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

M. Raju1†‡, A.P. Petrović1, A. Yagil2, K.S. Denisov3, N.K. Duong1, B. Göbel4,
E. Şaşıoğlu4, O.M. Auslaender2⋆, I. Mertig4, I.V. Rozhansky3, C.
Panagopoulos1†

1Division of Physics and Applied Physics, School of Physical and Mathematical
Sciences, Nanyang Technological University, 637371 Singapore
2Department of Physics, Technion, Haifa 32000, Israel
3Ioffe Institute, Politekhnicheskaya 26, 194021 St.Petersburg, Russia
4Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle
(Saale), Germany
‡Present address: Institute for Quantum Matter and Department of Physics and
Astronomy, Johns Hopkins University, Baltimore, MD, USA
⋆Present address: Neuroscience Institute and Tech4Health Institute, NYU
Langone Health, New York, NY
†Correspondence: mraju<EMAIL_ADDRESS>
##  1 Abstract

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.

##  2 Introduction

Magnetic skyrmions are topologically-charged nanoscale spin textures which
form due to competition between spin-rotating and spin-aligning magnetic
interactions. In thin film heterostructures these magnetic interactions can be
finely tuned via the multilayer geometry and composition, rendering skyrmion-
hosting films ideally suited for spintronic applications [1, 2, 3, 4, 5]. One
promising route towards functionalising skyrmions in metallic systems is to
utilise their intrinsic magnetoelectric coupling, which is manifested by a
topological Hall effect ($THE$) [6, 7, 8]. Charge carriers moving through a
skyrmion spin texture experience an emergent magnetic field ($B_{eff}$)
associated with the spin winding of a skyrmion. The transverse deflection of
charge carriers interacting with $B_{eff}$ results in a $THE$ [7]. Provided
that each skyrmion remains stationary with respect to incident charge
carriers, an array of skyrmions will exhibit a topological Hall resistivity
$\rho^{THE}$, given by

$\rho^{THE}=P\cdot R^{{}^{\prime}}_{0}\cdot(n_{sk}\cdot\Phi_{0}),$ (1)

where $P$ is the spin polarization of the charge carriers,
$R^{{}^{\prime}}_{0}$ is the Hall coefficient representing the effective
charge density contributing to the $THE$ (usually taken as the classical Hall
coefficient $R_{0}$), $B_{eff}\equiv~{}n_{sk}\cdot\Phi_{0}$ is the emergent
field associated with a given skyrmion density $n_{sk}$, and $\Phi_{0}=h/e$ is
the magnetic flux quantum, with $h$ Planck’s constant and $-e$ the electron
charge. This phenomenon is distinct from the classical and anomalous Hall
effects, which are proportional to the applied magnetic field $H$ and
magnetization $M(H)$, respectively [7]. Experimentally, $\rho^{THE}(H)$ can be
identified as the residual between the total measured Hall resistivity
$\rho_{yx}(H)$ and a fit to the classical $R_{0}(H)$ and anomalous $R_{S}M(H)$
Hall resistivities namely $\rho_{yx}^{fit}(H)=R_{0}H+R_{\mathrm{S}}M(H)$ [9]
(where $R_{\mathrm{0}}$ and $R_{\mathrm{S}}$ are the classical and anomalous
Hall coefficients respectively). The anomalous Hall resistivity
$\rho^{AHE}\equiv~{}R_{S}M(H)$ can be described by a superposition of terms
with linear and quadratic dependences on the longitudinal resistivity
$\rho_{xx}$, corresponding to the skew scattering and side jump terms
respectively [8]. However, the variation of $\rho_{xx}(H)$ in these
multilayers is extremely small ($<0.01\%$ for fields up to magnetic saturation
at $H_{S}$ and $<0.17\%$ for fields up to $\pm 5$ T [5, 9]). Consequently, the
treatment of $\rho^{AHE}$ as $[a\cdot\rho_{xx}(H)+b\cdot\rho_{xx}^{2}(H)]M(H)$
or simply $R_{S}M(H)$ has negligible influence on the estimated $\rho^{THE}$
[9]. A detailed scaling analysis of $\rho^{AHE}$ with $\rho_{xx}$ as a
function of temperature $T$ ($5-300$ K) and $H$ for these multilayers can be
found in Refs. [5, 9], together with a discussion of the validity and
reproducibility of the estimated $\rho^{THE}$.

Using equation ($1$) and our experimetally-determined $\rho^{THE}$, one can
hence estimate $n_{sk}$ from an electrical transport measurement as:

$n_{sk}(THE)=\mid\rho^{THE}\mid\div\mid(P\cdot R_{0}\cdot\Phi_{0})\mid.$ (2)

$n_{sk}$ may also be measured directly using real-space imaging techniques
such as magnetic force microscopy ($MFM$) [9, 10, 11], magnetic transmission
X-ray microscopy [4, 5] or Lorentz transmission electron microscopy [12].
Comparing these transport and imaging approaches can yield evidence for
adiabatic transport if $n_{sk}(THE)\approx n_{sk}(MFM)$[6, 7, 8], non-
adiabaticity if $n_{sk}(THE)<n_{sk}(MFM)$ [13, 14], or alternatively reveal
enhanced transverse scattering mechanisms if $n_{sk}(THE)>n_{sk}(MFM)$ [13,
15, 16, 17].

In bulk non-centrosymmetric materials which exhibit stable skyrmion lattices,
the values of $n_{sk}$ estimated from transport and imaging are in good
agreement [6, 7, 8]. However, this is not the case for thin film multilayers
with an interfacial Dzyaloshinskii-Moriya interaction ($DMI$), which can be
tuned to exhibit isolated or dense skyrmion configurations. Large, conflicting
values for $\rho^{THE}$ have been reported in magnetic thin films,
corresponding to $n_{sk}(THE)$ orders of magnitude larger than $n_{sk}(MFM)$
[5, 18, 10, 11, 12, 19].

Determining the mechanism leading to this extraordinary disagreement is
crucial for understanding the electrical response of chiral spin textures and
their detection in devices. Recently, a valuable clue has been provided by
predictions [15, 16, 17] and observations [20, 21] of spin fluctuation-induced
effects on charge transport in non-coplanar magnets. It is hence plausible
that quantum or thermal fluctuations may influence the Hall response of
materials with a chiral instability.

In this work, using temperature-dependent Hall transport and $MFM$, we track
the evolution of $THE$ across the transition between isolated skyrmions and a
disordered skyrmion lattice. We find that isolated skyrmion configurations
produce a larger $THE$ than dense arrays of skyrmions, with an enhancement of
up to three orders of magnitude at the transition. Our data reveals a power-
law behavior in $n_{sk}(THE)/n_{sk}(MFM)$ which we interpret in terms of
chiral spin fluctuations. Following universal scaling laws, we extract the
critical exponents governing this phase transition [22, 23, 24].

##  3 Results

Figure 1 shows measurements of $\rho^{THE}(H)$ at $300$ K for a set of
Ir/Fe($x$)/Co($y$)/Pt multilayers. $MFM$ images acquired at $H$ maximizing
$\rho^{THE}(H)$ display spin configurations ranging from isolated skyrmions
(Fig.1a-c) to dense, disordered skyrmion lattices (Fig.1d-g) (see section 1 of
the supplementary information for the evolution of magnetic textures). This
evolution in skyrmion configuration is driven by the $T$-dependent stability
parameter $\kappa\equiv~{}\pi D/4\sqrt{AK_{eff}}$, which describes the
competition between the three key magnetic interactions: exchange coupling
($A$), magnetic anisotropy ($K_{eff}$) and $DMI$ ($D$). Our multilayer
Ir/Fe($x$)/Co($y$)/Pt stacks allow us to systematically tune $\kappa$ (and
hence $n_{sk}$) via the Fe/Co layer thickness ratio: $x/y$ $<0.5$ yields
$\kappa<1$, whereas $\kappa\geq 1$ for $x/y$ $\geq 0.5$ [5]. Varying $T$
provides an additional handle to tune $\kappa$ for a given Fe/Co ratio,
$\kappa$ increases with $T$ due to $T$-dependent magnetic interactions (see
section $2$ of the supplementary information). At $H=0$, these chiral magnetic
films exhibit a labyrinthine stripe domain phase. Under a finite $H$, this
transforms into a metastable skyrmion phase if $\kappa<1$ (Fig.1(a)-(c)), or a
disordered skyrmion lattice phase if $\kappa\geq{1}$ (Fig.1(d)-(f)). For
$\kappa\geq{1}$, the skyrmion lattice dissolves into isolated skyrmions before
a uniformly polarized ferromagnetic ($FM$) phase develops for $H>H_{S}$. The
isolated skyrmion phase emerges between a lattice and a $FM$ phase; its
appearance can be regulated by $\kappa$, $T$, $H$, or a combination of these
parameters (see section 1 of the supplementary information). Indeed, in chiral
magnetic films the transformation of a polarized FM phase into an array of
isolated skyrmions and ultimately a skyrmion lattice occurs via a nucleation-
type second-order phase transition [22, 23, 24].

We find that isolated skyrmion configurations (Fig.1a-c) consistently generate
a larger $\rho^{THE}(H)$, despite their small $n_{sk}(MFM)$. Contrary to
expectations from equation (1), dense skyrmion arrays (Fig.1d-f) with larger
$n_{sk}(MFM)$ typically display a smaller $\rho^{THE}(H)$. The generic large
$\rho^{THE}(H)$ for isolated skyrmions can be further confirmed using the
evolution of a dense skyrmion array with increasing $H$, as shown in Fig.1d-f.
Here, $\rho^{THE}(H)$ rises as the skyrmion lattice dissolves into isolated
skyrmions before reaching a $FM$ phase with $\rho^{THE}(H)\approx 0$. To
further probe this discrepancy between transport and imaging experiments, we
evaluate $\Delta_{\rho/M}\equiv~{}n_{sk}(THE)/n_{sk}(MFM)$ [18, 9, 11, 12, 25,
19], which quantifies the effective topological charge contributing to the
measured $THE$, then compare this quantity for different skyrmion
configurations imaged by $MFM$. As $n_{sk}(MFM)$ increases, $\Delta_{\rho/M}$
is systematically reduced (Fig. 1g). Hence for a skyrmion lattice we have
$n_{sk}(THE)\approx n_{sk}(MFM)$ and for isolated skyrmions
$n_{sk}(THE)>n_{sk}(MFM)$. The strong influence of the skyrmion configuration
on $\rho^{THE}$ underlines the crucial role of $\kappa$ in determining the
magnitude of $\rho^{THE}(H)$. We therefore examine the transition between
isolated and dense arrays of skyrmions, and the influence of $\kappa$ on the
measured $THE$.

Figure 2a depicts the $T-\kappa$ space and hence the wide range of spin
textures which we can experimentally access by varying $T$ ($5-300$ K), the
Fe/Co thickness ratio and the number of repeats of [Fe/Co] in a multilayer
stack. Using our $MFM$ images we identify the transition (blue shaded region
in Fig. 2a) between isolated skyrmions and a dense, disordered skyrmion
lattice [5] and track its dependence on $T$ and $\kappa$ (see section 1 of the
supplementary information for the evolution of magnetic textures across $T$
and $\kappa$). We explore the impact of this transition on the $THE$ by
correlating the evolution of $\rho^{THE}$ across the $T-\kappa$ phase space.
Fig. 2b displays $\rho^{THE}(T,\kappa)$ curves for individual multilayers
studied as a function of $T$, while Fig. 2c shows $\rho^{THE}(\kappa)$ curves
at a fixed $T$ with varying Fe/Co. The location of the maximum in $\rho^{THE}$
(see Fig. 2b,c) tracks the phase boundary between isolated skyrmions and a
disordered skyrmion lattice tuned by $T$, $\kappa$. This confirms that the
degree of $THE$ enhancement is closely linked to proximity to the phase
boundary.

Here we note that the sign of $THE$ remains unchanged across the $T-\kappa$
phase diagram. However, $R_{0}$ changes from positive to negative with $T$ due
to multiband transport [9]. This crossover consistently follows the local
maximum in $\rho^{THE}$ and the transition from isolated skyrmions to the
lattice phase (for details see sections $4-6$ of the supplementary
information). Such correlation between the local enhancement of $\rho^{THE}$,
skyrmion configuration and sign reversal of $R_{0}$ suggests that these
factors are influenced by systematic changes in the occupancy of the
electronic bands while varying $T$ and the Fe/Co composition. To understand
the mechanism responsible for the $R_{0}$ sign reversal, we employed a tight-
binding model together with $ab-initio$ calculations of the electronic band
structure of our multilayer stack. When electron like spin-up states and hole-
like spin-down states are both present near the Fermi energy, a $T$ induced
change in the individual carrier densities can lead to a sign change of
$R_{0}$ due to compensation of the normal Hall signal from carriers with
opposite charge. However, due to their opposing spin alignment, the
interaction of these carriers with $B_{eff}$ from the skyrmions gives rise to
a $THE$ which maintains the same sign from $T=5-300$ K. A complete
quantitative analysis of the band structure is beyond the scope of this work;
however, these experimental observations provide valuable insight into the
links between thermodynamic stability and charge transport in chiral magnetic
textures.

We now return to the enhancement of the $THE$. Within the critical region
surrounding a second order phase transition, fluctuations of the incipient
order parameter ($\eta$) dominate the material response. Hence, chiral spin
fluctuations at all length scales may develop at the skyrmion lattice phase
boundary [7]. The spin chirality contribution to the $THE$ originates from a
non-zero scalar triple product $[S{{}_{i}}\cdot(S{{}_{j}}\times S{{}_{k}})]$,
where $S{{}_{i}}$, $S{{}_{j}}$, and $S{{}_{k}}$ form a cluster consisting of
three fluctuating spins. Such clusters can generate Hall signals via an
unconventional skew-scattering mechanism which can greatly exceed the
contribution from a stable skyrmion configuration [15, 16, 17, 26]. The size
of these clusters can be as small as several atomic spacings, which is much
smaller than the skyrmion radius determined by the macroscopic magnetic
interaction parameters in our films. Upon approaching the phase boundary, such
fluctuation-induced clusters are expected to proliferate throughout the
material, hence generating considerable topological charge [15, 16, 17]. The
correlation of these fluctuating spin clusters is associated with the
formation/destruction of long-range order, i.e. a stable skyrmion lattice
configuration. Within this scenario, the spin correlation length of the
skyrmion lattice (which for an infinite thermodynamic system would diverge at
the transition) is distinct from the short length scale spin fluctuations
responsible for a non-zero scalar spin chirality. A detailed microscopic
understanding of this fluctuating spin chirality in relation to the phase
transitions in chiral magnetic systems remain to be established theoretically
[26]. As we approach the transition the influence of such fluctuations should
be revealed by power-law behavior in material properties, including the spin
susceptibility and hence $THE$ [20]. The maximum observable $THE$ magnitude is
expected to saturate at a value corresponding to the maximal chiral spin
cluster density (imposed by the atomic spacing). Consequently, the power-law
enhancement is truncated close to the transition and there is no $THE$
singularity.

Our experimental observations reveal that the phase transition between
isolated skyrmions and a disordered skyrmion lattice is sensitive to $\kappa$,
$T$ and $H$. In the following we consider an effective temperature
$T^{{}^{\prime}}$ which accounts for the role of varying $T$ and $\kappa$ in
controlling this phase transition. Figure 2d illustrates how the critical
temperature $T_{c}$ and stability $\kappa_{c}$ defining the transition vary
with $\kappa$ and $T$ respectively, in accordance with experiments (Fig.2a).
The phase boundary between isolated skyrmions and a skyrmion lattice may be
considered a smooth function $T_{c}(\kappa)$ [22, 23, 24]. In the vicinity of
a phase transition ($\kappa=\kappa_{c}\sim 1$, within the transition region,
Fig. 2a) the boundary can be described as $T_{c}\cdot\kappa\approx$ constant
and hence,

$\displaystyle\frac{d\left(T_{c}\cdot\kappa\right)}{d\kappa}=\frac{d\left(constant\right)}{d\kappa}\Rightarrow\kappa\frac{dT_{c}}{d\kappa}+T_{c}=0\Rightarrow\frac{dT_{c}}{d\kappa}\approx-
T_{c},$

which in turn suggests an effective temperature $T^{\prime}=T\cdot\kappa$ with
a critical value $T_{c}^{\prime}=T_{c}\cdot\kappa_{c}$ defining the
transition. One may also consider the phase boundary in terms of varying $H$,
however, the interdependence of $\kappa$ and $H$, and its variation with $T$
remain unclear. We therefore adopt the simpler picture presented in Fig.2d
which allows a straightforward interpretation of our experimental observations
in Fig.1 and 2a-c.

Figure 3a shows a power-law behavior in $\Delta_{\rho/M}$ surrounding a
critical value of $T^{{}^{\prime}}_{c}\approx 110\pm 15$ K. This is consistent
with the presence of a second order phase transition driven by $T$ and
$\kappa$. According to Landau theory, the amplitude of $\eta$ grows as a
power-law on the low-symmetry side of any second order transition (which in
our case corresponds to $T^{\prime}>T_{c}^{\prime}$):

$\eta\propto|T^{\prime}-T_{c}^{\prime}|^{\beta},$

whereas fluctuations of $\eta$ on both sides of the transition scale as:

$\langle(\Delta\eta)^{2}\rangle\propto\frac{1}{|T^{\prime}-T_{c}^{\prime}|^{\gamma}}.$

Extracting the critical exponents $\beta,\gamma$ from our data requires the
identification of $\eta$ for the skyrmion lattice configuration. In the
ordered phase of a chiral magnet, a helical wavevector which describes the
spin rotation period emerges, capturing the symmetry broken by $\eta$ at the
transition. Discrete Fourier transforms of a skyrmion lattice yield a strong
peak in the structure factor at $k=l^{-1}$, where $l\propto
1/\sqrt{n_{sk}(MFM)}$ is the skyrmion lattice parameter and $k$ rises
continuously from zero with the emergence of a lattice. We therefore expect
$\eta$ to scale with $\sqrt{n_{sk}(MFM})$, allowing analysis of the transition
using the conventional scaling approach

${n_{sk}(MFM)}\approx{|T^{\prime}-T^{\prime}_{c}|^{2\beta}},\qquad
T^{\prime}>T_{c}^{\prime}.$ (3)

Fluctuations of $\eta$ in the vicinity of the phase transition create a
topological charge and hence contribute an effective skyrmion density
$n_{fl}\approx\langle(\Delta\eta)^{2}\rangle$ ($n_{fl}\gg n_{sk}(MFM)$) to the
total measured $n_{sk}(THE)$. Consequently,

$\Delta_{\rho/M}=\frac{n_{sk}(THE)}{n_{sk}(MFM)}=\frac{n_{sk}(MFM)+n_{fl}}{n_{sk}(MFM)}\approx\frac{n_{fl}}{n_{sk}(MFM)},$

therefore we may model the power-law rise in $\Delta_{\rho/M}$ as follows:

$\displaystyle\Delta_{\rho/M}\sim\frac{1}{|T^{\prime}-T^{\prime}_{c}|^{\gamma}},\qquad
T^{\prime}<T_{c}^{\prime}$ (4)
$\displaystyle\Delta_{\rho/M}\sim\frac{1}{|T^{\prime}-T^{\prime}_{c}|^{2\beta+\gamma}},\qquad
T^{\prime}>T_{c}^{\prime}$ (5)

where the extra $2\beta$ in the exponent above $T^{\prime}_{c}$ originates
from the appearance of a stable skyrmion lattice as indicated in the eqn.$3$.

Fits to eqns. $3$, $4$ to extract $\gamma$ and $\beta$ are shown as red lines
in Fig. 3a (for $T^{{}^{\prime}}<T_{c}^{{}^{\prime}}$) and 3b respectively,
where we use two independent data sets to obtain exponents $\gamma\approx
1.88\pm 0.43$, and $\beta\approx 0.31\pm 0.05$. These values suggest a three
dimensional Heisenberg spin system in which the exponents are modified by
competing spin interactions in a quasi-two-dimensional environment. This is in
agreement with earlier studies of thin film magnets [27], which consistently
reveal an increased $\gamma$ and a reduced $\beta$ with respect to the three
dimensional Heisenberg values $1.39$ and $0.36$ respectively [28]. To
independently cross-check our estimate of the exponents $\gamma$ and $\beta$
we also fit the discrepancy in the skyrmion lattice regime using eqn. $5$
(Fig. 3a for $T^{{}^{\prime}}>T_{c}^{{}^{\prime}}$). We obtain the combination
$(\gamma+2\beta)=2.61\pm 0.44$, which is consistent with the $\gamma$ and
$\beta$ extracted individually from our fluctuation analysis, adding credence
to the validity of these critical exponents.

As discussed above, the finite size of the short lengthscale fluctuating spin
clusters imposes an upper limit on the magnitude of $n_{fl}$, which is
expected to saturate as the system approaches $T^{{}^{\prime}}_{c}$.
Experimentally, we can only modulate $T^{{}^{\prime}}$ in discrete steps, so
we cannot tune $T^{{}^{\prime}}$ continuously through $T^{{}^{\prime}}_{c}$ to
reveal the expected saturation in $n_{fl}$ and hence $THE$. However, our
experimental $\Delta_{\rho/M}$ clearly demonstrates the anticipated power-law
scaling on either side of $T^{{}^{\prime}}_{c}$, systematically varying from
$1-10^{3}$. Using eqn.$1$ and the values of $|R_{0}|\approx 0.5-16$
n$\Omega$.cm/T measured in our multilayer films, we estimate that
$\Delta_{\rho/M}$ should saturate at a value $~{}10^{4}-10^{5}$ as
$|T^{{}^{\prime}}-T^{{}^{\prime}}_{c}|\rightarrow 0$. From the power-law
trends shown in Fig. 3a, we deduce that this saturation is only visible for
$|T^{{}^{\prime}}-T^{{}^{\prime}}_{c}|<20$ K, which lies beyond the minimum
$|T^{{}^{\prime}}-T^{{}^{\prime}}_{c}|\approx 40\pm 15$ K accessed during our
experiments. It may therefore be possible to engineer a further THE increase
of at least another order of magnitude, by tuning skyrmion-hosting multilayers
more closely towards their isolated skyrmion/disordered lattice transition.
Finally, we note that the presence of a finite population of skyrmions for
$T^{{}^{\prime}}<T^{{}^{\prime}}_{c}$ is consistent with the nucleation-type
transition by which the skyrmion lattice is established [22, 23, 24]. Some of
these isolated skyrmions may also be stabilised by local variations in the
magnetic interactions due to disorder in our films.

##  4 Discussion

Our results are summarised in Fig. 3c, which highlights the transition between
isolated skyrmions and a disordered skyrmion lattice, determined by real-space
imaging of the spin textures as well as magnetotransport. The critical
parameter governing the transition $T_{c}(\kappa)$ is identified by three
separate methods: the maximum in $\rho^{THE}$, $MFM$ imaging and the
fluctuation-induced rise in $\Delta_{\rho/M}$ at $T_{c}^{{}^{\prime}}=110\pm
15$ K. All three data-sets display considerable overlap, indicating an active
role of critical spin fluctuations in determining the magnitude of $THE$ in
chiral magnetic films. The ensuing discrepancy of up to three orders of
magnitude between $n_{sk}(THE)$ and $n_{sk}(MFM)$ in the vicinity of the phase
transition may account for the widely-varying magnitudes of $\rho^{THE}$
values previously reported in technologically-relevant chiral magnetic films
[18, 9, 10, 29, 11, 12, 25]. Our material platform allows the $THE$ to be
tuned from $n_{sk}(THE)\approx n_{sk}(MFM)$, which is typical for skyrmion
crystals of B$20$ compounds [6, 7, 8] to $n_{sk}(THE)>n_{sk}(MFM)$ in dilute
skyrmion configurations characteristic of interfacial systems [18, 9, 10, 29,
11, 12, 25]. This acute sensitivity of $\rho^{THE}$ to the magnetic skyrmion
configuration indicates the crucial role of chiral spin fluctuations and opens
a promising avenue towards controllable topological spintronics.

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Acknowledgments: This project in Singapore was supported by National Research
Foundation (NRF) Singapore, under NRF Investigatorship programme (Ref. No.
NRF-NRFI2015-04), the Ministry of Education (MOE) Singapore, Academic Research
Fund (AcRF) Tier 2 (Ref. No. MOE2014-T2-1-050) and MOE AcRF Tier 3 (Ref. No.
MOE2018-T3-1-002). M.R. thanks Data Storage Institute Singapore for sample
growth facilities. Analytical theory work at Ioffe Institute was supported by
the Russian Science Foundation (Ref. No. 17-12-01265). K.S.D. and I.V.R. also
thank the Foundation for the Advancement of Theoretical Physics and
Mathematics “BASIS”. B.G., E.S. and I.M. acknowledge support by CRC/TRR 227 of
Deutsche Forschungsgemeinschaft (DFG).

Author contributions: M.R. and C.P. conceived the research and coordinated the
project. M.R. carried out the deposition of films, magnetotransport
measurements and data analysis. A.P.P. provided input to the analysis of the
transport data. A.Y. performed low temperature MFM experiments and analysed
the data with O.M.A. N.K.D. performed room temperature MFM experiments and
analysed the data. K.S.D. and I.V.R. provided theoretical insights into the
topological Hall effect in disordered skyrmion systems. B.G. and I.M. provided
theoretical insights into the Hall effects in multiband systems for which E.S.
provided the ab-initio calculations. M.R. and C.P. wrote the manuscript with
discussions and contributions from all authors.

Additional Information: Supplementary Information accompanies this paper.

Competing Interests: The authors declare no competing interests.

Data availability: The authors declare that the data supporting the findings
of this study are available within the paper, and its supplementary
information.

Correspondence: Correspondence and requests for materials should be addressed
to:
mraju<EMAIL_ADDRESS>
Methods
Film deposition: Thin film multilayers consisting of
Ta($30$)/Pt($100$)/[Ir($10$)/Fe($x$)/Co($y$)/Pt($10$)]N/Pt($20$) (numbers in
the parentheses are layer thickness in Å and $N$ refers to the number of times
the Ir/Fe/Co/Pt stack is repeated, $x,y$ are varied between $0-6$ Å and $5-10$
Å respectively) were deposited on thermally oxidised Si wafers by dc magnetron
sputtering at room temperature, using a Chiron ultra-high vacuum multi-source
system. The optimal growth rates for individual layers are Ta: $0.55$ Å/s, Pt:
$0.47$ Å/s, Ir: $0.12$ Å/s, Fe: $0.13$ Å/s, Co: $0.2$ Å/s. The base vacuum of
the sputter chamber is $1\times 10^{-8}$ Torr and an argon gas pressure of
$1.5\times 10^{-3}$ Torr is maintained during sputtering.

Electrical transport: The magnetotransport measurements were performed using a
custom-built variable temperature insert (VTI) housed in a high-field magnet,
complemented by a Quantum Design Physical Property Measurement System (PPMS).
Current densities as low as $10^{4}$ A/m2 at $33$ Hz were used to avoid
current driven perturbation of spin textures. Detailed analysis for the
extraction of the topological Hall resistivity ($\rho^{THE}(H)$) can be found
in our earlier works [9, 5]. The non-zero value of $\rho^{THE}$ at $H>H_{S}$
serves as a conservative estimate of the error bar in the extracted
$\rho^{THE}(H)$ which is $\leq 2$ n$\Omega$.cm. This includes the systematic
errors involved in data analysis.

Magnetization: Magnetization measurements were performed in the region
$T=5-300$ K and a magnetic field of $\pm 4$ T using superconducting quantum
interference device (SQUID) magnetometry, in a Quantum Design Magnetic
Property Measurement System (MPMS) (see section 2 of the supplementary
informaton). $M(H)$ loops were recorded with the applied field in-plane (hard
axis) and out of the film plane (easy axis). The saturation magnetization
$M_{S}$ and the difference in saturation fields $H_{S}$ along the easy and
hard axes were used to estimate the effective magnetic anisotropy $K_{eff}$
[5, 9].

Magnetic interactions: $M_{S}$ and $K_{eff}$ were determined from SQUID
magnetometry measurements. The detailed methods for estimation of exchange
stiffness $A$ and $DMI$ were reported in our previous work [5]. In this work,
we use scaling laws,
$\frac{A(T)}{A(T=5~{}K)}=\left[\frac{M_{S}{(T)}}{M_{S}{(T=5~{}K)}}\right]^{1.5}$
and
$\frac{D(T)}{D(T=5~{}K)}=\left[\frac{M_{S}{(T)}}{M_{S}{(T=5~{}K)}}\right]^{1.5}$
involving the $T$-dependent saturation magnetization $M_{S}(T)$ to estimate
$A(T)$, $D(T)$, and the resulting $\kappa(T)$. Details can be found in section
2 of the supplementary information.

Magnetic force microscopy ($MFM$): Room temperature $MFM$ experiments were
carried out using a Veeco Dimension $3100$ Scanning Probe Microscope. The
$MFM$ tips used (Nanosensors SSS-MFMR ) were $\approx 30$ nm in diameter, with
low coercivity ($\approx 12$ mT) and ultralow magnetic moment ($\approx 80$
emu/cc). Samples were initially saturated in out of plane ($OP$) direction
using fields up to $H=-0.5$ T. The measurements were performed in $OP$ fields
starting from $H=0$ and incrementally approaching $H=+H_{S}$, with a typical
tip height of $20$ nm. The field evolution of spin textures is presented in
section 1 of the supplementary information. Low $T$ ($5-200$ K) $MFM$ imaging
is carried out using a cryogenic frequency modulated $MFM$ system [9]. We used
two commercial probes by Team Nanotec, model ML3 ($35-40$ nm Co alloy
coating), with $f_{0}\approx 75$ kHz and $k_{0}\approx 1$ N/m. The sample was
first stabilized at a given temperature and then magnetized in the $OP$
direction by applying $H>H_{S}$. After saturation, $MFM$ images were acquired
at various field values as $H$ was swept from $-H_{S}$ to $+H_{S}$. The
details of the $MFM$ image analysis are reported in our earlier reports [5, 9,
30].

Calibration of applied magnetic field: Extensive field calibrations were
performed using a reference Hall sensor and a reference Palladium sample to
minimize the field offsets between different experimental setups. Sensor
details, calibration data together with analysis for the extraction of
$\rho^{THE}(H)$ are reported in our earlier works [5, 9]. An additional field-
calibration crosscheck is performed for our $MFM$ images by estimating the
magnetization from the image itself. Here the magnetization of the image is
estimated as the ratio of the effective area of the imaged surface polarized
along the applied field to the total area of the image. The magnetization of
the $MFM$ image is then located on $M(H)$ data recorded by SQUID magnetometer.

Figure 1:  Enhanced topological Hall resistivity ($\rho^{THE}$) for isolated
skyrmions compared with dense skyrmion arrays: (a)-(c) $\rho^{THE}(H)$ of
isolated skyrmion configurations observed in [Fe2/Co8]20, [Fe2/Co6]20 and
[Fe2/Co5]20. (d)-(f) $\rho^{THE}(H)$ of dense skyrmion arrays observed in
[Fe5/Co5]20, [Fe4/Co6]20, and [Fe4/Co4]20. (g)
$\Delta_{\rho/M}\equiv~{}n_{sk}(THE)/n_{sk}(MFM)$ estimated from transport and
imaging experiments shown in (a-f) and in section 1 of the supplementary
information. $n_{sk}(THE)$ is given by $\rho^{THE}/(P\cdot
R_{0}\cdot\Phi_{0})$ with $P=0.56$ and $n_{sk}(MFM)$ is estimated from $MFM$
images; details of our image analysis methods can be found in our earlier
reports [5, 9, 30]. The detailed evolution of magnetic textures with $H$ can
be found in section $1$ of the supplementary information. Scale bar in white
($MFM$ images corresponding to panels (a)-(e)) are $1$ $\mu$m and in black
($MFM$ images corresponding to panel (f)) $0.5$ $\mu$m. All the images are
acquired at a scan height of $20$ nm. The color scale represents the phase
shift in the $MFM$ signal due to the magnetic force acting on the tip. The
$\rho^{THE}(H)$ profiles show a small non-zero offset $\leq 2$
n$\Omega\cdot$cm above the saturation field of magnetization ($H>H_{s}$): this
is a systematic offset due to our fitting procedure (which is designed to
avoid any “false positive” $THE$ detection) and hence reflects the maximum
error in the magnitude of $\rho^{THE}(H)$ shown in Fig.$1$g.

Figure 2: Evolution of $\rho^{THE}$ across the phase transition between
isolated skyrmions and a disordered skyrmion lattice in $T-\kappa$ space. (a)
Phase diagram of skyrmion configurations in $T-\kappa$ space obtained by
varying $T$ and Fe/Co composition. The shaded area in blue is the transition
region from isolated skyrmions to a disordered skyrmion lattice revealed by
$MFM$ imaging. The spatial evolution of skyrmion configuration is analysed
using Delaunay triangulation: details of the analysis including the nearest-
neighbour (NN) coordination number and NN angular orientation can be found in
section $1$ of the supplementary information [5]. Details of our estimation of
the $T$-dependent $\kappa$ are presented in section $2$ of the supplementary
information. (b) Evolution of $\rho^{THE}$ with $(T,\kappa)$. Arrows with
dotted lines point to the specific samples in panel (a) for which $\rho^{THE}$
is studied as a function of $T$. (c) Evolution of $\rho^{THE}$ with $\kappa$
at a fixed $T$. Arrows with dotted lines point to the various samples in panel
(a) for which $\rho^{THE}(\kappa)$ is studied at a fixed $T$. (d) Schematic
$T-\kappa$ phase diagram illustrating the phase boundary $T_{c}(\kappa)$ and
the use of the effective temperature $T^{{}^{\prime}}=T\cdot\kappa$ to
describe the phase transition. The error bars on $\rho^{THE}$ in panels (b)
and (c) are $\leq 2$ n$\Omega\cdot$cm.

Figure 3: Insights into the critical behavior captured by the topological Hall
effect ($THE$). (a) Power-law behavior of
$\Delta_{\rho/M}\equiv~{}n_{sk}(THE)/n_{sk}(MFM)$ with effective temperature
$T^{{}^{\prime}}=T\cdot\kappa$, and a critical point at
$T_{c}^{{}^{\prime}}=110\pm 15~{}K$. Insets show the estimation of critical
exponents, $\gamma$ for $T^{{}^{\prime}}<T^{{}^{\prime}}_{c}$ and
$\gamma+2\beta$ for $T^{{}^{\prime}}>T^{{}^{\prime}}_{c}$. (b) Evolution of
$n_{sk}(MFM)$ with $T^{{}^{\prime}}$, showing a power-law behavior for
$T^{{}^{\prime}}>T^{{}^{\prime}}_{c}$ and the estimation of the critical
exponent $2\beta$. (c) Identification of the critical region $T_{c}(\kappa)$
using three different approaches, namely, the local maximum in $\rho^{THE}$
with $T$ and $\kappa$, the power-law rise in $\Delta_{\rho/M}$ and the
transition region (from isolated skyrmions to a disordered skyrmion lattice)
indicated by $MFM$. The error bar in the estimated exponents reflects the
error in the slope of the linear fits shown in the insets to panel $3$a and
$3$b. The same error bar is reflected through the shaded regions in the power-
law fits. Error bars on $\rho^{THE}_{max}$ in panel $3$c reflect the variation
in the peak position of $\rho^{THE}$ between Fig.$2$b and $2$c. The shaded
region for $T_{c}(\kappa)$ from $\Delta_{\rho/M}$ in panel $3$c reflects a
conservative estimate of $\pm 35\%$ variation in $\kappa$ resulting from an
upper bound in the estimation of the exchange ($A$) and $DMI$ (D) constants as
well as $K_{eff}$ from magnetization measurements [5].