equation} \frac{d \vec{M}}{dt} = \gamma \mu_0 \left(\vec{M} \times \vec{H}_{\text{eff}}\right) + \frac{\alpha}{M_s}\left(\vec{M} \times \frac{d\vec{M}}{dt}\right) \end{equation} Where $\gamma = \frac{-|e|}{2m_e}\left|g_e\right| \approx -1.7609\cdot10^{11}\frac{\text{rad}}{\text{T s}}$ is the gyromagnetic ratio, with $g_e \approx -2$ as the electron g-factor. The vaccuum permeability is given by $\mu_0 \approx 1.2566 \cdot 10^{-6}\frac{\text{T m}}{\text{A}}$. The first term is the torque on the magnetization by the total magnetic field $\vec{H}_{\text{eff}}$, and the second term is the introduced damping, scaled by the Gilbert damping $\alpha$. Finally, the saturation magnetization of the material is given by $M_s$. The magnetization and the effective magnetic field are both given in units of A/m. This equation can be rewritten in an explicit form for small $\alpha$: \begin{equation} \frac{d \vec{M}_n}{dt} = \frac{\gamma \mu_0}{1 + \alpha^2}\left(\left(\vec{M}_n \times \vec{H}_{n}^{\text{eff}}\right) + \frac{\alpha}{M_s} \left[\vec{M}_n \times \left(\vec{M}_n \times \vec{H}_{n}^{\text{eff}}\right)\right]\right) \end{equation} Where an index n is already introduced for the layer number. This system of equations can be solved using iterative methods such as Runge-Kutta.

The only thing left before this equation can be solved is to find the effective magnetic field $\vec{H}_{\text{eff}}$ on each layer, which is simply the sum of all magnetic fields acting on the layer. The first contribution comes from the externally applied magnetic field $\vec{H}_{\text{ext}}(t)$. The second field is the demagnetizing field, given by $\vec{H}_{D} = -\mathbf{N} \vec{M}_n$, where we assume the ferromagnetic layer to be infinitely thin: $\mathbf{N} = \text{diag}(0, 0, 1)$. The next term in the effective field comes from the uniaxial anisotropy along the z-axis, given by $\vec{H}_A = \frac{2 K^{\text{eff}}_n}{\mu_0 M_s^2}\left(\vec{M}_n \cdot \hat{z}\right)\hat{z}$. The effective anisotropy term used here is given by: $K^{\text{eff}}_n = K_v + \frac{K_s}{t_n}$, where $K_v$ and $K_s$ are the volume and surface anisotropy, respectively. The units of the anisotropy constant are given by $[K^{\text{eff}}_n] = \frac{J}{m^3}$. Furthermore, $t_n$ is the thickness of the layer along the z-axis. Note that I stated that the layers have an out-of-plane anisotropy axis, which means that the anisotropy field must be stronger than the demagnetizing field.

Finally, we have the magnetic field acting on the layer due to the exchange couplings with the layers on top and on the bottom, called the RKKY interaction [3]. This field is given by: \begin{equation} \vec{H}_{\text{ex}} = \frac{J_{n-1,n}}{\mu_0 M_s^2 t_n} \vec{M}_{n-1} + \frac{J_{n,n+1}}{\mu_0 M_s^2 t_n} \vec{M}_{n+1} \end{equation} Where the units of the coupling strength are given by $[J] = \frac{J}{m^2}$. The coupling strengths are defined to be negative for antiferromagnetic coupling. This gives the final equation for the effective magnetic field acting on layer n in the soliton ratchet: \begin{equation} \vec{H}_{n}^{\text{eff}} =\vec{H}_{\text{ext}}(t) - \mathbf{N} \vec{M}_n + \frac{2 K^{\text{eff}}_n}{\mu_0 M_s^2} \left(\vec{M}_n \cdot \hat{z}\right)\hat{z} + \frac{J_{n-1,n}}{\mu_0 M_s^2 t_n} \vec{M}_{n-1} + \frac{J_{n,n+1}}{\mu_0 M_s^2 t_n} \vec{M}_{n+1} \end{equation} Now that we know which fields act on the layers, we can start the analysis of the soliton ratchet using the Ising model. For this, we have a stack of 8 layers with a down soliton inserted in the middle, where all the magnetizations are perfectly aligned with the z-axis.

We can now calculate the effective magnetic fields acting on these layers in this configuration, in the absence of an externally applied magnetic field. For now, we only look at the fields acting on layers 4 and 5, which make up the soliton. This gives the following effective fields in the z-direction: \begin{equation} \begin{split} H^{\text{eff}}_4 &= -\frac{2 K(t_2)}{\mu_0 M_s} + M_s - \frac{\Delta J}{\mu_0 M_s t_2}\\ H^{\text{eff}}_5 &= -\frac{2 K(t_1)}{\mu_0 M_s} + M_s - \frac{\Delta J}{\mu_0 M_s t_1} \end{split} \end{equation} With $K(t) = K^{\text{eff}} = K_v + \frac{K_s}{t}$. Furthermore, $\Delta J \equiv |J_1| - |J_2| > 0$. We also define the ratchet such that $t_1 < t_2$, which means that $K(t_1) > K(t_2)$. These fields are both negative for any value of $\Delta J$, as we have an out-of-plane anisotropy axis, given by the following requirements: \begin{equation} \begin{split} \frac{2 K(t_1)}{\mu_0 M_s} &> M_s\\ \frac{2 K(t_2)}{\mu_0 M_s} &> M_s \end{split} \end{equation} We now want to apply an external field in the positive z direction, to flip layer 4 and move the soliton one layer upwards. This is possible, as the field keeping layer 4 pointed downwards is weaker than the field acting on layer 5. The field applied is given by: \begin{equation} H^{\text{ext}}_1 = H^{\text{s}}_4 + \varepsilon_1 \Delta H_1 \end{equation} Where the switching field $H^{\text{s}}$ is defined as the absolute value of $H^{\text{eff}}$, and $\Delta H_1 \equiv H^{\text{s}}_5-H^{\text{s}}_4$. Furthermore, the field strength of the first ratchet step is given by $0 \leq \varepsilon_1 < 1$.

The second ratchet step is a bit more involved, as we now want layer 3 to flip downwards, instead of that layer 4 flips back. We again calculate the effective magnetic fields, but now on layers 3 and 4: \begin{equation} \begin{split} H_3^{\text{eff}} &= +\frac{2 K(t_1)}{\mu_0 M_s} - M_s - \frac{\Delta J}{\mu_0 M_s t_1}\\ H_4^{\text{eff}} &= +\frac{2 K(t_2)}{\mu_0 M_s} - M_s - \frac{\Delta J}{\mu_0 M_s t_2}\\ \end{split} \end{equation} We now require that these layers are unstable without an external field, which enables us to flip layer 3 instead of layer 4, and this means that $H_3^{\text{eff}}$ and $H_4^{\text{eff}}$ must be negative. This gives the following requirements: \begin{equation} \Delta J > \left(2 K(t_1) - \mu_0 M_s^2\right) t_1 \text{ and } \Delta J > \left(2 K(t_2) - \mu_0 M_s^2\right) t_2 \end{equation} We now define these destabilizing fields to be the absolute values of $H_3^{\text{eff}}$ and $H_4^{\text{eff}}$ again. By reducing the field $H^{\text{ext}}_1$ to just below the destabilizing field of layer 3, this layer will flip. This means that the destabilizing field of layer 3 must be stronger than that on layer 4, which gives the final requirement: \begin{equation} \Delta J > 2 \Delta K \frac{t_1 t_2}{\Delta t} \end{equation} With $\Delta K \equiv K(t_1) - K(t_2)$ and $\Delta t \equiv t_2 - t_1$. The field applied during the second ratchet step is given by: \begin{equation} H^{\text{ext}}_2 = H^{\text{d}}_3 - \varepsilon_2 \Delta H_2 \end{equation} With $\Delta H_2 \equiv H^{\text{d}}_3 - H^{\text{d}}_4$, and $0 \leq \varepsilon_2 < 1$. To summarize, the Ising model predicts that we have a working soliton ratchet if the difference in coupling strengths $\Delta J$ is large enough. However, the Ising model can't make predictions about the speed and stability with which the ratchet will function for certain J values. For instance, one would expect that the ratchet will suffer greatly under stability issues for large coupling strengths J. The same holds for the field strengths $\varepsilon$, which can take on any value between zero and one.

The LLG equation can be used to make predictions about the stability and speed of the soliton ratchet, as it calculates the time evolution of all magnetizations of the layers. A parameter that needs to be added when making use of the LLG equation is $\theta_{\text{ext}}$, which is the angle of the external field with the z-axis. When all magnetizations are perfectly aligned with the z-axis at the start, the external field needs to be applied at an angle to exert a torque on the magnetizations. This angle has great effects on the stability and speed of the ratchet.

To start the stability analysis, we choose to first fix the thicknesses $t_1$ and $t_2$ of the layers. This in turn fixes the anisotropy constants and the saturation magnetization. The chosen constants are given by [1]: \begin{equation} \begin{split} &t_1 = 0.7 \text{ nm, }t_2 = 0.8 \text{ nm}\\ &K(t_1) = 1.66 \text{ MJ/m}^3\text{, }K(t_2) = 1.59 \text{ MJ/m}^3\\ &M_s = 1.29 \text{ MA/m} \end{split} \end{equation} Where we assume that the dependence of the saturation magnetization on the thickness of the layer is negligible. For these constants, we can use the Ising model to predict for which values of $J_1$ and $J_2$ we should have a working ratchet:

The difference in coupling strength $\Delta J$ must be large enough to fulfill the requirements introduced before, so the yellow region in the graph has a working ratchet, according to the Ising model. However, when trying out some values, we experienced that only in a small region within the yellow plane the ratchet actually functioned. To investigate this further, we need to quantify the stability of the ratchet. For this, we make use of the following quantity: \begin{equation} Q = \sum_{n\neq n_{\text{switch}}}\langle (M_n - \langle M_n \rangle)^2\rangle \end{equation} Which simply adds the variances of the magnetizations of all non-switching layers. Now we choose a few values in the yellow domain where the ratchet did function when modelled using the LLG equation, and we perform the first ratchet step. While doing this, we vary the field strength $\varepsilon_1$ and the field angle $\theta_{\text{ext}}$. The obtained graphs show nice stability patterns, with yellow hotspots of high instability. We immediately see that the ratchet is highly unstable for higher values of $\Delta J$. To see what causes this, we take point 3 from the $J_1$ against $J_2$ plane and choose values for $\varepsilon_1$ and $\theta_{\text{ext}}$ in the highly unstable regime. We see that layer 4 switches like it should, but a bit later, layer 5 also switches. This can be explained by the fact that the field $H^{\text{ext}}_1$ needed to switch layer 4 comes closer and closer to the switching field of the bulk. It will never exceed the switching fields of the other layers, but unlike in the Ising model, the LLG equation does not require an applied field to exceed the switching field, especially if we apply the field at an angle. We can now also investigate the switching times of the first ratchet step for the 9 chosen values of $J_1$ and $J_2$. We define a layer to be switched if $M_z > \beta M_s$, with $\beta$ arbitrarily chosen to be 0.98. As expected, if we move toward higher $|J_1| + |J_2|$ values, the switching times get shorter. However, point 7, 8 and 9 suddenly shows a large region where the switching times get very large. We now take point 7 and choose values for $\varepsilon_1$ and $\theta_{\text{ext}}$ in the non-switching region. The simulation actually shows that the layer flips very fast, as expected, but that it never actually aligns completely with the z-axis. This can be explained by the fact that the exchange fields are much stronger than the anisotropy field, which tries to align the spin with the z-axis. This means that the magnetization stays aligned with the external field, which is applied at an angle. To summarize, using the LLG equation we can optimize the ratchet for the coupling strengths, field strengths and the field angle. High values for $\Delta J$ are unstable, because the bulk layers will start switching too. High $|J_1| + |J_2|$ values are unstable as the magnetizations will not align with the z-axis anymore.

The second ratchet step can also be investigated, but the results are less interesting. The stability of the step barely depends on both $\varepsilon_1$ and $\theta_{\text{ext}}$. This could be explained by the fact that the field that performs the switch is not the external field, but the intrinsic destabilizing field, which is not applied at an angle. However, the switching times only depend on the angle. The switch of a layer starts with a slow precession around the z-axis, which gradually increases until the layer completely flips. This first part is where the external field, which is applied at an angle, can speed up the process.

Finally, the optimized parameters can be chosen for our fixed layer thicknesses. Point 1 in the $J_1$ against $J_2$ diagram is chosen for its high stability, and the other parameters are chosen as far in the top left of the stability diagram as possible. Here, the stability is high and the switching time is low. \begin{equation} \begin{split} &J_1 = -0.00122\text{ J/m}^2\text{, }J_2 = -0.00034\text{ J/m}^2\\ &\varepsilon_1 = 0.8\text{, }\varepsilon_2 = 1.0\text{, }\theta_{\text{ext}} = 0.00095 \text{ rad} \end{split} \end{equation} The ratchet can now be simulated using the LLG equation, which shows a reliable ratchet for 32 layers and 4 full ratchet cycles. The two ratchet steps take 2 nanoseconds each, which corresponds to a data transfer speed of 0.5 GHz.

To conclude, the Ising model can be used to derive requirements for a soliton ratchet to function. However, a model such as the LLG equation must be used to optimize the ratchet, as the Ising model does not take the time dynamics of the system into account.

References

[1] R. Lavrijsen, J.-H. Lee, A. Fern ́andez-Pacheco, D. C. M. C. Petit, R. Mansell, and R. P. Cowburn, Nature 493, 647 (2013).
[2] J. M. D. Coey, Magnetism and magnetic materials (Cambridge University Press, 2010).
[3] M. A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954).