1 Topological Phase Transitions and Berry -Phase Hysteresis in Exchange -Coupled Nanomagnets

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Topological Phase Transitions and Berry-Phase Hysteresis in

Exchange-Coupled Nanomagnets

Ahsan Ullah,* Xin Li, Yunlong Jin, Rabindra Pahari, Lanping Yue, Xiaoshan Xu, Balamurugan

Balasubramanian, David J. Sellmyer, ‡ and Ralph Skomski‡

Department of Physics & Astronomy and Nebraska Center for Materials and Nanoscience, University of

Nebraska, Lincoln, NE 68588

*E-mail: aullah@huskers.unl.edu

‡ Deceased

Abstract

Topological phase in magnetic materials yields a quantized contribution to the Hall effect

known as the topological Hall effect (THE), which is often caused by skyrmions, with each

skyrmion creating a magnetic flux quantum ±h/e. The control and understanding of topological

properties in nanostructured materials is the subject of immense interest for both fundamental

science and technological applications, especially in spintronics. In this article, the electron-

transport properties and spin structure of exchange-coupled Co nanoparticles with an average

particle size of 13.7 nm are studied experimentally and theoretically. Magnetic and Hall effect

measurements identify topological phase transitions in the exchange-coupled Co nanoparticles and

discover a qualitatively new type of hysteresis in the topological Hall effect, namely Berry-phase

hysteresis. Micromagnetic simulations reveal the origin of the topological Hall effect, namely the

chiral domains with domain-wall chirality quantified by an integer skyrmion number. These spin

structures are different from the skyrmions formed due to Dzyaloshinskii-Moriya interactions in

B20 crystals and multilayered thin films and caused by cooperative magnetization reversal in the

exchange-coupled Co nanoparticles. An analytical model is developed to explain the underlying

physics of the Berry-phase hysteresis, which is strikingly different from the iconic magnetic

hysteresis and constitutes one aspect of 21st century reshaping of our view on nature at the

borderline of physics, chemistry, mathematics, and materials science.

I. INTRODUCTION

Topological phase transitions (TPTs) permeate areas such as superfluid and superconductors

[1, 2], basic quantum mechanics [3, 4], fractional quantum-Hall effect [5], and topological

insulators [6] and therefore have gained significant interest in both science and technology. TPTs

are very different from ordinary Landau-type phase transitions [4, 7-9]. Rather than involving

symmetry breaking and order-parameter changes, they are characterized by changes in topological

numbers. For example, coffee cups have one hole, located in the handle, and are therefore

characterized by the topological number (Euler genus) g = 1. A flat pancake has no holes (g = 0)

so that the piercing of a number of holes into a pancake is a trivial example of a TPT.

Topological phase transition is in contrast to magnetic hysteresis, which is based on a phase

transition between an ordered low-temperature and a disordered high-temperature [1-6, 10, 11,

12]. An intriguing aspect of magnetic hysteresis is its relation to magnetic phase transitions.

Figures 1(a-b) compare the atomic-scale origin of ferromagnetism with the nanoscale or 'micro-

magnetic' origin of hysteresis. When a ferromagnet is cooled below the Curie temperature Tc, it

develops a spontaneous magnetization Ms (a). This process is a Landau-type phase transition,

defined as a singular change of a local order parameter (M) due to spontaneous symmetry breaking.

The ordered phase has the character of a k = 0 Goldstone mode whose magnetization can point in

any direction (a). This degeneracy is removed by symmetry-violating terms in the Hamiltonian,

such as magnetic anisotropy [13].

Magnetic hysteresis, Fig. 1(b), is on top of the Landau transition (a). When a magnetic

material is subjected to an external field H, then its magnetization M(H) is generally not single-

valued but splits into ascending and descending branches. A well-known example is small

nanoparticles of volume V and anisotropy energy K1V sin2



in a magnetic field H = Hz. The color

coding throughout this article is Mz(r) = +Ms (red), Mz(r) = –Ms (blue) and intermediate (yellow).

For positive K1,



= 0 (red) and



= 180º (blue) are energetically favorable but separated by an

energy barrier K1V (



= 90º). This energy barrier needs some external field to be overcome and is

therefore the reason for the hysteresis.

Fig. 1. Phase transitions: (a) Curie transition (magnetic Landau transition), (b) magnetic hysteresis, (c)

Lifshitz transition in metal, and (d-i) topological phase transition in a magnetic thin film with

perpendicular anisotropy. In (c), the gray areas denote the k-space region occupied by electrons at

the Fermi level. In (d-i), red and blue regions indicate positive (↑) and negative (↓) magnetizations

with respect to the film plane. Topological phase transitions are characterized by topological

numbers Q. The topological protection in the micromagnetic case is experimentally established, for

example through the "blowing" of skyrmions [14 - 16].

While topology has a long history, the idea of topological phase transition goes back to the

Lifshitz transition [4, 7]. Figure 1(c) shows the k-space meaning of the Lifshitz transition in metals.

Itinerant electrons fill the available electron states until the Fermi level is reached. The occupancy

at the Fermi level (gray) depends on the number of electrons, and there are several scenarios that

change the topological quantum number Q, such as external mechanical pressure and chemical

addition of electrons. Each Fermi-surface region (gray) yields an integer contribution to Q,

irrespective of the size and shape of the pocket. Topological concepts are now applied to many

areas of physics, from skyrmions [7, 8, 13, 15, 16, 17- 20] to topological insulators and other

quantum materials [21-30], all of them fascinating research topics in their own rights.

Figures 1(d-i) show the magnetic analog of the Lifshitz transition in a thin film. The field H

is perpendicular to the film and affects ↑ (red) and ↓ (blue) regions separated by domain walls

(yellow). The underlying micromagnetism is very similar to that of magnetic skyrmions [31, 32,

33-36] and to XY-model transitions [8]. When an electrical current flows through the film, then

the spins of the conduction electrons exchange-interact with the local magnetization M(r) and

become, in general, noncoplanar noncollinear. This noncollinearity creates a Berry curvature [5],

an emergent magnetic field, and subsequently a Hall-effect contribution known as the topological

Hall effect (THE) [5, 33]. These effects are proportional to the skyrmion density [31, 33, 37, 38]



= 1

4 m · 









∂m

∂x  ∂m

∂y (1)

where m = M(r)/Ms is the normalized magnetization and the x-y-plane is the film plane. The

emergent magnetic flux that corresponds to the THE is equal to Q h/e, where Q = ∫



dxdy is the

skyrmion number and h/e is the magnetic flux quantum. In granular thin films, such as the one

considered in the present paper, there are also nonzero derivatives ∂m/∂z. By virtue of

measurement geometry, ∂m/∂z does not contribute to the THE [33], but it is one source of noise

[39]. The skyrmion density is nonzero for spins m(r) that are both noncollinear and noncoplanar,

and Eq. (1) is actually a continuum version of the triple product or spin chirality



s = mi·(mj × mk),

where mi = m(Ri) describes the atomic spins that cause the conduction electrons to develop their

Berry phase.

Much of the fascination with topological phase transitions originate from the great simplicity

of the mathematics conveyed by Eq. (1), which is summarized in Supplement A [45]. In

skyrmionic structures such as those of Figs. 1(d-i), the spins inside the red and blue regions are

parallel (m = ±ez), so that ∂m/∂x, ∂m/∂y, and



are zero. The integral over



therefore reduces to

an integral over the yellow domain-boundary regions in Figs. 1(d-i). It can be shown that

Q = 12𝜋

⁄ ∮



 dl (2)

where



is the curvature of the region's yellow boundary and the integral in Eq. (7) has the value

2π [40]. This integral is equal to ±1 for any area enclosed by a single yellow boundary [40]. While

Eq. (2) is valid for arbitrary domain shapes, it requires domain walls free of internal singularities

such as Bloch lines [33, 41]. Mathematically, M(r) is a fiber bundle [42] on the base space r and

therefore locally flat but globally nontrivial [43]. In fact, Fig. 1(d-i) provides a simple example of

a bulk-boundary equivalence, a feature that forms a cornerstone of topological physics [22]. The

sign of Q depends on the vorticity [44] of the spin structure, that is, on whether the region enclosed

by the yellow boundary is red (Q = +1) or blue (Q = -1). In particular, Q is independent of the

clockwise or counterclockwise chirality of Bloch walls in the yellow region (Figs. S1, S2 in the

Supplemental Material [45]).

As discussed above, TPTs do not increase or decrease order parameters but consist of

changes in topological numbers. This leads to the question of whether such transitions lead to

hysteretic features beyond magnetic hysteresis. This hysteresis was not recognized in earlier

research, because available systems had micron-size rather than nanoscale feature sizes, which

makes it very difficult to detect the Berry curvature in Hall-effect measurements. In this paper, we

have fabricated exchange-coupled Co nanoparticle films having a much smaller average size of

about 13.7 nm and show topological phase transitions and berry phase hysteresis using

experiments. The experimental results and the underlying physics are also explained using

micromagnetic simulations and an analytical model.

II. EXPERIMENTAL AND COMPUTATIONAL METHODS

An inert gas condensation-type cluster-deposition method, schematically shown in Fig. S3a in

Supplement B, as described elsewhere [45, 46]. First, Co nanoparticles were produced by a DC

magnetron sputtering using a mixture of argon and helium with a power of 200 W in a gas-

aggregation chamber. After the formation, the nanoparticles were extracted towards the deposition

chamber and deposited as a dense film on a Si (100) substrate having a Hall bar. The base pressure

of the gas-aggregation chamber was 6  10-8 Torr and the respective Ar and He flow rates were

maintained at 400 and 100 SCCM (standard cubic centimeter per minute), respectively. The

pressure in the cluster-formation chamber during the deposition was 0.7 Torr.

The Co nanoparticles were deposited with a low coverage density on a thin carbon film

supported by copper grids for transmission-electron-microscopy measurements using an FEI

Technai Osiris STEM. For magnetic and electron-transport measurements, the cluster-deposited

nanoparticles were deposited for an extended time as a dense film as discussed in our previous

works [46, 47]. The above measurements were performed using a superconducting quantum

interference device (SQUID) and physical property measurement system (PPMS), respectively. A

schematic of a dense nanoparticle film is shown in Fig S3(d) [45], and therefore they are exchange

coupled and conducting. The thickness of the Co nanoparticle film is about 270 nm. The

conduction channels for the Hall contacts were fabricated before depositing the Co nanoparticles,

as described in Ref. 46. To prevent oxidation upon exposure to air, the Co nanoparticle film was

capped with a SiO2 layer of about 10 nm thickness immediately after deposition, using a radio-

frequency magnetron sputtering. The SiO2 cap layer is thinner (about 10 nm) as compared to the

Co nanoparticle film of about 270 nm thickness and is also diamagnetic. Therefore, the film-SiO2

interface is not expected to affect the magnetic and transport properties of the Co nanoparticle

films. The particles have an average size of 13.7 nm with a narrow size distribution, see Fig S3 (b,

c) and crystallize in the hcp structure, as shown in Fig. S4 [45]. A commercial AFM/MFM (Atto

AFM/MFM Ixs; Attocube Systems) was used to map the topography and magnetic images at 200K. During

the measurement, the MFM was performed in constant height mode (single pass) with PPP-MFMR tip from

NANOSENSORS. The lift height is 250 nm and the scan speed is 5μm/s.

To numerically model the magnetic and Berry-phase hysteresis, we have performed micro-

magnetic simulations using ubermag supported by OOMMF [48, 49]. We have numerically

extracted skyrmion number Q from the spin structure. A densely packed film of 1000 Co

nanoparticles has been considered. The Co particles have sizes of about 13.7 nm and the total size

of the simulated system, shown in Fig. S10, is 240 nm  240 nm  60 nm. We have used a compu-

tational cell size of 1.8 nm, which is well below the exchange length lex [12], coherence radius

5.099 lex of Co (10 nm) [12], and the domain-wall width (14 nm) of Co [12], and the current particle

size. This cell size ensures a reasonable real-space resolution of M(r).

Aside from the numerical cell size, our continuum approach is valid on length scales much

larger than the Co-Co interatomic distance of 0.25 nm. This makes it possible to consider the thin

film as a fiber bundle M(r) with the base space r, allowing us to define quantities such as the

boundary curvature



. A refined atomistic analysis, not considered here, would yield corrections

due to the discrete nature of the atoms at the particle's surfaces and near contact points, see e.g.

Sect. 4.5 in Ref. 12. In particular, the crystal structure of Co is inversion-symmetric, so there are

no bulk Dzyaloshinskii-Moriya interactions (DMI). Corrections to the magnetization angles

caused by the DMI of surface atoms are likely but probably very small. Note that the Co nano-

particles exhibit nanoscale inversion symmetry, as contrasted, for example, to the magnetism of

Co/Pt bilayers [27].

The cluster-deposition method yields isotropic nanoparticles with random grain orientation

and therefore a random orientation of the easy magnetization axes n of the hcp Co particles,

obeying <nx> = <ny> = <nz> = 0 and <n2> = 1. This randomness in simulations, clearly visible in

Fig. S10(b), was implemented by using python np.random.uniform [45, 49]. The particles touch

each other as shown in Fig S3(d), so that the exchange stiffness A near the contact points is the

same as in bulk Co.

Temperature-dependent micromagnetic effects are included in the lowest order, that is, by

considering the intrinsic materials parameters Ms, K1, and A as temperature-dependent. This

approach accounts for the atomic spin disorder outlined in Fig. 1(a). Other finite-temperature

corrections, caused for example by magnetic viscosity [12], have been ignored. In our simulations,

we have taken values of Ms = 1300 kA/m, K1 = 0.58 MJ/m3, and A = 10.3 pJ/m [12].

III. RESULTS AND DISCUSSION

Transmission electron microscope and the corresponding particle-size histogram show an

average particle size of 13.7 nm with a standard deviation



/d  0.15 (Figs. S3b and S3c) for the

Co nanoparticles. We have conducted magnetic, electron-transport, and Hall-effect measurements

at temperatures from 10 K to 300 K for the dense Co nanoparticle films as schematically shown in

Fig. S 3(d). The magnetic hysteresis loops are shown in Fig. S5, and the measured coercivities are

0.18 T at 10 K and 0.04 T at 300 K.

Figure 2 compares the experimental data on a Co nanoparticle thin film (a) with numerical

predictions (b). The THE was extracted from Hall-effect measurements, Fig. S6, as explained in

Supplement B. We see that the Berry-phase hysteresis loops (colored) look qualitatively different

from the magnetic hysteresis loops (black) and that they are much broader than the magnetic ones.

Figure 2(a-b) also shows that Berry-phase hysteresis loops contain more features than magnetic

hysteresis loops. There are both mathematical and physical explanations for these differences.

Mathematically, Eq. (1) contains derivatives, which amounts to a numerical amplification of

details. Physically, Berry-phase hysteresis loops exhibit a more complicated dependence on the

spin structure, because



is more complicated than m.

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1TopologicalPhaseTransitionsandBerry-PhaseHysteresisinExchange-CoupledNanomagnetsAhsanUllah,*XinLi,YunlongJin,RabindraPahari,LanpingYue,XiaoshanXu,BalamuruganBalasubramanian,DavidJ.Sellmyer,‡andRalphSkomski‡DepartmentofPhysics&AstronomyandNebraskaCenterforMaterialsandNanoscience,UniversityofNebraska...

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