Perpendicular reading of single confined magnetic skyrmions

Thin-film sub-5 nm magnetic skyrmions constitute an ultimate scaling alternative for future digital data storage. Skyrmions are robust noncollinear spin textures that can be moved and manipulated by small electrical currents. Here we show here a technique to detect isolated nanoskyrmions with a current perpendicular-to-plane geometry, which has immediate implications for device concepts. We explore the physics behind such a mechanism by studying the atomistic electronic structure of the magnetic quasiparticles. We investigate from first principles how the isolated skyrmion local-density-of-states which tunnels into the vacuum, when compared with the ferromagnetic background, is modified by the site-dependent spin mixing of electronic states with different relative canting angles. Local transport properties are sensitive to this effect, as we report an atomistic conductance anisotropy of up to ∼20% for magnetic skyrmions in Pd/Fe/Ir(111) thin films. In single skyrmions, engineering this spin-mixing magnetoresistance could possibly be incorporated in future magnetic storage technologies.

Electronic structure in the magnetically-active Fe-layer resolved into minority (solid) and majority (dashed) spin-channels. The splitted-structure of the FM-background (green) is modified due to quasi-AFM interactions when approaching the center of the skyrmion (black).    However, the charge transfer of this inner Pd-layer is now much smaller. Since the surface Pd-layer interacts with Fe indirectly through the inner Pd-layer, the induced spin magnetic moment is rather small (0.08 µ B ) on the surface.

Supplementary Note 2 -Angular dependence of the LDOS in nano-skyrmions
We now analytically derive the change in the LDOS at site r, denoted ∆LDOS(r), inside an axially symmetric skyrmion measured from the origin at r = 0 as a function of energy and magnetization rotation direction defined by the unit vector of the magnetic moment s(r) = sin θ(r) cos φ(r), sin θ(r) sin φ(r), cos θ(r) . Examples of rotation parameters θ and dθ for given skyrmion magnetic moments can be found in Supplementary Figure 2d. When compared to the ferromagnetic state (with all moments pointing out-of-plane), there are two contributions to ∆LDOS(r), one due to spin-orbit interaction (SOI) and one due to noncollinearity (NC): ∆LDOS(r) = ∆LDOS SOI (r) + ∆LDOS N C (r). The contribution from SOI, the so-called anisotropic magnetoresistance, is well known 3−5 : where A(r, E) is a coefficient depending on the site r and energy E, andŝ z (r) is the zcomponent of the spin-moment at site r. Thus upon including SOI, we expect, for example, a sin 2 θ(r) dependence, which contributes to ∆LDOS(r) in second-order.
The contribution from NC, intuitively, comes from the change in the electronic structure upon rotation of the magnetic moments at consecutive sites (i, j). For homogeneous magnetic spirals, a constant deviation in the LDOS from the ferromagnetic (FM) state will be observed for each atom in the spiral (ignoring SOI). In such a spiral, the smooth rotation of moments θ(i) → θ(j) = θ(i) + dθ for each atom pair is a symmetry operation commuting with a Hamiltonian having translational invariance, making each atom equivalent, and the electronic structure the same for each atom in the spiral. However, upon transforming the spiral's pitch dθ → dθ , one would find a different constant deviation of the electronic structure from the FM-state, such that spirals of different pitch can be identified by their different magnitudes in ∆LDOS N C .
In skyrmions, however, the rotation of the magnetic moments is not homogeneous, i.e.
dθ is not constant for all nearest-neighbor atom pairs while traversing the skyrmion's diameter. Thus there will be a site-dependent deviation in the LDOS among the atoms inside a skyrmion with respect to each other. In what follows, we demonstrate that these deviations are a complex function of the rotation angles, which will depend on the details of the electronic structure, the energy probed, and even the size of the skyrmions.
To derive ∆LDOS N C (r), we utilize multiple scattering theory: G is the Green function describing the whole system upon rotation of the magnetic moments, and g is the Green function describing the initial FM-state. The Green function will be used to evaluate the change in the LDOS induced by the rotation of the magnetic moments: as given in a matrix notation where a trace over orbital and spin angular momenta has to be performed. G can be evaluated via the Dyson equation connecting the non-collinear state to the ferromagnetic one: where ∆V describes the change of the potential upon rotation of the magnetic moments. It can be expressed as: where σ is the vector of Pauli matrices, and V diff is the difference of the two spin components We execute a similar expansion for the ferromagnetic initial Green function matrix g: where 1 2 is the 2 × 2 identity matrix.
Let us evaluate the first-order and second-order terms contributing to the Dyson equation (Supplementary Equation 3): where i and j are sites surrounding site r, or can be the site r itself. Since the trace over spin has to be performed, we will focus only on the terms that in the end will contribute to Supplementary Equation 2. We use a pair of useful properties of the Pauli matrices: and where i is the imaginary unit. After simplifying, we find the following result: where the coefficient B is given by In other words, the first sum in Supplementary Equation 6 leads to a behavior like (1−cos θ i ).
The second-order term is given by where the coefficient C is related to the Green functions and V diff by: Thus we obtain a dependence on the dot product of the unit vectors of the magnetic moments (1 − cos dθ cos dφ) and a contribution depending only on the z-components of the unit vectors of the magnetic moments.
We have thus demonstrated that due to NC, the dependence of the change in the LDOS with respect to the ferromagnetic state upon rotation of the magnetic moments is not trivial, and will have terms depending on the dot product between magnetic moments, contrary to the contribution coming from SOI. The non-collinear contribution is then where {s} is the spin configuration. Of course, depending on the details of the electronic structure and strength of perturbation related to the non-collinearity, higher-order terms can be important and have to be included in Supplementary Equation 13.
Combining ∆LDOS N C and ∆LDOS SOI , in the next section we will fit the change in the LDOS in terms of trigonometrical functions that depend on the rotation angles of the magnetic moments. We will apply these fits to our ab initio results as well as to an extended Alexander-Anderson model used to interpret the variation of the LDOS resonance-splitting upon rotation of the magnetic moments on neighboring sites.

Supplementary Note 3 -Two-atom extended Alexander-Anderson model
We wish to estimate the change in the LDOS and qualitatively understand the shifting in energy of resonant d-states in Fe as a function of the rotation angle between adjacent moments. To this end, we consider for simplification two magnetic atoms (i, j) = (1, 2) each having one localized orbital d z 2 whose single-particle eigenenergy is centered about The initial Hamiltonian describing this model is diagonal in spin-space. We could also consider an orbital of the type d xz in order to address the coupling induced by SOI between the d z 2 and d xz , as done by Caffrey et al 6 . However, since the impact of SOI on the LDOS has already been discussed by others, we focus here on the impact of NC on the LDOS. We study the ∆LDOS as we vary dθ = θ 1 − θ 2 between the two atoms. We restrict the hopping from atom-to-atom to non-spin-flip processes, characterized by the interaction parameter V hop .
In terms of Green functions, the following equation gives the LDOS for site 1: where Γ takes care of the broadening of the states. Instead of solving exactly the previous equation, one could also use perturbation theory, as described in the previous Supplementary Note 2, simplifying Supplementary Equation 13 to: where D = B 121 + C 1221 .
The energy of the resonant d-states, their width, and splittings come from our firstprinciples calculations, e.g. Supplementary Figure 3a  If the magnetic state is antiferromagnetic (dθ = 180°), there is repulsion between the minority and majority spin-states leading to a shift given by  and Pd/Pd/Fe/Ir ( Supplementary Figures 4a and 5a).
Next, we wish to estimate the change in the LDOS as a function of rotation as previously discussed, which leads to the TXMR signal. In Supplementary Figure 2 Figure 6b), the TXMR has the strongest signal near −0.8 eV, as before. Therefore one could infer from Supplementary Figure 6 that as the diameter of skyrmionic quasiparticles is increased, the spin-mixing effect not only survives, but also that specific locations of strong TXMR signals remain in similar energy windows as in smaller structures.

Supplementary Note 7 -Skyrmion racetracks for dense magnetic memories
Spin-transfer torque magnetic random access memory (STT-MRAM) circuits reliably read-out bit-states depending on a tunneling magnetoresistance anisotropy of ∼30-50% in some structures 9 , with a hope to achieve a magnetoresistance ratio R ON /R OFF ≈ 200% by 2022 10 . A TXMR effect as large as ∼20% as we have shown in this work should be enough to provide adequate read-margin for scaled technologies, and is larger than the < 2% change in resistance found in widespread commercially-used hard disk read heads based on anisotropic magnetoresistance alone 11 . Smaller changes in magnetoresistance just means there should be a more sensitive read-out circuit. Typically this means a few extra control-and boosttransistors and does not substantially increase the footprint of the memory, i.e. incorporating more sensitive read hardware does not degrade packing density considerably.
In potential skyrmion-based devices using CPP-TXMR, a R ON /R OFF ≈ 120% could feasably be well-worth the tradeoff when considering the possible performance gains with regards to: (1) potentially very low power dissipation due to small currents needed to manipulate the magnetic textures; (2) fast speed operation due to reduced read/write latencies associated with nano-scopic size; and (3) large increases in packing density. Let's consider points (2) and (3) in greater detail.
One issue with racetrack memories is that they are not random access. In a random access memory (RAM), any read/write operation can access any bit with roughly the same access time since the word and bit access lines (WL and BL), which are connected to the set/reset elements and read elements, are also connected in parallel with the individual memory cells (see Supplementary Figure 7a). In a racetrack memory, the situation is different. In practice, if a read/write were requested for an address whose representative bit were at the a small number compared to the total access time needed to complete the read or write operation, which in modern dynamic RAM (DRAM) is in the range 20-50 ns 13 . We do note, however, that larger in-plane currents will be required to accelerate the quasiparticles up to a velocity such as v Sk = 100 m/s, meaning there will be a tradeoff between t delay and power consumption.
Regardless, by incorporating skyrmion racetracks based on spin-mixing, there seems to be at first glance negligible additional acquired access latency -in fact, we may learn to find in the end that racetracks can be potentially faster than traditional RAM in certain geometries and biasing conditions, due to the nano-scopic size of the skyrmion quasiparticles.
With regards to circuit layouts, it is clear that moving to racetracks will provide large gains in packing density. As an example, let's compare the workhouse 1-transistor 1capacitor (1-T 1-C) DRAM unit to our racetrack-based spin-mixing magneto-memory. Con- Even including more intricate sense amplifiers for read/write operations in skyrmion racetracks, it is clear that incorporation of sub-5 nm skyrmions as data-carriers would substantially increase packing density compared to current state-of-the-art technologies.
Possibly the gains in packing density could even be larger introducing vertical racetracks 14 , which are difficult to fabricate thus far.