Colossal topological Hall effect at the transition between isolated and lattice-phase interfacial skyrmions

The topological Hall effect is used extensively to study chiral spin textures in various materials. However, the factors controlling its magnitude in technologically-relevant thin films remain uncertain. Using variable-temperature magnetotransport and real-space magnetic imaging in a series of Ir/Fe/Co/Pt heterostructures, here we report that the chiral spin fluctuations at the phase boundary between isolated skyrmions and a disordered skyrmion lattice result in a power-law enhancement of the topological Hall resistivity by up to three orders of magnitude. Our work reveals the dominant role of skyrmion stability and configuration in determining the magnitude of the topological Hall effect.

1 Evolution of magnetic textures 1

.1 Evolution of isolated skyrmions and skyrmion lattices with applied field
Figures S1-S3 show the evolution of isolated and dense skyrmion configurations with applied magnetic field (H) for the samples discussed in figure 1 of the main paper. These multilayers show labyrinthine stripe domains at H = 0. For κ < 1 textures evolve with H as: stripe domains to isolated skyrmions and then to a uniform polarized phase. For κ ≥ 1 textures evolve as: stripe domains to a skyrmion lattice, then to isolated skyrmions and finally to a uniform polarized phase.

Quantitative analysis of skyrmion configurations
(1) We highlight that variation of T = 5 − 300 K results in a larger change in K ef f (T ) compared to A(T ) and D(T ). As a result, . As shown in figure S14, depending on sample composition, D    Figure S15 shows the T-dependent ρ T HE (H) profiles for samples presented in figure 2a of the main paper. Please note that the profiles show a small non-zero offset above the saturation field of magnetization (H > H s ), as explained in the methods section of the paper. This is a systematic offset due to the fitting procedure. The THE does not change sign.   4 Correlation between the enhancement in ρ T HE , skyrmion configuration transition and sign reversal of R 0 In Fig. S17 we show how the local enhancement in ρ T HE discussed in figure 2b-c of the main paper correlates with the sign reversal of R 0 . We find that the local enhancement in ρ T HE always occurs at a temperature higher than or equal to the sign reversal of R 0 (S17a-c). Similar behaviour is observed upon varying κ (via the Fe/Co composition) at fixed temperature (S17d-e), with the ρ T HE peak occurring at higher κ and R 0 values as large as −3 to −11 nΩ.cm/T, which are considerably larger than the tiny values recorded close to the sign reversal. This systematic behaviour in ρ T HE and sign reversal in R 0 can also be correlated with the transition in skyrmion configuration from an isolated to a lattice phase. The close connection between electrical transport and emerging magnetic textures suggests that both these factors are influenced by systematic variations in the occupancy of the electronic bands while varying the temperature and composition of our films. It is tremendously difficult to extract quantitative information from the band structure of these multilayer stacks, due to the presence of large number of bands. However, in the following sections we use a tight binding model and DFT calculations to qualitatively explain the sign reversal of R 0 without affecting the sign of ρ T HE , as well as the systematic band shift with respect to the Fermi energy upon changing the Fe/Co composition.  As shown in Fig. S18, our T-dependent Hall transport suggests a sign change of the classical Hall coefficient (R 0 ) when T is increased, and it is sensitive to the relative thickness of the Fe/Co layers. However, we find a topological Hall effect (T HE) that still has the same sign (see above section and Fig. S15). This will be explained in the following.
We have conducted ab-initio calculations of the measured systems (shown in the next section). However, the band structure of a multi-stack is highly complicated. Even for a single stack of Pt(5)/Co(3)/Fe(2)/Ir(5) the unit cell consists of 30 atoms, leading to many overlapping bands near the Fermi energy. This renders a direct calculation of the Hall resistivity impossible. However, we can extract the fundamental features of the carriers from the band structure to qualitatively understand the origin of the unconventional sign reversal behavior in R 0 and T HE.

Semiclassical argumentation
The existence of a large number of bands provides a variety of effective masses of the carriers. In particular, we find electron and hole states with various magnitudes of effective masses near the Fermi energy. This was a motivation to construct a two-orbital tight-binding model that allows to explain the experimental findings qualitatively. We start our explanation with semi-classical considerations for the transport in semiconductors.
We construct the model such that near the Fermi energy E F = 0, electron states and hole states with different effective masses are present (Fig. S19a). The spins in these two bands are oriented parallel (↑↑) and anti-parallel (↑↓) with respect to the texture, respectively. This is accounted for in the tight-binding matrix via a Hund's coupling term by analogy with Ref. [7] (see the following section for more details). As the spin texture we consider a skyrmionic phase for the T HE and a collinear ferromagnet for the classical Hall effect.
Typically, the Hall resistivity is given by when electrons (carrier density n and mobility µ e ) and holes (carrier density p and mobility µ h ) are present. In the presence of an external magnetic field and a ferromagnetic spin texture, as in the ordinary Hall effect scenario, the effective magnetic field is given exclusively by the actual field. Both carrier species feel the same magnetic field B h = B e = B.
For this reason, the sign of the resistivity is determined by the sign of pµ 2 h −nµ 2 e . Under variation of the T , the dominating carrier density can change from hole-like to electron-like. For two bands, this is only possible if both carriers have different effective masses. Simply speaking, for elevated T , more higher energetic states (i. e. electronic states) are occupied than lower energetic states (i. e. hole states).
Concerning the THE, the emergent field of the skyrmion acts differently on the two different types of carriers that are characterized by spin parallel and spin anti-parallel states, respectively, in our model. In the adiabatic limit, the electron spins align with the spin texture upon traversing a skyrmion. Therefore, the different spin species align oppositely and feel opposite emergent fields. Considering only the skyrmion texture and no external field, the effective magnetic fields are B e = −B h = B em . For this reason, the sign of the Hall resistivity is given by −(pµ 2 h + nµ 2 e )B em which does not change its sign under variation of the two carrier densities.

Explicit transport calculation using Berry curvature
Next, we show explicit calculations using the four-band tight-binding model to add credence to the above interpretation.
The model has been constructed such that it resembles a system which exhibits a sign change in the classical Hall con- where c † i , c i are creation and annihilation operators at site i with orbital α. In all calculated systems, we consider two orbitals {α, β} = {1, 2} which -for simplicity -are assumed to not hybridize: |t In the following, the classical and the topological Hall scenarios will be discussed. In both cases, the tightbinding Hamiltonian is diagonalized where the eigenvalues are the band energies E n (k) and the eigenvectors |n(k) are important for the calculation of the reciprocal-space Berry curvature This quantity determines the intrinsic contribution to the Hall conductivity f is the Fermi distribution function that accounts for the reoccupation of the electronic states upon variation of the tem- Ordinary Hall conductivity. First, we discuss the results for a ferromagnetic texture in the presence of an external magnetic field. This scenario brings about an ordinary Hall effect. In this case, the texture is m i ≡ e z . Without the presence of an external magnetic field, the unit cell of the considered square lattice comprises a single lattice site with two orbitals. The resulting band structure exhibits four bands as shown in Fig. S19a.
The presence of an external magnetic field modifies the hoppings Here, the vector potential A can be expressed in Landau gauge to fulfill B = ∇ × A. As a detail, it is worth mentioning that the Peierls phase in the hopping has to be periodic. For certain magnetic field strengths a magnetic unit cell can be constructed. Here, p and q have to be coprime integers. The smaller q, the smaller the unit cell and the faster the calculation. For this reason, we chose p/q = 1/36 which would correspond to a large field. This is unproblematic, since the strength of the magnetic field does not affect the energy and T dependence of the Hall conductivity qualitatively but only scales its magnitude.
The resulting band structure consists of many Landau levels (cf. Ref. [7] for more details). The energy de- Topological Hall conductivity. Secondly, we discuss the topological Hall effect of electrons in skyrmion lattice without taking into account the magnetic field, in order to calculate the isolated topological contribution. This time the magnetic texture is space dependent. The texture in the magnetic unit cell is given as for r i < r 0 , where r i = x 2 i + y 2 i is measured with respect to the skyrmion's center. This magnetization resembles circular Néel skyrmions (topological charge N Sk = −1) in a square lattice. For the calculations we assume a skyrmion radius of r 0 = 5a (here a is the lattice constant) and a square shaped unit cell comprising 100 lattice sites. occurs upon variation of the T (Fig. S19f), different to the behavior of the ordinary Hall effect (Fig. S19d), as discussed above.
In order to corroborate the tight-binding model discussed above, we performed ab-initio calculations of different stacks.
We used the QuantumATK package [8] together with the GGA-PBE exchange-correlation functional [9] and the SG15 combination of norm-conserving pseudopotentials and LCAO basis sets [10]. We used a dense 24 × 24 × 1 k-point grid and a density mesh cutoff of 160 Hartree. The total energy and forces have been converged at least to 10 −4 eV and 0.01 eV/Å, respectively. Spin-orbit coupling is included in band structure calculations. The minimal setup is a single stack consisting of Pt(10)/Co(6)/Fe(4)/Ir (10). Additionally, we add another Pt(10) since in the experiment no free Ir surface is present. We calculate a collinear ferromagnet; the self-consistent calculation of a skyrmion texture is too demanding by todays standards. Still, the unit cell consists of many atoms leading to many bands in the band structure (Fig. S20). This renders a direct calculation of the Hall conductivity impossible.
In Fig. S20, the bands have been colored blue if the Co character is dominant and red if Fe is dominant. The bands also hybridize with Ir and Pt to a large degree but we decided to focus here on the magnetic elements. Near the K point the motive of the tight-binding model becomes visible. We observe electron and hole bands with different effective masses and different magnetic compositions. This explains that the peculiar sign change of the Hall coefficient can indeed occur upon variation of the T . Of course, here the situation is more complex than in the four-band tight-binding model but the general arguments remain.
Next, we will show how the motive in the band structure changes upon changing the stack. In the experiment, the multistack consists of several repetitions. In the calculations, we can only calculate up to three repetitions in a reasonable time. As is shown in Fig. S21, a repetition of the stack adds more bands to the band structure. However, the copies look Figure S20: Ab-initio band structure of a single stack Pt(10)/Co(6)/Fe(4)/Ir(10)/Pt (10). Blue bands are dominated by Co and red bands are dominated by Fe. The highlighted region exhibits electron and hole bands with different masses.
The evolution of these bands under variation of the stack will be investigated in Figs. S21 and S22.
rather similar to the initial bands implying that the repetition rate plays a minor role for the Hall conductivity.
This changes when the composition of the stack is changed. Exemplarily, we vary the Fe thickness (Fig. S22), where panel b is the highlighted region of Fig. S20. Compared to Fe(4), a decrease of the Fe thickness leads to a shift of the Fe bands in the negative energy direction (Fig. S22a). Reversely, an increase in the thickness shifts the Fe bands slightly upwards in energy (Fig. S22c).