1. Introduction
Magnetic topological states, e.g. skyrmions, being distinct magnetic quasiparticles with nontrivial topology [1–3] and chiral magnetic structures, have been intensively investigated for prospective applications in spintronics [4, 5] including non-volatile memory devices [6–9], nano-oscillators [10, 11], race-track memory [12] and logic gates [13], etc. For the
practical implementation of magnetic skyrmions in spintronic devices, such as random access memory, their creation, stabi- lization and manipulation properties at room temperature are very important [14–20].
Towards this purpose, various techniques such as using external magnetic fields [21], electric currents [22], local heating [23] and more recently spin-transfer torque (STT) effects using spin polarized currents [24] have been attempted in the past few years. Unfortunately, these techniques suffer from the drawbacks of high energy consumption as well as
Journal of Physics D: Applied Physics
Strain-mediated electric manipulation of magnetic skyrmion and other topological states in geometric confined nanodiscs
Nasir Mehmood1 , Xiao Song1, Guo Tian1, Zhipeng Hou1, Deyang Chen1, Zhen Fan1, Minghui Qin1 , Xingsen Gao1,3 and Jun-Ming Liu1,2
1 Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials and Institute for Advanced Materials, South China Academy of Advanced Optoelectronics, South China Normal University, Guangzhou 510006, People’s Republic of China
2 Laboratory of Solid State Microstructures and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, People’s Republic of China
E-mail: [email protected]
Received 31 July 2019, revised 12 September 2019 Accepted for publication 25 September 2019 Published 17 October 2019
Abstract
Magnetic skyrmions in multilayer stack structures are promising magnetic memory bits for applications in race-track storage devices or random-access memories. For practical implementation in energy efficient memory devices, it is essential to achieve a deterministic switch of skyrmion states, e.g. nucleation and elimination, triggered by electric fields. In this work, by means of micromagnetic simulation, we demonstrate the strain-mediated electric field driven switching of magnetic skyrmion and other topological states without the presence of a magnetic field in a proposed device consisting of nanodiscs of trilayer Pt/Co/
Ta stacks on a piezoelectric substrate. It is revealed that the multiple magnetic state switching behaviors, including the nucleation and elimination of skyrmion or labyrinth stripe domain states, as well as topological transitions between skyrmion, vortex, skyrmionium and vertical single domain states, can be triggered by applying an external electric field mediated by strains. The corresponding critical strains and their dependence on the geometric parameters of the nanodiscs (e.g. size and number of stacks) for triggering such magnetic switching are also explored. The stability of these topological domain states and possible switching behaviors (e.g. volatile and non-volatile) between the different states are discussed as well.
The capability for controlled switching of these topological states provides a new pathway to energy efficient high-density magnetoelectric memory and logic devices.
Keywords: skyrmion, electric driven magnetic switching, nanomagnets, high-density memory (Some figures may appear in colour only in the online journal)
N Mehmood et al
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© 2019 IOP Publishing Ltd 53
J. Phys. D: Appl. Phys.
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10.1088/1361-6463/ab47bd
Paper
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Journal of Physics D: Applied Physics IOP
2020
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3 Author to whom any correspondence should be addressed.
https://doi.org/10.1088/1361-6463/ab47bd J. Phys. D: Appl. Phys. 53 (2020) 014007 (9pp)
large heat dissipation. One possible way to overcome these disadvantages is to reduce the threshold current through the modulation of perpendicular magnetic anisotropy (PMA) by means of an external electric field or voltage [25–28]. This method is highly energy efficient since the energy consump- tion to switch the magnetic states is negligible due to the very small amount of current required. This was also demonstrated by numerous observations of magnetic state switching based on electric field-induced magnetic anisotropy manipulation mediated by magneto-elastic coupling in multiferroic hetero- structures consisting of nanomagnets elastically coupled with a piezoelectric substrate [29, 30]. Recently, strain-mediated manipulation of magnetic skyrmions has also been reported [31, 32].
On the other hand, nanoscale multiferroic heterostruc- tures might present a more favorable pathway to the effective tuning of the magnetic states, due to the enhanced mechanical coupling as well as remarkable shape anisotropy effect [33– 35]. This was illustrated by the prediction of strain-mediated controlled creation and annihilation of magnetic skyrmions in nanodiscs [32]. However, a more comprehensive investigation of electric tuning of magnetic skyrmions and other topolog- ical magnetic states in nanoscale multiferroic heterostructures has not yet been attempted. For instance, both skyrmion and skyrmionium (also known as the target state) have been pre- dicted to be stabilized in confined nanoscale geometries [36– 41] which further manifests that nanoscale size confinement might provide more opportunities for the electric switching of multiple topological states, holding promise for energy effi- cient high-density memory devices.
In this work, by means of a micromagnetic simulation study we demonstrate the reversible strain-mediated electric manipulation of isolated magnetic topological states including the creation and elimination of a helical stripe domain and switching between skyrmion, vortex, skyrmionium and single domain states in magnetic nanodiscs (see figures 1(a) and (b)).
Here, the proposed device consists of magnetic nanodiscs with Pt/Co/Ta trilayer stacks grown on a piezoelectric substrate, which enables the stabilization of various topological states without involving any external magnetic fields. The observa- tion of multiple magnetic state switching demonstrates the potential for energy efficient high-density magnetoelectric memory and logic devices, with a simple architecture that is compatible with modern microelectronic processes for inte- grated devices.
2. Micromagnetic simulation
The device structure proposed for the micromagnetic simula- tion in this work is shown in figure 1(a), which consists of a nanodisc with stacks of trilayered Pt(3 nm)/Co(1.8 nm)/
Ta(1.9 nm), assumed to be deposited on a piezoelectric sub- strate. The trilayer stacks repetition number (n) of the nano- discs varies from 1 to 16 and the disc diameter (ddisc) varies from 100 nm to 1 µm. Using suitable parameters, we were able to produce various topological domain states (see figures 1(a)
and (b)) in a controlled manner and investigate the switching behaviors between these states.
In skyrmion based spintronic device applications, micro- magnetic simulation has been widely implemented to model the dynamic behaviors of the magnetic states as well as the functionality and performance of devices [42]. The computa- tional study presented here was performed using the MuMax3 micromagnetic simulation software package [43–45] which calculates the magnetic switching dynamics on the basis of the finite difference discretization technique using the fol- lowing explicit form of the Landau–Lifshitz equation: [43]
∂m
∂t = γLL
1+α2(m×Beff+α(m×(m×Beff))) (1) where γLL and α are the gyromagnetic ratio and dimension- less damping parameter respectively while Beff is the effec- tive field consisting of the Heisenberg exchange (Bexch), the Dzyaloshinskii-Moriya exchange (Bdm), magnetostatic (Bdemag) and magneto-crystalline anisotropy (Banis), the Zeeman (external applied field) (Bext) and thermal (Btherm) field contributions. The simulation was conducted at room temperature (300 K). The finite temperature simulation in the MuMax3 software is provided by means of this fluctuating thermal field (Btherm) [43]. Mesh sizes of 2 × 2 nm2 for diam- eters up to 600 nm and 4 × 4 nm2 for larger diameters were used. In order to reduce the computational resources and speed up the simulations, the effective medium approach, proposed in [46], was implemented, in which the multilayer film system has been modeled as a single homogeneous magnetic layer.
This model is able to reproduce the static and dynamic behav- iors of the more complicated multilayer system, according to specific scaling laws. These scaling laws provide the volume- average of the saturation magnetization (Ms), Heisenberg (Aex) and interfacial DMI (Di) exchange and effective aniso- tropy (Keff) constants in terms of the following equation: [46]
Ms
Ms = Aex Aex =Di
Di = Keff Keff =tCo
tth
(2) where stands for the volume-average, while tCo and tth are the thickness of the Co layer and the total thickness of the multilayer stacks respectively. The strain-induced PMA (Kε) generated by the electric field-induced in-plane strain (εip) is given by: [32]
Kε=−3 2
Å
c11+c12−2c212 c11
ã λsεip
(3) where c11, c12 and λs are the elastic stiffness and magnetostric- tion coefficients of the magnetic material respectively. This strain-induced PMA (Kε) contributes to the uniaxial effective PMA (Keff) as:
Keff=K0−Ks+Kε.
(4) where K0 refers to the residual anisotropy and Ks = (1/2)µ0Ms2 Nz is the shape anisotropy in which µ0, Ms and Nz are the free- space permeability, saturation magnetization and demag- netization factor, respectively. In order to make an estimated relation between Kε and the electric field-induced in-plane
strain (εip), the elastic stiffness (c11 and c12) and magnetostric- tion (λs) coefficients of Co (given in table 1) were taken into consideration in this work.
3. Phase diagram
Spin structure phase diagrams can help to choose a specific route for the desired magnetization switching, generated as a result of tuning the in-plane strain triggered by the externally applied electric field. Firstly, we calculated a magnetic domain structure phase diagram for the states, as a function of the lat- eral dimensions of the nanodisc (i.e. ddisc ranging from 100 nm to 1 µm) and the uniaxial effective PMA constant (Keff), as shown in figure 1(b). The states were relaxed from the initial out-of-plane uniform domain configuration, i.e. vertical single domain state with Mz/Ms ~ +1, by running the simulation for a sufficiently long time to gain stable (or metastable) states.
It can be seen from the spin structure phase diagram that the single domain magnetic configuration remains stable at rela- tively large effective PMA, i.e. Keff ⩾ 0. On the other hand, for smaller PMA values (Keff < 0) the single domain state con- verts into either the labyrinth stripe domain or into the vortex due to the larger shape anisotropy (Ks). Isolated single sky- rmion states can also be stabilized with moderate PMA (Keff)
values. Nonetheless, the magnitude of Keff for the existence of isolated skyrmion states decreases with the increase in size of the nanodisc (from 100 nm to 300 nm in figure 1(b)). As far as the lateral dimensions of the geometry are concerned, it can be noted that the domain structures in larger nanodiscs tend to appear in the form of stripe or multidomain states.
Moreover, a spin structure phase diagram for 200 nm nano- discs, as a function of the uniaxial effective PMA (Keff) and layer repetition number (n) of the trilayer stacks, was also calculated and is shown in figure 1(c). Variation of n for the trilayer stacks changes the equivalent thickness in the confined
Figure 1. (a) Proposed device structure consisting of a nanodisc of Pt/Co/Ta trilayer stacks residing on a piezoelectric substrate and various magnetic topological states that can be stabilized in the nanodisc at zero magnetic field. HM1, HM2 and FM in (a) refer to the first heavy metal layer, second heavy metal layer and ferromagnetic layer respectively. (b), (c) Phase diagrams for the spin structure within the nanodisc (derived from initial out-of-plane magnetization single domain state) as a function of uniaxial effective perpendicular anisotropy (Keff) and disc diameter ddisc (b), as well as Keff and layer repetition number n of the multilayer stacks (c). n = 6 for (b) and ddisc = 200 nm for (c). Local normalized perpendicular magnetization (Mz/Ms) in the color contours in (b) and (c) are represented by the color scale bar and the arrows inside the contours show the directions of the in-plane magnetic moments.
Table 1. Material parameters used in the simulations and strain calculations.
Parameter Value
Ms 1.003 MA m−1
Aex 10 pJ m−1
Di 1.3 mJ m−2
K0 0.67 MJ m−3
c11a 295 Gpa
c12a 159 Gpa
λsb –60 ppm
a Tromans [47].
b Klokholm and Aboaf [48].
geometry. It can be seen from the figures that with the increase of n, most of the magnetic moments tend to align themselves in the out-of-plane direction and favor the stabilization of vertical single domain and skyrmion states even at a relatively lower Keff. For a fixed value of Keff, larger number of stacks causes the magnetic moments to align in the out-of-plane direction and tends to stabilize the out-of-plane single domain configuration even at relatively weak PMA (Keff ⩾ −6.8 × 105 erg cm−3).
The in-plane orientated magnetic moments are favored at weaker PMA (Keff ⩽ −17.0 × 105 erg cm−3) so the magn etic configurations take the form of vortex states due to the geometric confinement effect. However in the case of small n values (n = 1 and 2) and a weak PMA regime (Keff ⩽ −17.0 × 105 erg cm−3), the magnetic moments align themselves in the form of in-plane single domain states rather than the vortex that is stabilized at relatively large n (>2).
4. Controlled switching of various magnetic states As mentioned earlier, applying an electric field to the piezo- electric substrate can create an in-plane strain which in turns produces perpendicular anisotropy (Kε). This Kε contributes to the uniaxial effective PMA (Keff) of the system. This strain- induced PMA can therefore be tuned by the application of an
external electric field to switch the domain structures. The contribution of the in-plane strain to the magnitude of Keff can be estimated by using equations (3) and (4). We employed this key factor for obtaining robust, deterministic and repeat- able multistate magnetization switching within the proposed device architecture based on the multilayer stack nanodiscs.
In order to check the repeatability of each switching scenario, we performed at least five repetition cycles. The skyrmion number, also called the topological charge is given as [49]:
Q= 1 4π
¨
m·(∂xm×∂ym)dxdy.
(5) For skyrmion and vortex states in confined geometric structures the values of topological charge (Q) are approxi- mately ±1 and ±0.5, respectively, while both the skyrmio- nium and single domain states have almost zero topological charge, i.e. Q ~ 0 [37, 50, 51]. Therefore, in this sense the switching or transition of topological states can be categorized as the variation in topological charge.
4.1. Switching between helical stripe and single domain states Firstly, we investigated the switching behaviors between the labyrinth stripe domains and the single domain state triggered
Figure 2. Switching between stripe and single domain states. (a) Time sequence evolution of the local magnetization (Mz/Ms) in response to a strain pulse on the nanodisc (with ddisc = 200 nm and n = 1). Magnetic domain states at different stages are shown at the top and the color scale bar represents the distribution of Mz/Ms in the color contours. (b), (c): Variation of the estimated critical strain for stripe domain creation and annihilation with respect to disc diameter ddisc (for n = 1) (b) and layer repetition number n (for ddisc = 200 nm) (c) of the multilayer stacks.
by a strain pulse in a nanodisc with ddisc = 200 nm and n = 1.
Figure 2(a) shows the time sequence evolution curve of mag- netization (M–t). The corresponding magnetic domain states at different instants of switching are shown at the top of the switching curve. The initial single domain state was chosen from the phase diagram corresponding to Keff = 6.8 × 105 erg cm−3 (εi ~ 0.134%) in figure 1(c). Once a negative elec- tric field was applied to induce a compressive strain (to a value of ∆ε ~ −0.118% in this case), the labyrinth stripe domains nucleate within the nanodisc and remains stable after the pulse is removed. Note that ∆ε = εf − εi, where εi and εf stand for the values before and after the applica- tion of strain pulse respectively. The nucleation of the stripe domain state initiates at the edge of the nanodisc where the magnetic moments start to switch from the +Mz direction to zero, as shown in the magnetic domain structures given at the top of figure 2(a). Elimination of the labyrinth stripe domains in the nanodisc are performed by applying an electric field- induced tensile strain pulse (∆ε ~ 0.747%) which converts the stripe domain into the single domain state, as reflected by the change of nor malized Mz/Ms from nearly zero to −1.
Once the single domain state is achieved, it remains stable even after the removal of the pulse.
Furthermore, we also estimated the values for the critical strain (∆εc) i.e. the minimum electric field-induced strain required for the elimination of the labyrinth stripe domains for different ddisc and n, as shown in figures 2(b) and (c). It is evident from these figures that a larger strain is required to eliminate the stripe domains in relatively larger discs and a similar trend for the cases with larger n can also be seen.
Variation of critical strains for the creation of the helical stripe domain from the single domain state is also shown in these two figures, which indicates that the nucleation strain shows an increase first and then drops with respect to diameter, pro- ducing a maximum critical strain (smallest negative strain) below −0.1%. On the other hand, the creation curve with respect to n exhibits a monotonous decrease in strain (increase in magnitude |∆εc|).
4.2. Skyrmion creation and annihilation
Another good example of topological transition is the cre- ation and annihilation of the skyrmion state. Figure 3(a) shows the time sequence evolution of the perpendicular nor- malized magnetization component (Mz/Ms) and topological charge (Q) during the creation of the skyrmion state from the single domain state and their annihilation, regaining the initial single domain state, triggered by a strain pulse for a nano- disc of ddisc = 200 nm and n = 4. The initial single domain state was chosen from the phase diagram corresponding to Keff = 6.8 × 105 erg cm−3 (εi ~ 0.134%) in figure 1(c).
Magnetic domain configurations at different transition instants are also given at the top. It can be seen from the domain struc- tures that on the application of a compressive strain pulse to the nanodisc with an initial out-of-plane single domain state (Q ~ 0 and Mz/Ms ~ +1), skyrmion nucleation is initiated at the center of the nanodisc to form a bubble structure and
expands to cover a large portion of the nanodisc. Before stabi- lizing to the skyrmion state, the diameter of the inner domain shows some extent of fluctuation, which is also evident from the oscillation in the Mz/Ms curve. After stopping the applied strain pulse, this isolated skyrmion state remains stable with zero magnetic field. On the application of an opposite field i.e. tensile strain pulse to the nanodisc, the inner domain of the skyrmion starts to shrink until it completely vanishes, and the initial single domain magnetic configuration is regained which remains stable after the ending of the applied pulse.
This topological transition is also evident from the time sequence evolution of the topological charge between Q ~ 0 and −1, which represent the single domain state and skyrmion state respectively.
Next, we also estimated the critical strain values (∆εc), i.e. the minimum magnitude of the strain required for skyr- mion creation and annihilation, for nanodiscs with different ddisc and n. Figures 3(b) and (c) show the variation of critical strain values for the creation and annihilation of the isolated skyrmion state with respect to ddisc and n respectively. Here n for figure 3(b) is 4 and ddisc for figure 3(c) is 200 nm. It can be seen from figure 3(b) that the critical strain magnitude
|∆εc| for skyrmion creation decreases monotonously with the increase in ddisc up to 600 nm above which the skyrmion state becomes unstable. The value of ∆εc for skyrmion annihilation increases with the increase in ddisc, as shown in figure 3(b).
The variation of ∆εc for skyrmion creation with n, shown in
Figure 3. Skyrmion creation and annihilation i.e. switching between skyrmion (Q ~ ±1) and single domain state (Q ~ 0). (a) Time evolution of local normalized perpendicular magnetization (Mz/Ms) and topological charge (Q) in response to the applied strain pulse to a nanodisc (with ddisc = 200 nm and n = 4). Magnetic domain structures at different stages are shown on the top of the M-t curve and local normalized perpendicular magnetization (Mz/Ms) is represented by the color scale bar. (b), (c) Variation of the estimated critical strain for annihilation and creation of the isolated skyrmion state against disc diameter ddisc (b) and layer repetition number n (c) of the multilayer stack. Here n = 4 for (b) and ddisc = 200 nm for (c).
figure 3(c), depicts that the magnitude of the critical strain i.e.
|∆εc| for skyrmion creation increases with the increase in n.
On the other hand, the value of ∆εc for skyrmion annihila- tion increases with increasing n before arriving a maximum at n = 8 and then starts decreasing.
4.3. Switching between skyrmion/skyrmionium and vortex states
Figure 4(a) shows the time sequence evolution of the topolog- ical charge (Q) for magnetic switching between skyrmion and vortex states (at ddisc = 200 nm and n = 6). Some instanta- neous domain structures at intermediate stages are also shown at the top of the figure. Switching of skyrmion (Q ~ ±1) into a vortex (Q ~ ±0.5) was caused by applying a compressive strain (∆ε ~ −0.342%) to the nanodisc with an initial sky- rmion state (initial Keff = −6.8 × 105 (εi ~ −0.4%)). During this switching, the magnetic moments within the nanodisc start to align in in-plane orientation in the form of a vortex after a few sequences of vibrations through an intermediate state (i.e. skyrmionium state). The skyrmion creation from the vortex occurs in such a way that the skyrmion nucleation
begins at the center of the nanodisc and then the size of the skyrmion inner domain expands to a maximum size. The diameter of this inner domain oscillates before attaining a stable size. The vortex is unstable once the strain pulse ends and the initial skyrmion state is recovered therefore indicating that the switching is volatile.
It can also be noted that the vortex is only stable at small Keff and if it increases the vortex will switch into an isolated skyrmion, skyrmionium or labyrinth stripe domain state depending on the value of final strain. We implemented this to achieve an isolated skyrmionium state from the vortex state.
During switching between vortex and skyrmionium, the topo- logical charge (Q) shows some fluctuations (possibly due to the merging of domain walls) and finally becomes stable (see figure 4(b)). It can be seen from the magnetic states at the transition instants on the way of switching from vortex into skyrmionium that vortex first goes to a skyrmion like domain state with Q nearly equal to −1 and then converts into a skyrmionium state (Q ~ 0). However, in the next step, i.e. the conversion of skyrmionium into vortex, the magnetic domain structure fluctuates between skyrmion and skyrmionium like states before reaching a stable vortex state. This is evident from the curve of the topological charge in figure 4(b) that Q fluctu- ates between +1 (for skyrmion) and 0 (for skyrmionium) before getting to a stable value of 0.5 (for vortex). The esti- mated value of ∆ε for which the skyrmionium appears in the nanodisc with ddisc = 200 nm and n = 14 was found to be
~0.197% (the initial vortex state was chosen from the phase diagram corresponding to Keff = −17.0 × 105 erg cm−3 (εi
~ −0.8%) in figure 1(c)). It is noted that the sign of the topo- logical charge (Q) or nor malized perpendicular magnetization (Mz/Ms) at the core of the vortex is also reversed after one complete switching cycle, which corresponds to an opposite magnetic moment at the vortex core.
4.4. Tuning of multiple topological states
From the above observations, we find that it is possible to tune multiple topological states using strain in certain nanodiscs.
An example of the temporal evolution of various topological states, induced by external strain in a specific nanodisc with a diameter of 200 nm and n = 6, is shown in figure 5(a), in which the variations of topological states are also reflected by the time sequence evolution of topological charge (Q) and normalized perpendicular magnetization (Mz/Ms) in response to strain (∆ε). Here, we started from an initial single domain state with vertical magnetic moments, i.e. Mz/Ms ~ +1, chosen from the phase diagram corresponding to Keff = −3.4 × 105 erg cm−3 (εi ~ −0.27%) in figure 1(c). It can be seen that, initially, the decrease in strain ∆ε results in the creation of a skyrmion state with a reversed inner core domain. Further decrease in strain produces a vortex state (see figure 5(a)).
On increasing the strain towards the tensile strain regime, the vortex state first evolves into a skyrmionium state. Further increase in strain leads to the elimination of the inner domain forming a skyrmion state. Finally, the inner core domain of the skyrmion state is eliminated again resulting in a vertical single domain state (see figure 5(a)). In other words, on increasing
Figure 4. Volatile magnetization switching between skyrmion and vortex states. (a) Switching between skyrmion (Q ~ ±1) and vortex (Q ~ ±0.5) and time sequence evolution of topological charge in response to the applied strain to the nanodisc (with ddisc = 200 nm and n = 6). (b) Switching between vortex (Q
~ ±0.5) and skyrmionium (Q ~ 0) and time sequence evolution of topological charge in response to the applied strain to a nanodisc (with ddisc = 200 nm and n = 14). Magnetic domain states at different stages are shown at the top of each case. Local normalized perpendicular magnetization (Mz/Ms) in the color contours is represented by the color scale bar and the arrows inside the contours show the directions of in-plane magnetization.
the strain from the vortex state, the magnetic moments start to align perpendicular to the nanodisc plane leading to a sequen- tial occurrence of skyrmionium, skyrmion and vertical single domain state.
More interestingly, one can see that after gradually decreasing and then increasing the strain, the magnetization of the vertical single domain state switches to the opposite direc- tion, i.e. Mz/Ms from +1 to −1, accompanied by the reversal in topological charge of the intermediate states. This can be understood by analyzing the specific switching paths, wherein the appearance of the skyrmionium state plays important role in causing the reversal of magnetization. When reducing the strain from the initial vertical single domain state (positive magnetiza- tion, i.e. Mz/Ms ~ +1), a skyrmion first nucleates with a reversed core domain (with negative magnetization) and subsequently switches to a vortex state with a negative vortex core. With the increase of external strain, the vortex state first switches to a skyrmionium state with a core domain of the same magnetiza- tion as the vortex core (negative). On further increase in the
strain, the core diminishes and a skyrmion state appears with a new core domain of the opposite magnetization (positive).
Finally, the new core domain disappears resulting in a new vertical single domain with negative magnetization, i.e. Mz/Ms
~ −1. The observed reversible switching in both magnetization and topological charge is quite repeatable and can maintain the same switching sequence for more than five cycles.
The evolution of total energy density (Etotal) and its indi- vidual contributions from exchange (Eexc), demagnetization (Edem) and anisotropy (Eanis) energy densities, in the course of the tuning of multiple topological magnetic states is also shown in figure 5(a). It can be seen that the demagnetization energy favors the vortex state, while both the anisotropy and exchange energies favor single domain states. Both the skyr- mion and skyrmionium states occur with moderate exchange, demagnetization and anisotropy energy density values.
Therefore, it is the competition between these energy densi- ties which stabilizes different topological magnetic states at different strain values.
Figure 5. (a) Evolution of topological charge, normalized perpendicular magnetization (Mz/Ms), total energy density (in J m−3) and its contributing components (exchange, demagnetization and anisotropy) in response to applied strain in a nanodisc with ddisc = 200 nm and n = 6. The color bar represents the out-of-plane magnetization (Mz/Ms) in the color contours of magnetic states and the arrows inside the contours show the directions of in-plane magnetization. (b) Stability diagram for various topological magnetic states in a nanodisc with respect to applied strain with the same geometric parameters as (a).
The stability of the different topological magnetic states (including metastable states) against the applied strain was also examined and the results are shown in figure 5(b). It is clear from this stability diagram that the vortex state stabilizes at a lower strain range, while the single domain state exists at a higher strain range. On the other hand, both the skyrmion and skyrmionium states lie in an intermediate strain range. It is also interesting to see that there exists an overlap in the sta- bility regions of the single domain and skyrmion states, which demonstrates that bi-stable switching can occur between these two states in a specific strain range. Therefore, non-volatile magnetic switching can be triggered by using proper strain pulses, depending on the choice of a proper Keff (see example in figure 3). In contrast, there is no overlap between the sta- bility regions of vortex and skyrmion/skyrmionium states, thus the switching between these states should be volatile, as exemplified in figure 4.
5. Conclusion
In summary, by means of a micromagnetic simulation study we have demonstrated strain-mediated electric field control of reversible switching between multiple magnetic topological states including skyrmion, vortex, skyrmionium, helical stripe and single domain states in nanodiscs of Pt/Co/Ta trilayer stacks. It was found that the conversion from one topological state to other states strongly depends on the magnitude of the applied strains. In this way, the switching becomes somehow more deterministic because one can toggle between different states by simply selecting an appropriate external strain. The critical strains required for different switching types have also been calculated for different geometric parameters of the mul- tilayer stack nanodiscs and the stability of the various topo- logical states was also discussed. The present study indicates the potential for implementation of these topological states in high-density magnetoelectric memory and logic device applications.
Acknowledgments
The authors would like to thank the National Key Research Program of China (Nos. 2016YFA0201002, 2016YFA0300101), the State Key Program for Basic Researches of China (No.
2015CB921202), the Natural Science Foundation of China (Nos. 11674108, 51272078 and 11574091), the Project for Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme (2014), the Natural Science Founda- tion of Guangdong Province (No. 2016A030308019).
ORCID iDs
Nasir Mehmood https://orcid.org/0000-0001-8608-0416 Minghui Qin https://orcid.org/0000-0002-8306-125X Xingsen Gao https://orcid.org/0000-0002-2725-0785
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