SYSTEM AND METHOD

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application No. 62/895,044, filed on September 3, 2019, entitled “DIRECT IMAGING OF AN INHOMOGENEOUS ELECTRIC CURRENT DISTRIBUTION,” the disclosure of which is incorporated herein by reference in its entirety.

BACKGROUND

TECHNICAL FIELD

[0002] Embodiments of the subject matter disclosed herein generally relate to a system and method for imagining an electrical current distribution for a given geometry configuration of a conductor, and more particularly, to a direct imagining of an inhomogeneous electric current distribution using trajectories of magnetic half sky rmions.

DISCUSSION OF THE BACKGROUND

[0003] The accurate determination of the electric current distribution in a conductor, including its magnitude and direction, is extremely desired for designing micro/nano-devices and integrated circuits, particularly in the case of an inhomogeneous current distribution due to the geometric constraint of the conductor. Over the years, various techniques, such as the scanning SQUID probe, scanning Hall probe, and scanning tunneling magnetoresistance probes have been developed to map the current density distributions. These techniques probe a magnetic stray field generated by the electrical current in the conductor, and then reconstruct the current density distribution based on the Biot-Savart law. All these approaches do not provide a direct observation of the electrical current, and the complex relation between the local stray field vector and the current density vector hampers the accurate mapping of the current density and direction. Moreover, these techniques are costly, and require specialized instruments to be developed for the industry. Some existing scanning probes also need extreme conditions, such as very low temperatures, for example, in the case of the SQUID probes.

[0004] Alternately, the electrical current distribution in a conductor, which includes both the density and the direction can, in principle, be numerically simulated using finite element calculations. However, due to the over-simplified model and difficulties to take into account the non-uniformity or tiny cracks in the real conductor, the results may not be convincing and trustable.

[0005] Thus, currently, it is still experimentally impossible to directly observe and map the inhomogeneous current distribution in a given conductor. Therefore, there is a need to find a new method and system that is capable of imagining the current distribution in a given conductor, accurately, and without the need of highly specialized equipment and/or extreme conditions. BRIEF SUMMARY OF THE INVENTION

[0006] According to an embodiment, there is a method for generating an electrical current distribution that is associated with a target conductor structure. The method includes receiving information about the target conductor structure, which has a given geometry that includes at least first and second electrodes, providing a multi-layer structure having two heavy-metal layers and a ferromagnetic layer so that a spin current is generated within the heavy-metal layers and injected into the ferromagnetic layer, adding corresponding at least first and second electrodes to a surface of the multi-layer structure so that a same geometry is obtained as for the target conductor structure, injecting one or more electrical current pulses into the at least first and second electrodes to generate half-skyrmions into the multi-layer structure, imagining a trajectory of the half-skyrmion with a Kerr effect microscope, and mapping the trajectory of the half-skyrmion to the target conductor structure after correcting the trajectory with a Hall angle corresponding to the half-skyrmion. The corrected trajectory represents the electrical current distribution of the target conductor structure.

[0007] According to another embodiment, there is a system for generating an electrical current distribution that is associated with a target conductor structure. The system includes a processor configured to receive information about the target conductor structure, which has a given geometry that includes at least first and second electrodes, a multi-layer structure having two heavy-metal layers and a ferromagnetic layer so that a spin current is generated within the heavy-metal layers and injected into the ferromagnetic layer, corresponding at least first and second electrodes connected to a surface of the multi-layer structure so that a same geometry is obtained as for the target conductor structure, a power source configured to inject one or more electrical current pulses into the at least first and second electrodes to generate and drive half-skyrmions into the multi-layer structure, and a memory for receiving Kerr effect microscope images of a trajectory of the half- skyrmion. The processor is configured to map the trajectory of the half-skyrmion to the target conductor structure after correcting the trajectory with a Hall angle corresponding to the half-skyrmion. The corrected trajectory represents the electrical current distribution of the target conductor structure.

[0008] According to still another embodiment, there is a multi-layer structure for determining an electrical current distribution in a target conductor structure. The multi-layer structure includes a first heavy metal layer, a ferromagnetic layer formed over the first heavy-metal layer, and a second heavy-metal layer formed over the ferromagnetic layer so that the ferromagnetic layer is sandwiched between the first and second heavy-metal layers. The heavy-metal layers are configured to generate a spin current to be injected into the ferromagnetic layer and the ferromagnetic layer is configured to generate a half-skyrmion due to the spin current. A thickness of the ferromagnetic layer is selected so that a Hall angle of the half-skyrmion is 0 degrees.

BRIEF DESCRIPTION OF THE DRAWINGS

[0009] For a more complete understanding of the present invention, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which:

[0010] Figure 1 is a schematic diagram of a target conductor structure for which an electrical current distribution is desired to be determined;

[0011] Figure 2 illustrates a multi-layer structure that is used to generate half- skyrmions whose trajectories approximate the electrical current distribution in the target conductor structure of Figure 1;

[0012] Figures 3A and 3B illustrate the normalized out-of-plane and in-plane hysteresis loops, respectively, of the multi-layer structure of Figure 2;

[0013] Figure 4 shows the variation of the magnetic anisotropic field with the thickness of the ferromagnetic layer of the multi-layer structure;

[0014] Figure 5A illustrates a magnetic domain formed in the ferromagnetic layer of the multi-layer structure when no current is applied, and Figure 5B illustrates the modified magnetic domain of the ferromagnetic layer of the multi-layer structure when a current is applied;

[0015] Figures 6A to 6C illustrate the evolution of a magnetic domain induced by magnetic field pulses in a 1.2 nm thick ferromagnetic layer of the multi-layer structure; [0016] Figures 7 A and 7B show the original magnetic domain in thicker ferromagnetic layers than the layer of the structures of Figures 6A to 6C;

[0017] Figures 8A and 8B schematically illustrate a system including the multi layer structure for determining an electric current distribution of a target conductor structure;

[0018] Figure 9 illustrates the dependency of the Hall angle of the half- skyrmions with the effective magnetic anisotropy field in the ferromagnetic layer; [0019] Figures 10A and 10B illustrate the dependence of the effective perpendicular magnetic anisotropy and the size of the half-skyrmion (or width of the strip domain) on the effective perpendicular magnetic anisotropic field, respectively; [0020] Figures 11 A to 11 D illustrate the evolution of the paths of the half- skyrmions with the time during which the electrical current is applied to the multi layer probe;

[0021] Figure 12 illustrates the dependency of the Hall angle with the current density;

[0022] Figures 13A to 13D illustrate partial strip domain patterns and the reconstructed current density distribution for a given target conductor structure determined with a multi-layer structure having the same geometry as the target conductor structure;

[0023] Figure 14 illustrates the current density distribution of the target conductor structure calculated based on a theoretical model;

[0024] Figure 15 illustrates the full current density distribution of the target conductor structure determined with a small Hall angle; and [0025] Figure 16 is a flowchart of a method for determining the current distribution in a target conductor structure by imagining the trajectories of half- skyrmions that move in a ferromagnetic layer.

DETAILED DESCRIPTION OF THE INVENTION

[0026] The following description of the embodiments refers to the accompanying drawings. The same reference numbers in different drawings identify the same or similar elements. The following detailed description does not limit the invention. Instead, the scope of the invention is defined by the appended claims. The following embodiments are discussed, for simplicity, with regard to a multi-layer structure that generates half-skyrmions. However, the embodiments to be discussed next are not limited to half-skyrmions, but may be applied to other skyrmions with different charges.

[0027] Reference throughout the specification to “one embodiment” or “an embodiment” means that a particular feature, structure or characteristic described in connection with an embodiment is included in at least one embodiment of the subject matter disclosed. Thus, the appearance of the phrases “in one embodiment” or “in an embodiment” in various places throughout the specification is not necessarily referring to the same embodiment. Further, the particular features, structures or characteristics may be combined in any suitable manner in one or more embodiments.

[0028] According to an embodiment, a magneto-optical Kerr effect (MOKE) microscopy is used to directly map the inhomogeneous current density distribution vector (or distribution of electric field lines) in thin films. The mechanism involved in this novel process is the motion of magnetic half-skyrmions driven by an electrical current in heavy-metal/ferromagnetic-metal/heavy-metal trilayer structures. The inhomogeneous current distribution in a target conductor structure can be mapped directly by the trajectories of the magnetic half-skyrmions in the trilayer structure, which are driven by an electrical current applied to the trilayer structure. In one application, no external field is required for the stabilization of the half-skyrmions as well as the observation of the half-skyrmion Hall effect. It is found, as discussed later, that the Hall angle of the half-skyrmions is current density independent and can be reduced to be substantially 4 degrees by tuning a thickness of the ferromagnetic- metal layer of the trilayer structure. The Hall angle is so small that an elongation path of the half-skyrmion approximately delineates the invisible current flow in the target conductor structure as demonstrated in both a continuous film and a curved track example. The method developed herein provides a practical and feasible technique to directly map the inhomogeneous current distribution even in complex geometries for both fundamental research and industrial applications.

[0029] More specifically, suppose that one is interested in determining the electrical current distribution for the target conductor structure 100 illustrated in Figure 1. The target conductor structure 100 has a given geometry. For simplicity, this geometry includes a first electrode 110, having a certain shape, and a second electrode 120, having another certain shape. The shape of the electrodes can be different or the same. The number of electrodes can be larger than two. The electrodes can be made of any conductor material and a distance between the electrodes may have any value. The two electrodes 110 and 120 are attached to a substrate 102, which is also made of the conductor material. Given the specific geometry of the target conductor structure 100, it is desired to determine the electric current distribution between the two electrodes 110 and 120 when a current is injected at one electrode and extracted at the other electrode. The method and system discussed herein for solving this problem can be applied to any number of electrodes, any size of the electrodes, any shape of the electrodes, etc., i.e. , any geometry of the target conductor structure.

[0030] According to an embodiment as illustrated in Figure 2, a multi-layer structure 200, to be used for determining the electrical current distribution in the target conductor structure 100, includes a seed layer 202, for example, Ta, but other elements may be used, a heavy metal layer 204, for example, Pt, but other elements may be used, a ferromagnetic-metal layer 206, for example, Co, but other elements may be used, and another heavy-metal layer 208, for example, Ta, but other elements may be used. A current may be injected into this multi-layer structure, at a contact plate 210 and extracted to another contact plate 212. In one application, the contact plates 210 and 212 are configured so that each plate is in direct contact to each of the layers 202 to 208. A heavy-metal is any metal with relatively high densities, atomic weights, or atomic numbers, for example, Ta.

[0031] When an in-plane current I flows into the heavy-metal/ferromagnetic- metal hetero-structure 200, through the contact plate 210, a spin current is generated within the heavy-metal layers 204/208, and the spin current is injected into the ferromagnetic layer 206, due to the spin Hall effect (SHE). The orientation of the spins p injected into the ferromagnetic layer 204/206 is determined by the following relationship: p = -sgne _SH (z xj), (1) where Q _5H is the spin Hall angle of the heavy-metal, z is the unit vector describing the normal to the interface between the layers of the multi-layer structure 200, and / is the unit vector of the electrical current density. If the polarization of the injected spins is not parallel to the magnetization in the ferromagnetic layer 206, this misalignment will generate spin torques that, in turn, through their accumulated actions, will induce the magnetization direction of the ferromagnetic layer to be switched, or bringing the magnetic domain walls into motion, when the current density of the current I is large enough. This phenomenon is responsible for the generation and motion of half-skyrmions in the multi-layer structure 200.

[0032] Various studies have reported a current-driven motion of the skyrmions in a heavy-metal/ferromagnetic-metal hetero-structure due to the SHE [1-3] The paths of the skyrmions are governed by the skyrmion Hall effect [2] In principle, if the Hall angle is a known entity, the electrical current’s direction in a target geometry can be extracted from the trajectories of the skyrmions when moving through the target geometry. However, the experimental results show that the skyrmion Hall angle depends significantly on the current density due to the presence of disorders [2, 3], and that a large number of images have to be taken during the application of the electrical current in order to record the paths of the skyrmions [1-3] Such procedure is not an effective way of recording the moving paths of the skyrmions. Moreover, such procedure requires the application of an external magnetic field to stabilize the isolated skyrmions.

[0033] The inventors have observed that the growth of very narrow strip magnetic domains in the structure 200 is driven by electric current pulses, when the structure includes Pt/Co/Ta films, at zero magnetic field. The growth of these strip domains is assumed to be determined by the current-driven motion of the half- skyrmions [4] Additionally, the inventors have found that the Hall angle of the half- skyrmions is independent of the electrical current density, which is different from the full skyrmions, which as discussed above, depend significantly on the current density.

[0034] Therefore, as discussed next, the inventors have found that it is possible to extract the electrical current direction in a given target conductor from the images of the magnetic domain patterns, in a multi-layer structure, generated by an electrical current, particularly in the case of an inhomogeneous current density. The ability to reduce the Hall angle to a small value (substantially 4°, where the term “substantially” is used herein to mean up to +/- 50% of the value characterized by the term “substantially”, i.e. , in this case, between 2° and 6°) by carefully tuning the effective perpendicular magnetic anisotropy (PMA) (obtained by varying the thickness of the Co layer) allows this novel process to directly map the electrical current distribution in the target conductor by using the moving paths of the half- skyrmions in the multi-layer structure 200, which is shaped to have the same geometry as the target conductor.

[0035] A practical implementation of the novel process discussed above is now presented. The multi-layer structure 200 is made in this embodiment by depositing Ta (5 nm)/Pt (3 nm)/Co (f)/ Ta (2 nm) films on top of each other as shown in Figure 2. Note that the layer 206 of Co is made with different thicknesses for the various embodiments discussed herein. The various layers were deposited on thermally oxidized Si substrate 201 , using a DC magnetron sputtering at room temperature. The pressure of the Ar gas is set at 0.4 Pa, with a base pressure lower than about 2 ^c 10 ^-5 Pa. The patterned straight and curved tracks discussed later were prepared using lithography and Ar-ion milling. Differential polar MOKE measurements were performed using a MOKE microscope and voltage pulses were applied by an arbitrary waveform generator and amplified using a bipolar operational power supply/amplifier.

[0036] Figures 3A and 3B show the normalized out-of-plane and in-plane hysteresis loops, respectively, of the multi-layer structure 200, for the various thickness t of the Co layer 206 (the thickness t is listed in each of the figure on the left hand side), at 300 K. The loops in Figures 3A and 3B indicate that all the multi layer structure samples 200 possess a PMA with a remnant ratio of ~ 100%. However, it is noted that the effective magnetic anisotropy field moHk decreases as the thickness of the Co layer increases, as shown in Figure 4.

[0037] To fully characterize the magnetization reversal behaviors of the multi layer structure samples 200, their magnetic domain structures are investigated (directly observed) using a polar MOKE microscope under different stimuli (magnetic field and current). First, a multi-layer structure 200 with t = 1.9 nm for the Co layer 206 is selected, and the process is started with a single domain structure by saturating the multi-layer structure 200 with a positive (upward) perpendicular dc magnetic field of H= 200 Oe, which is much larger than the saturation field H _s . Then a series of negative (downward) field pulses with an amplitude slightly larger than the nucleation field (-60 Oe) and the pulse width r = 0.1 s is applied to reverse the magnetization of the domain.

[0038] The evolution of the magnetic domain structures is imaged and recorded, using a quasi-static technique, by which each image is taken in a zero field after each magnetic field pulse. Note that the domain size and morphology remain almost unchanged, when a zero external field is applied during the imaging. The inventors have found that the reversed domain (the black patterns), with a downward magnetization ( Mz < 0), is formed at one nucleation site first, and then it grows during each magnetic field pulse. The dendritic growth of the magnetic domains may be explained due to the existence of the Dzyaloshinskii-Moriya interaction (DMI) [1] in the sample multi-layer structure 200 and the relatively weaker PMA. After 10 magnetic field pulses, a pattern of narrow upward domains are formed. A magnetic domain 500 created with a magnetic field is shown in Figure 5A. When electric current pulses are applied to the magnetic domain 500, strip domains 502 are generated as shown in Figure 5B.

[0039] The expanding behavior of the magnetic domain due to the magnetic field pulses discussed above is different from that in the multi-layer sample having the Co layer with a thickness of t = 1.2 nm, as this sample possess a very strong PMA, in which a perfect circular bubble domain 600 is created as shown in Figure 6A, and this bubble then gradually expands to cover the whole sample to form a single-domain state, as illustrated in Figures 6B and 6C.

[0040] It is noted that, with a field amplitude of -60 Oe, the upward domains do not vanish, even after applying many more pulses. By further increasing the downward field from -70 Oe to -140 Oe, the length and the width of the upward domains decrease gradually to finally vanish, leading to a saturated, ferromagnetic state in the opposite direction. This may explain why the out-of-plane hysteresis loop shown in Figure 3A is not perfectly rectangular, and that it reaches saturation gradually, especially for the multi-layer structure with larger Co thicknesses (weaker PMAs). Also note that the density of the efficient nucleation sites in the studied films is very low, due to the high homogeneity of the film, and that because of this, finding a nucleation site sometimes takes a very long time.

[0041] Figures 7A and 7B show two typical magnetic domain patterns 700 and 710, captured after a few field pulses, for multi-layer structure samples 200 with t = 1.8 and 2.0 nm. It is noted that the width of each downward domain decreases as the Co thickness increases, due to a decrease of the PMA with the increasing t (see Figure 4). When the thickness t of the ferromagnetic layer is larger than 2.0 nm, the sample’s PMA becomes very weak, and the domain sizes are too small to be distinguished using a MOKE microscope (not shown here). Eventually, the magnetization will lie in the film’s plane, as the PMA further decreases.

[0042] The magnetic field-induced dynamics of the magnetic domain structures discussed above is now further discussed with regard to a practical implementation into a system 400, as illustrated in Figures 8A and 8B. To study the magnetic domain structures, the structure 200 was patterned into a long strip track 402, which was formed on a substrate 401. The long strip track 402 is connected to two electrodes 404 and 406 on both ends, as shown in Figure 8A, where the strip track 402 and the two electrodes 404 and 406 have the same composition and structure, as illustrated in Figure 8B. A cross-section of the structure 200 of Figure 8A is illustrated in Figure 8B. Note that the strip track 402 has a width much smaller than the electrodes 404 and 406, so that a MOKE image 420 can be obtained for a given portion 403 of the strip track 402. A current pulse is applied with a power source 430, between the electrodes 404 and 406.

[0043] A Kerr microscope 440 can be positioned next to and above the given portion 403 for obtaining the MOKE image 420. A microprocessor 450 is connected to the Kerr microscope 440 and also to the power source 430 for controlling them. The microprocessor 450 may include a memory 452, that stores various instructions associated with the method for determining the current distribution in the target structure 100. These instructions are discussed later in more detail.

[0044] Labyrinth-like domain patterns 500 are generated in an upward saturated multi-layer structure 200 by applying a few downward magnetic field pulses on a sample with t = 1.95 nm, as previously discussed, and shown in Figure 5A. Current pulses with a density j = 5.2x10 ¹⁰ A nr ² are then applied to the strip track 402, with the power source 430, for a total duration r = 0.1 s, along the +x direction where the current density is normalized by the total thickness of the structure 200, that includes Ta (5 nm)/Pt (3 nm)/Co (1.95 nm)/Ta (2 nm). Figure 5B shows the Kerr image of the magnetic domain pattern after two current pulses. In this figure, it can be seen the formation of new parallel, narrow, straight-strip magnetic domains 502 with Mz < 0, which grow from the existing labyrinth-like domains 500.

[0045] The growing direction of the narrow magnetic domains 502 has an almost fixed angle 0 _HaU between a longitudinal axis of the narrow domains 502 and the current direction (+x). Because of the well-defined chirality of the spin textures of the domain walls resulting from the DM I, the front-ends 504 of the strip domains 502 can be considered as half-skyrmions with a topology number of Q =1/2 [4] Therefore, this phenomenon is called herein a “half-skyrmion Hall effect,” with an estimated Hall angle 0 _HaU (between the current direction x and the elongation directions of the domains 502) of about 38°. Although the observed phenomenon is slightly different from the current-induced skyrmion Hall effect, where skyrmions move as rigid objects without changing in size, the physics behind both phenomena are likely to be similar.

[0046] To confirm this observation, the inventors have investigated the relationship between the electrical current directions in the structure 200 in Figure 8A, and the half-skyrmion 504’s moving directions in the narrow domains 502. In total, four configurations were studied by varying the directions of magnetization (up and down) and that of the current (+x and -x). It was observed that by reversing the magnetization direction, the moving direction of the half-skyrmions is also reversed, due to the opposite topological charge. The inventors have concluded from these investigations that the longitudinal component (along the x direction) of the velocity of the half-skyrmions always follows the direction of the current, i.e., extends against the direction of the electrons flow. Such behavior is very similar to that of Neel-type skyrmions in Pt-based trilayer films, where the Pt is the bottom layer [3, 5]

Moreover, the absolute values of the Hall angles are almost identical for all four configurations. [0047] A model based on Thiele approach has been used to analyze the motion of the half-skyrmions driven by the spin-orbit torque induced by the spin Hall effect. Based on this model, the Hall angle 0 of a half-skyrmion (i.e., Q=1/2) has been found to be governed by the equation: where Q =1/2 is the skyrmion number, A is the exchange constant, eff is the effective perpendicular magnetic anisotropy, d is the half-skyrmion size, and a is the Gilbert damping constant. According to the above equation, the dependence of the Hall angles of the half-skyrmions on the eff was investigated by changing the thickness t of the Co layer, from 1.5 to 2 nm, as illustrated in Figure 9. The results are indicated by curve 900 in the figure. The inventors have found that the Hall angle of the half-skyrmions decreases monotonically, as the effective magnetic anisotropy field Hk increases, without an indication of saturation in the low H regime. It is observed that the Hall angle of the half-skyrmions can be much larger in films with an even weaker PMA. However, the inventors found that the magnetic domains in the multi-layer structures 200 with a weaker PMA are too narrow to be distinguished by the used MOKE system, due to its resolution limits. Figure 9 also shows that the Hall angle can be reduced down to about 4°, by increasing the effective PMA, which feature is used later to directly map the current flow in thin films.

[0048] Using the data shown in Figures 10A and 10B, i.e., the dependence of eff and the size of the half-skyrmion (or width of the strip domain) on the perpendicular magnetic anisotropic field (moHk), respectively, and assuming that A = 1 ^c 10 ^-11 J nr ¹ , the Hall angles of the half-skyrmions were calculated as a function of _jU oHk, using Eq. (2), for a = 0.01, 0.018 and 0.1, respectively. These results were also plotted in Figure 9, together with the experimental data 900. The experimental data 900 appear to be well-described by Eq. (2) for a = 0.018.

[0049] The current density dependence on the Half-Skyrmion Hall angle is now investigated. The current density is also a desired parameter for applications that investigate the current distribution in a target material. As discussed above, equation (2) predicts that the Hall angle of the half-skyrmion is independent of the driving electrical current. To confirm this prediction experimentally, the current density j, applied to a sample with t = 1.95 nm, was varied from 0 to 6.54x10 ¹⁰ A nr ² . The inventors also varied the duration of the applied current, r, from 1 ms to 10 s.

The obtained results are presented in Figures 11 A to 11 D. In these figures, it can be seen that even for r = 10 s (a nearly constant current), there is a current threshold, jih = 3.76x10 ¹⁰ A nr ² , to bring the half-skyrmions into motion and consequently trigger the growth of the strip domains 502 (see Figure 11 A). With a smaller r and j < jt , no response of the domain structure was observed to the applied current. For j ³ jih, the speed of the half-skyrmions (calculated using the fastest half-skyrmion) increases very slowly, from 1.14 to 68 pm s ^-1 , as j is increased from 3.76x10 ¹⁰ to5.2x10 ¹⁰ A nr ² . When j becomes larger than 5.2x10 ¹⁰ A nr ² , the speed increases sharply, as shown in Figure 12. However, even the maximum speed, at j = 6.1 ^c 10 ¹⁰ A nr ² , is only 5 mm s ^-1 , being in the creep-motion regime. The Hall angles under different current densities are estimated and summarized in Figure 12. Thus, the inventors have found that the Hall angle is almost independent of the current density and that it remains constant, at a value of approximately 38°, which confirms the above hypothesis.

[0050] Having all this information, the inhomogeneous electric current distribution in a target conductor can now be reconstructed based on the half- skyrmions in the multi-layer structure 200 and their Hall angles, as now discussed. Because the Hall angle of the half-skyrmions 504 is independent of the applied current density, and because the trajectories of the half-skyrmions 504 are accurately recorded/reflected by the elongation of the very narrow strip domains 502, the images of the narrow domains 502 for the structure 200 can be used to reconstruct the distribution of the inhomogeneous electric current into the target conductor structure 100, which is similar to the multi-layer structure 200. The term “similar” is used herein to mean having the same geometry.

[0051] To demonstrate the feasibility of this novel technique, the target conductor structure 100 show in Figure 1 is simulated by bonding two copper electrodes 1310 and 1320, as shown in Figure 13A, directly to a multilayer film 1301, having the same layer structure as the multi-layer structure 200, and having the Co layer with a thickness t = 1.95 nm. To simulate the electrical current distribution on the target conductor structure 100, the geometry of the film 1301 and the electrodes 1310 and 1320, is made identical to the target conductor structure 100, i.e. , the sizes, orientations, and configuration of the various elements is the same. In this way, the electrical current distribution obtained for the system 1300 is expected to be identical to the electrical current distribution for the target conductor structure 100. [0052] The resistance between the two electrodes 1310 and 1320 was found for this specific configuration to be 64.43 W. The film 1301 is first magnetized to saturation, using a positive magnetic field. When a +9 V voltage pulse with r = 10 ms is applied between the two electrodes 1310 and 1320, a well-shaped pattern 1330 can be observed on the Kerr image 1360. This image shows that many strip domains 1302 are generated near the two electrodes 1310 and 1320, in areas with the highest current density. Subsequently, the strip domains 1302 grow from the left electrode 1310 to the right electrode 1320, following a path that is governed by the half-skyrmions Hall effect as shown in Figure 13A. In this figure, the directions in which the strip domains 1302 grow (or the moving direction of the half-skyrmions) are shown at four random positions A to D, using arrows 1340. Considering that the Hall angle is 38° for the film 1301 , as indicated in Figure 9, it is possible to obtain the true current directions (arrows 1350) at those four random positions A to D, by rotating the arrows 1340 by 38° clockwise, to obtain the arrows 1350. When the polarity of the voltage (or current flow direction) is reversed, the strip domains 1302 grow from the right electrode to the left electrode as illustrated in Figure 13B. Both figures show the saturating magnetic field 1308 coming out of the picture.

[0053] If the film 1301 is saturated with a negative magnetic field, shown as magnetic field 1308 entering into the page in Figures 13C and 13D, it leads to the creation of upward strip domains that grow from the positive electrode to the negative one with the Hall angles opposite to those observed in Figures 13A and 13B. The direction 1340 of the strip domains 1302 growth and the corresponding directions 1350 of the electric current at four random positions are also indicated in Figures 13C and 13D.

[0054] To verify the validity of the current distribution 1350 constructed based on the Kerr images 1360, the corresponding current distribution was also calculated for the target conductor structure 100 using the traditional finite element modeling implemented by COMSOL Multiphysics based on the experimental configuration used in Figure 1. It was found, as illustrated in Figure 14, that the experimental results (arrows 1350 in Figures 13A to 13D) reconstructed from the Kerr images 1360 accurately match the numerical results (arrows 1450 in Figure 14). Thus, the inventors concluded that the electric current distribution in any target conductor having a given geometry may be simulated by using the multi-layer structure 200 200, configured to have the same geometry as the target conductor, by applying an electrical current between two or more electrodes placed on the trilayer structure, observing the paths of the half-skyrmions in corresponding MOKE images, and correcting these paths with the associated Hall angle, which is a function of the selected sizes of the materials of the trilayer structure.

[0055] The half-skyrmions paths observed in Figures 13A to 13D did not fully develop in the magnetic domains of the structure 1301, i.e. , a given number of current pulses have been applied to the electrodes 1310 and 1320, but not enough to fully form all the magnetic domains. If enough current pulses are applied to the electrodes, as illustrated in Figure 15, and a thickness of the ferromagnetic layer in the trilayer structure 1301 is appropriately selected to obtain the Hall angle of about 4°, then it is possible to directly obtain and observe the directions of the electrical current that would take place in the target structure 100 in Figure 1, by imagining the system 1300 as illustrated in Figure 15.

[0056] More specifically, as shown in Figure 9, the Hall angle decreases sharply as the effective PMA increases, and the Hall angle can be reduced down to 4° for the structure 200 with t = 1.5 nm for the Co layer. It has been shown that the strip domain grows in the structure 200 almost in alignment with the current distribution directions in the target conductor structure 100. Therefore, the Kerr images 1360 of the domain patterns of the multi-layer structure 200 can be considered as a viable direct mapping technique of an inhomogeneous current distribution of the structure 100. Based on these observations, an electrical current has been sent through the film 1301, using the t = 1.5 nm Co layer. The Kerr images 1360 of the domains obtained with \/= 9 V and r= 10 ms are shown in Figure 15, and one can observe that the magnetic domain pattern in the structure 200 is very similar to the distribution of the simulated electric current distribution in the target 100 calculated as shown in Figure 14. Thus, by choosing the film 1301 to achieve the Hall angle of close to 0° (such as 4°), it is possible to directly observe, through Kerr images, the electric current in the target conductor.

[0057] The same novel method was further applied to a film 200 with t = 1.5 nm, where the film has been shaped into a curved track, and the trajectories of the half-skyrmions driven by current pulses were investigated. The obtained images show the initial magnetic domains generated by applying a few negative magnetic field pulses to the positively saturated curved track. Some half-skyrmions are then created from the left entrance of the curved structure, when the current pulse with j = 7.83x10 ¹⁰ A nr ² is applied from the left to the right. It was found that the half- skyrmions with Q = 1/2 move along the curved conduction path. Note that because the initial state is a wide domain wall, the width of the created strip domain at the rear part is also large, and that a pinning site splits the strip domain and creates one more half-skyrmion. The obtained images also show that a half-skyrmion with Q = - 1/2 moves into the left side of the curved track. Then, a strip domain with a uniform width created by the trajectory of the half-skyrmion along the curved track is observed.

[0058] All these results demonstrate that the half-skyrmion paths can be used to directly map an inhomogeneous current distribution. For practical applications, in order to map the current distribution arising from a complex set of the electrodes, one can deposit a heterostructure, for example, the structure Pt/Co/Ta 200 as discussed above, but other similar structure may be used, that has the same geometry as the complex set of electrodes, and image the magnetic domain pattern in this structure using a Kerr microscope. Depending on the thickness of the various layers in the structure, the corresponding Hall effect may be calculated and this angle is applied to rotate the trajectories of the half-skyrmions visualized with the Kerr microscope for obtaining the current distribution in the target structure.

[0059] As has been discussed above, the inventors have found that the moving direction of the half skyrmion does not depend on the current density. This finding is in stark contrast to the fact that the Hall angle of an individual moving skyrmion depends nearly linearly on the current density, in the low-current-density regime, and saturates at a threshold current density, as observed in a Ta/CoFeB/TaOx system by [2] This unexpected result relative to the existing art [2] can be understood as following.

[0060] For an individual skyrmion, its motion can be described by Newton’s law, where the skyrmion is considered as a rigid particle and its internal structure and energy are constant. The forces acting on the skyrmion include, the effective driving force induced by the current that has a fixed direction (includes both the SOT- induced force and the Magnus force), the pinning force caused by the defects in the material, and the scattering by the defects in the materials. Under a small current density, the driving force may just be slightly stronger than the pining force. After each de-pining process, the kinetic energy (or velocity) of the skyrmion would be quite small. It can be then scattered easily by the random potentials originated from defects again and again, which leads to a random change in velocity (direction and magnitude), most likely, similar to the scattering of electrons by the defects in a conductor. Therefore, the motion of the individual skyrmions should be relatively random. With increasing the current density, the driving force will gradually dominate over other forces, and consequently the motion of individual skyrmions will be also gradually governed by the driving force induced by the current, and therefore, the average direction should gradually move to the driving force direction. Eventually, as the current density increases to a threshold value, the moving direction of the individual skyrmions will be fully aligned with the force direction. In this saturated case, the pining force and scattering become a perturbation in comparison with the large driving force. [0061] From the energy point of view, the total energy of the system is composed of mainly the internal energy of the skyrmion, its kinetic energy, pinning energy, demagnetization energy of the system, and thermal energy. When the current density is small, the kinetic energy of the skyrmion acquired from the electrical current is not significantly large in comparison to the others. The dynamics of the skyrmion will be dominated by random scattering potential and the thermal effect, which lead to a Brownian-like motion of the particle-like skyrmion with a small effective mass. More importantly, during the random motion of the skyrmion, the internal energy of the skyrmion and the energy of the whole system will not change much, if the kinetic energy of the skyrmion is not considered. Therefore, it is not difficult to understand the current density dependence of the moving direction of the individual particle-like skyrmions in the weak current regime.

[0062] However, during the elongation of the skyrmion though the motion of a half-skyrmion driven by a current, more spins will be flipped along the direction of the effective driving force, which changes the internal structure and energy of this skyrmion, although the skyrmion number is constant. Since the motion of the half- skyrmion is a consequence of the spin flipping caused by the current force, only the spins in the direction of the effective driving force can be flipped. The spins away from the effective driving force direction cannot flip, because there is not enough energy (or torque) to de-pin the domain walls. Therefore, it is difficult for the half- skyrmion to move away from the driving force direction. Another important characteristic of the effective driving force of the current is that it is extremely anisotropic, that is, it is unidirectional, which is in sharp contrast to the magnetic field induced domain expansion that is an isotropic process.

[0063] The system 400 discussed above with regard to Figures 8A and 8B can be used for generating an electrical current distribution that is associated with a target conductor structure 100 as discussed herein. Such system 400 includes the processor 450, which is configured to receive the target conductor structure 100, which has a given geometry that includes at least first and second electrodes 110, 120, the multi-layer structure 200 having a heavy-metal layer and a ferromagnetic layer so that a spin current is generated within the heavy-metal layer and injected into the ferromagnetic layer, corresponding at least first and second electrodes 404, 406 connected to a surface of the multi-layer structure 200 so that a same geometry is obtained as for the target conductor structure 100, the power source 430 configured to inject one or more electrical current pulses into the at least first and second electrodes 404, 406 to generate and drive a half-skyrmion 504 into the multi layer structure 200, and a memory 452 for receiving Kerr effect microscope images 420 of a trajectory of the half-skyrmion 504. The processor 450 is configured to map the trajectory of the half-skyrmion 504 into the target conductor structure 100 after correcting the trajectory with a Hall angle corresponding to the half-skyrmion 504. The correct trajectory represents the electrical current distribution of the target conductor structure.

[0064] In one embodiment, the multi-layer structure includes a first layer of Ta, a second layer of Pt, a third layer of Co, and a fourth layer of Ta, and wherein the second layer of Pt and fourth layers of Ta sandwich the third layer of Co. In one application, the processor is further configured to calculate the Hall angle for the half- skyrmion based on a configuration of the multi-layer structure, and/or to rotate plural tangents to the trajectory of the half-skyrmion with the Hall angle to obtain the electrical current distribution in the target conductor structure. The Hall angle depends on the effective perpendicular magnetic anisotropy of the multi-layer structure, which in turn depends on a thickness of the ferromagnetic layer and the thickness of the ferromagnetic layer is selected so that the Hall angle is substantially 4 degrees.

[0065] In one embodiment, the multi-layer structure 200 for determining an electrical current distribution in a target conductor structure 100 includes a first heavy-metal layer 204, a ferromagnetic layer 206 formed over the first heavy-metal layer 204, and a second heavy-metal layer 208 formed over the ferromagnetic layer 206 so that the ferromagnetic layer is sandwiched between the first and second heavy-metal layers. The heavy-metal layers are configured to generate a spin current to be injected into the ferromagnetic layer. The ferromagnetic layer is configured to generate a half-skyrmion due to the spin current. A thickness of the ferromagnetic layer is selected so that a Hall angle of the half-skyrmion is as small as 4 degrees.

[0066] The structure 200 includes as many electrodes as the target conductor structure, where a geometry of the electrodes of the multi-layer structure is identical to a geometry of the electrodes of the target conductor structure. The half-skyrmion moves in the ferromagnetic layer along a trajectory as one or more electrical current pulses are applied to the electrodes of the multi-layer structure. [0067] The system 400 discussed above may be programmed, by storing instructions into the memory 452, to implement a method for generating an electrical current distribution that is associated with a target conductor structure 100. The method is illustrated in Figure 16 and includes a step 1600 of receiving the target conductor structure 100, which has a given geometry that includes at least first and second electrodes 110, 120, a step 1602 of providing a multi-layer structure 200 having a heavy-metal layer 204/208 and a ferromagnetic layer 206 so that a spin current is generated within the heavy-metal layer 204/208 and injected into the ferromagnetic layer 206, a step 1604 of adding corresponding at least first and second electrodes 404, 406 to a surface of the multi-layer structure 200 so that a same geometry is obtained as for the target conductor structure 100, a step 1606 of injecting one or more electrical current pulses into the at least first and second electrodes 404, 406 to generate and drive a half-skyrmion 504 into the multi-layer structure 200, a step 1608 of imagining a trajectory of the half-skyrmion 504 with a Kerr effect microscope, where this step may be performed simultaneously with the step 1606, and a step 1610 of mapping the trajectory of the half-skyrmion 504 into the target conductor structure 100 after correcting the trajectory with a Hall angle corresponding to the half-skyrmion 504. The correct trajectory represents the electrical current distribution of the target conductor structure.

[0068] In one application, the multi-layer structure includes a first layer of Ta, a second layer of Pt, a third layer of Co, and a fourth layer of Ta, and wherein the second layer of Pt and the fourth layer of Ta sandwich the third layer of Co. A charge Q of the half-skyrmion is ½. The trajectory of the half-skyrmion lies entirely into the multi-layer structure.

[0069] The method may further include a step of calculating the Hall angle for the half-skyrmion based on a configuration of the multi-layer structure, and/or choosing a configuration of the multi-layer structure to produce a smallest Hall angle for the half-skyrmion. In one application, the smallest Hall angle is substantially 4 degrees. The method may further includes a step of rotating plural tangents to the trajectory of the half-skyrmion with the Hall angle to obtain the electrical current distribution in the target conductor structure, and/or applying no external magnetic field while generating the half-skyrmion, and/or generating the half-skyrmion as the Hall angle of the half-skyrmion is current density independent while a Hall angle of a full skyrmions is current density dependent. The electrical current distribution is inhomogeneous. The Hall angle depends on the effective perpendicular magnetic anisotropy of the multi-layer structure, which in turn depends on a thickness of the ferromagnetic layer.

[0070] The disclosed embodiments provide a method for visualizing the electrical current distribution in a target structure by using a Kerr microscope and a multi-layer structure that generates half-skyrmions. It should be understood that this description is not intended to limit the invention. On the contrary, the embodiments are intended to cover alternatives, modifications and equivalents, which are included in the spirit and scope of the invention as defined by the appended claims. Further, in the detailed description of the embodiments, numerous specific details are set forth in order to provide a comprehensive understanding of the claimed invention. However, one skilled in the art would understand that various embodiments may be practiced without such specific details.

[0071] Although the features and elements of the present embodiments are described in the embodiments in particular combinations, each feature or element can be used alone without the other features and elements of the embodiments or in various combinations with or without other features and elements disclosed herein. [0072] This written description uses examples of the subject matter disclosed to enable any person skilled in the art to practice the same, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the subject matter is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims.

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