Generation of magnetic skyrmions through pinning effect
On the basis of analytical argument and lattice simulation, we propose that magnetic skyrmions can be generated through the pinning effect in 2D chiral magnetic materials. In our simulation, we find that magnetic skyrmions are generated in the pinning areas and remain stable for a long time. We have studied the properties of the skyrmions with various values of ferromagnetic exchange strength and the Dzyaloshinskii-Moriya interaction strength . By using the pinning effect, magnetic skyrmions can be generated in the absence of external magnetic field or magnetic anisotropy at desired positions. This method is useful for the potential applications in magnetic information storage.
pacs:75.70.Kw, 66.30.Lw, 75.10.Hk, 75.40.Mg
The topologically protected structure called skyrmion can be formed in a chiral magnet nagaosa (); Fert (), as discovered in the bulk MnSi by using neutron scattering Muhlbauer (), and also observed by using Lorentz transmission electron microscopy Yu1 () and by using spin-resolved scanning tunnelling microscopy (STM) 2DSkyrmion2 (). They exist in magnetic materials that lack inversion symmetry and are due to Dzyaloshinskii-Moriya (DM) interactions DMI (). Magnetic skyrmions can be driven by spin current, and the critical current density to manipulate magnetic skyrmions is lower than for magnetic domain-walls ultralow (), thus they are promising as future information carriers in magnetic information storage and processing devices. Therefore it is very interesting to find out the efficient means of creation and manipulation of magnetic skyrmions.
Single skyrmions can be created and deleted on a surface with the help of local spin-polarized STM Romming (). A large number of magnetic skyrmons were created in a short time by constructing a geometrical constriction bubble (); thetarho (). An important feature in these works is the presence of magnetic field. On the other hand, skyrmions can also be generated without magnetic field, but using a circulating current Tchoe (), and can be generated and stabilized with the help of magnetic anisotropy anisotropy (); bilayer1 (). They can also be created at desired position with the help of direct current (DC) and with modified magnetic anisotropy even in the absence of DM interaction dccurrent ().
Defects and impurities can lead to inhomogeneities of the ferromagnetic exchange coupling , the DM interacion , and the magnetic anisotropy pin (); pin2 (). This is the effect of pinning. The modified magnetic anisotropy dccurrent () is an example of the pinning effect on the creation of skyrmions. The creation of skyrmions by using STM Romming () can also be regarded as the effect of inhomogeneity of . Previously, the external magnetic field, the magnetic anisotropy, or a current is needed to generate skyrmions.
In this paper, however, we propose a novel method to generate magnetic skyrmions, only using the inhomogeneity of . The radii and other properties of the skyrmions are controlled by the parameters of the model, for example, the exchange coupling and the DM interaction .
Ii Basic Idea
It is known that the skyrmions can be generated with the help of magnetic anisotropy, so we only consider the case without magnetic anisotropy. In this case, in the presence of an external magnetic field, the Hamiltonian can be written as as nagaosa (); FreeEnergy ()
where is the orientation of the local magnetic moment, is the local ferromagnetic exchange strength, is the local strength of DM interaction, is the external magnetic field along direction. The local magnetic moment of a skyrmion can be parameterized as nagaosa ().
where and are polar coordinates of the 2D position vector , with the origin of the polar coordinate system at the center of the skyrmion, , , is an arbitrary angle, is a function describing the shape of a skyrmion, with and . The skyrmion charge is defined as nagaosa (); Tchoe ()
It was found previously that for a skyrmion, can be approximately described as arctan ()
Using this expression with , some examples of are shown in Fig. 1.
Two kinds of pinning were considered previously. In Ref. pin (), both and are inhomogeneous, while is kept constant. In Ref. pin2 (), is constant while is inhomogeneous. For simplicity, we only consider the latter case. Moreover, we only consider the case of . From Eq. (1), for , we find the Euler-Lagrange equation
where , , is a pinning function dependent only on , and it is assumed that the skyrmion center is at the center of the pinning and the pinning is rotationally symmetric. When there is no pinning while an external magnetic field is applied, Eq. (5) becomes
It is known that, in this case the skyrmions can be generated.
We note that when pinning is present, term in Eq. (5) may play a role as the term. For example, suppose but the pinning has the form of
which is very close to the Eq. (6) except that the is position dependent in (5) while is position independent in (6). Using the numerical method introduced in Ref. pin (), we solved Eq. (5) with given in Eq. (7) and with parameter values , , , , . As shown in Fig. 2, the solution is very close to skyrmion ansatz (4), suggesting that it is possible to create a skyrmion by using pinning effect.
Iii Lattice simulation
The above investigation, with an artificial pinning effect, suggests the possibility of creating a skyrmion using pinning effect only. We now consider a more realistic pinning effect. As a local structure, the effect of pinning should be suppressed very quickly in deviating from the pinning center. An exponentially decaying function is assumed in Ref. pin2 (), while a Gaussian function is assumed in Ref. pin (). We follow Ref. pin () to assume has Gaussian form,
where , and are undetermined coefficients with , and . The radius of the pinning is denoted as , and . In this paper, we use dimensionless parameters, and always set for simplicity.
The effective parameter depends on not only the DM interaction but also the parameters of the skyrmions, and whether a solution of can be identified as a skyrmion is technically subtle. Therefore rather than solving the Euler-Lagrange equation, we perform a lattice simulation, which directly provides evidence of skyrmions. The lattice simulation is based on the Landau-Lifshitz-Gilbert (LLG) equation nagaosa (); pin (); LLG ()
where is the local magnetic momentum at grid , is the Gilbert damping constant, is the effective magnetic field
where the discrete Hamiltonian can be written as discreteH ()
where refers to each neighbour, and on a square lattice. So pin ()
We run the simulation on a square lattice with open boundary condition, and with . The center of the pinning is set to be at the point . In the following, we denote the time step as , and the steps the simulation take is denoted as . The simulation is running on the GPU. Compared with the programs running on the CPU, the GPU has a great advantage on this problem.
iii.1 The case of .
We first run the simulation for , and various values of , starting with randomized and stopping when become stable. Previously, the Gilbert constant is taken as to pin (); arctan (); alpha1 (); alpha2 (); alpha3 (); bilayer (). We find that the larger the value of , the more rapid the simulation can be done, and that the smaller the value of , the longer it takes for to become stable. Hence we use alpha2 () and for , while use pin (); alpha3 () and for .
iii.1.1 Skyrmions can be generated.
The results are shown in Fig. 3. We run the simulation for . Among each of these runs, a skyrmion is generated at the center of the pinning except when or . When is smaller, it takes longer time to generate the skyrmion. Generally, it takes longer time to generate a skyrmion compared with the generation of skyrmions by using the external magnetic field.
In our simulation, it is easy to generate a skyrmion, but it does not always appear in each run of the simulation, because the initial state is randomized. Fig. 4 shows two examples that the simulation failed to generate a skyrmion at the pinning center, as the skyrmion-like structure in the center cannot be ‘cut off’ from the helicity phase. Although the skyrmion is not generated, the structure of in the central region is significantly different from elsewhere.
We also study the properties of the skyrmions generated in the simulation. We only study the skyrmions nearest to of the pinning centers. The value of is determined by at the center of the skyrmions. The radius of each skyrmion is estimated from the radius of the iso-height contour with for . The angle and is determined by using and at the iso-height contour with .
iii.1.2 Radii of the skyrmions.
We first investigate the relation between the skyrmion radius and the DM interaction strength . We find that the larger the value of , the smaller . The relation between and can be fitted by , as shown in Fig. 5.
Then we study the relation between and the radius of the pinning, as well as that between the distance between the pinning center and the skrymion center. As shown in Fig. 6, we run the simulation for and different values of , and find that the skyrmion radius is more relevant with than , and is smaller when is larger. For the same value of , decreases with the decrease of pinning radius .
iii.1.3 Skyrmion numbers and .
We find for all the skyrmions appearing in the simulation, then with , the skyrmion number only depends on the sign of . As shown in Fig. 3, both skyrmions can be generated. For a skyrmion generated by an external magnetic field, depends on the direction of . But for a skyrmion generated by using the pinning effect in absence of a magnetic field, the sign of becomes a free choice and depends on the initial state (Figs. 7, 8).
For , we find that for , , while for , . Hence in both cases. We also run simulations with (Fig. 9), in which is also valid for the skyrmions generated. This can be understood from
which differs from the case nagaosa () by replacing as . As a result, the sign of is a free choice, while the sign of is determined by . Hence the energy is lowest when , and , with the sign of determined by . when , when .
iii.1.4 The process of the skyrmion generation.
The process of the skyrmion generation is interesting. We show two examples with and in Figs. 10 and 11, respectively. When is too small, the width of the strip in the helicity state is large compared with the radius of the pinning, and the pinning effect is small. On the other hand, when is large, it is known that the external magnetic field to generate skyrmions is approximately border () when is large. Since the pinning plays role as , a large value of is needed to generate skyrmions when is large. As a result, we conclude that, with fixed, the skyrmions can only be generated when the value of is appropriate.
When the value of is appropriate, and when the strip of the helicity state is formed over the region of the pinning, different parts of the strip receive different effects from the pinning, i.e., the strip is thicker at the center of the pinning and is thinner at the edge of the pinning. As a result, an olive-shaped structure can be formed near the center of the pinning. It is known that the skyrmions can be dragged to the center of the pinning pin (). When it is dragged to the center, the olive-shaped structure can possibly be cut off from the stripand becomes a skyrmion. As shown in Figs. 10 and 11, at the beginning of simulation, several olive-shaped structures are generated, and one of them is dragged into the center of the pinning and its shape becomes more symmetric because of the rotational symmetry of .
iii.2 The case of .
iii.2.1 The properties of the skyrmions when .
We also study the case of . Using , , , , we find the skyrmions generated in the simulation (Fig. 12). In this case, the possible range of in which skyrmions can be generated is narrower than the case of and . As in the case of , the radius of the skyrmion increases with the decrease of . However, the skyrmions generated in large distance from the pinning center is larger than in the case of . Same as the case when , we also find and for all the skyrmions generated in case .
iii.2.2 The bound state phenomenon.
We also observe an interesting phenomenon when is very small in the case of (Figs. 13 and 14), which does not show up in the case of . When is very small, two skyrmions with and are generated and move to the pinning center. They keep rotating around each other with the distance shrinking till annihilation. Previously, it was found that the interaction between two skyrmions on two layers with opposite skyrmion charge can form a bound state bilayer (). The situation we consider may provide another stage to study skyrmion bound states.
In this paper, we propose a novel mechanism to generate magnetic skyrmions without needing the external magnetic field or magnetic anisotropy. We find that skyrmions can be generated through the pinning effect only, i.e., with the inhomogeneous magnetic exchange strength . Our lattice simulation has verified this idea. In the simulation, we study the properties of the skyrmions generated under various parameter values. We find that the radius of the skyrmion increases when decrease. For , the closer the skyrmions to the pinning center, the larger the sizes of the skyrmions. For , the sizes are smaller. We also find that all skyrmions generated have and , while the sign of depend on the initial state. For , we also find the generation of a pair of skyrmions with opposite charges at the pinning center.
The phenomenon that the skyrmions can be generated by using the pinning effect only is useful for practical applications of the magnetic skyrmions. Through the engineering of pinning in the designated site, we can generate a skyrmion on this site. It is hoped that experiments and applications be made by using this method.
This work is supported by National Natural Science Foundation of China (Grant No. 11374060 and No. 11574054).
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