Supplementary Information:Robust skyrmion-bubble textures in SrRuO3 thin filmsstabilized by magnetic anisotropy

P. Zhang,1, a) A. Das,1 E. Barts,1 M. Azhar,1 L. Si,2 K. Held,2 M. Mostovoy,1 and T.Banerjee1, b)1)University of Groningen, Zernike Institute for Advanced Materials, 9747 AGGroningen, The Netherlands2)Institut fur Festkorperphysik, TU Wien, Wiedner Hauptstraße 8-10, 1040 Vienna,Austria

(Dated: 8 June 2020)

I. GROWTH AND CHARACTERIZATION OF SRO FILMS

Sample Substrate T (C) Thickness (nm)A 600 8.9 ±0.09B 600 8.6 ±0.16C 650 7.5 ±0.12

TABLE S1. SRO thin film parameters: the thickness was fitted from XRR.

Epitaxial SRO films were deposited on terminated and annealed (100) SrTiO3 (STO)single-crystal substrates by pulsed laser deposition (PLD) using a 248 nm-wavelength KrFexcimer laser. Reflection high-energy electron diffraction (RHEED) was used to monitorthe growth of SRO thin films in-situ.

The laser fluence was 1.35 J/cm2 with a pulse repetition rate of 1 Hz. During the deposi-tion process, the target-to-substrate distance was kept at 58 mm, the substrate temperatureswere varied from 600 to 650 C and the oxygen partial pressure was controlled to be around0.13 mbar. Reflection high-energy electron diffraction (RHEED) was used to monitor thegrowth of SRO thin films in-situ.

Post-deposition, the films were cooled to room temperature at the rate of 10 C perminute in an oxygen pressure of 100 mbar. The growth parameters of the different filmsand their thicknesses as measured by x-ray reflectivity (XRR) are tabulated in the table1. The single crystalline STO substrates used in this work have miscut angles α varyingbetween 0.05 to 0.1 and were chemically treated using the standard BHF protocol fol-lowed by annealing in oxygen at 960 C to achieve a uniform TiO2 terminating plane1. Thestructural characterization of SRO films (shown in Fig. S1) were studied by x-ray diffrac-tion (XRD) using a Panalytical X’pert diffractometer whereas the surface morphology androughness were determined by an atomic force microscopy (AFM) in tapping mode. TheAFM topology before and after the thin film deposition (Fig. S2) reveals the presence ofboth TiO2 and SrO surface termination for all substrates used. Such double terminatedsubstrates were found to result in local differences in structural and electronic propertiesat the different terminating sites2. Resistivity studies on unpatterned films (A and B) weredone in standard four-terminal van der Pauw geometry for temperatures between 5 K to300 K and shown in Fig. S3a. The differences in the temperature dependence of ρxx forsuch thick SRO films (A and B), in spite of similar deposition conditions and thickness,are remarkable, and underscore the role of the local differences in substrate termination to

a)Electronic mail: p.zhang@rug.nlb)Electronic mail: t.banerjee@rug.nl

2

42 44 46 48 501E0

1E1

1E2

1E3

1E4

1E5

1E6

1E7

SR

O (

00

2)

ST

O (

00

2)

Inte

nsi

ty (

a.u

.)

2 Theta (Deg.)

A

B

FIG. S1. X-ray diffraction patterns of epitaxial SRO thin films deposited on STO (100) substrates.

electronic transport and ferromagnetic transition temperature (Tc=115 K for film A and120 K for film B). The magnetization of the thin films was studied at variable temperaturesby sweeping the applied magnetic field along the in-plane and out-of-plane directions usinga Quantum Design superconducting quantum interference device (SQUID). One such mea-surement, for the as deposited film B, shown in Fig. 1a (main text). A nonmonotonousdependence of σAHE is found for both samples with temperature upto Tc. Film A exhibits asign reversal with temperature and magnetization, whereas Film B shows no such reversal.

II. SCALING OF THE ANOMALOUS HALL RESISTIVITY IN SRO FILMS

The transverse resistivity (ρxy) in the Hall transport geometry is given by :

ρxy = RoB⊥ +RsM⊥(B⊥) + ρTHE (S1)

where the first term is due to the ordinary Hall effect (OHE). The second and third termsare due to the anomalous Hall effect (AHE) and the topological Hall effect (THE) re-spectively. The figures in the main text (Figs. 1d and 2b,c,d), displaying ρxy - ρOHE intheir y-axis labels, refer to the AHE and THE contributions after subtraction of the OHEbackground.Anomalous Hall effect provides the link between the spin dependent band structures, crys-talline symmetry and magnetocrystalline anisotropy in SRO films with the Berry phasemechanism in crystal momentum space3–5. Fig. 1b and Fig. S3b, shows the scaling ofthe anomalous hall resistivity (conductivity) for films A and B with temperature. Thetwo films show different scaling behavior with temperature and magnetization. Film A

3

(a) (b) (c)

FIG. S2. AFM images of the surface morphology for STO substrate (top panel) and SRO film(bottom panel). (a) Film A, (b) Film B, (c) Film C.

shows a sign change in ρAHE with temperature whereas no such sign change is observedin film B. In the intrinsic regime, σAHE is written as -ρAHE/ρ

2xx. The resistivity and the

conductivity vanishes around Tc where the magnetization of the films are zero. ρAHE isobtained from the zero-field resistivity after the subtraction of the ordinary hall background.

A sign change in the anomalous Hall resistivity (ρAHE) is commonly observed in SROfilms and attributed to the cross-over between the intrinsic and extrinsic mechanisms ofanomalous Hall effect6. Increase in temperature, increases the resistivity (ρxx) of the filmsand the AHE is dominated by side jump mechanism. We observe in Fig. 1a (main text) thatfilm A with a larger resistivity than film B shows a sign change in ρAHE , in contradictionto the side jump mechanism that usually dominates at a larger value of ρxx. On the otherhand, σAHE depends on the electronic band structure and the avoided band crossings thatact as magnetic monopoles in the crystal momentum space. A larger conductivity at highermagnetization in both films indicate a plausible Berry phase connection to the Hall con-ductivity in the intrinsic regime. This indicates that the conductivity in film B also scaleswith magnetization despite of no sign reversal in the conductivity. From the magnetizationmeasurements with field in film B (Fig. 1a in main text), we observe a difference in thesaturation magnetization for magnetic field applied along the in and out-of-plane direction.This indicates the existence and competition between the multiaxial anisotropies giving riseto such a difference and are different for films A and B. All Hall transport measurementsare performed with the magnetic field applied in the direction normal to the plane of thetransport, thus such differences in the multiaxial anisotropies between the two films willbe reflected in the temperature variation of ρAHE and σAHE , as seen in Fig. 1b in maintext and Fig. S3b. As discussed above, and shown in Fig. S2, the SRO films are grown ona mixed terminated surface of STO (001), thus σAHE is sensitive to local changes in the

4

0 100 200 300

200

300

400

500A

ρxx

(μΩ

cm

)

T (K)

B

(a) (b)

σAH

E (S

/cm

)

1.2 0.9 0.6 0.3

-15

0

15

M (µB / Ru)

FIG. S3. (a) Temperature dependence of the resistivity of SRO thin films deposited on nominallydifferent substrate terminations but of similar thickness. (b) Scaling of Anomalous Hall conductiv-ity with magnetization of the samples A (red symbols) and B (black symbols). The conductivitysign as the magnetization decreases (on approaching Tc) for sample A, similar to ρAHE (Fig. 1bin main text)

band structure and electronic properties resulting in such a contrast.Anomalous Hall resistivity (ρAHE) is obtained by subtracting the ordinary Hall effect

-4 -2 0 2 4

-0.5

0.0

0.5 ρB (+)

ρ AHE (

µΩ c

m)

B⊥ (T)

ρA (-)

10 K

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3 ρA (+)

ρ AHE (

µΩ c

m)

B⊥ (T)

ρB (-)

90 K

FIG. S4. Anomalous Hall resistivity ρAHE with out of plane magnetic field sweeps in film A.The arrows indicate the trace and retrace directions, indicating the sign change at the two differenttemperatures. The solid blue line passing through origin is the ordinary Hall background subtractedfrom the ρxy (raw data).

background (linear). The background is subtracted with the line passing through origin asindicated with the blue solid line shown in Fig. S4. Next, the difference in the value of theresistivity at zero field is extracted as ρAHE . As indicated in the figure, at low temperatures(10 K) for film A, the value of the resistivity while tracing the field from negative to positiveis positive (ρA in Fig. S3), whereas on retrace the value is negative (ρB). The differenceis then written as ρAHE = ρA − ρB . This changes sign at 90 K as shown in Fig. S4 rightpanelThe Anomalous Hall coefficient, Rs is written as the sum of the contributions due to intrin-sic mechanism, side jump and skew scattering, that scales with the sheet resistivity (ρxx)

5

of the SRO film6 :

Rs = Aρxx +B

∆2 + (~/τ)2ρxx

2 + Cρxx2 (S2)

the linear term of Rs with ρxx is due to the skew scattering, the quadratic terms are dueto both intrinsic and side jump contribution. ∆ is associated with the band structure andspin splitting which is taken as 0.2 eV, τ is the scattering time which is taken as 6.6 x 10−14

s7 and C is the characteristic side jump length which is taken as 10 A8. The coefficient Bdeals with the sign change in Rs as shown in Fig.S5b. Rs is obtained from ρAHE/M⊥ andis plotted with ρxx in Fig. S5b. The solid red line is the fit to Eq. S2.Fig. S5a shows the magnetization of the films A and B with temperature. The solid blue

0 50 100 150

0.0

0.3

0.6

0.9

1.2 A B

260 280 300 320

-0.05

0.00

0.05

0.10Sample A

150 200 250 300-1.0

-0.5

0.0

0.5

1.0 A B C

ρxx (µΩ cm)

ρxx (µΩ cm)

ρ AH

E(µΩ

cm)

Rs

(µΩ

cm/ T

)

T (K)

M (µ

B/R

u)

(a) (b)

(c)

FIG. S5. (a) M-T variation for films A and B. The magnetization is studied using SQUID mea-surements with field cooling (3 T). The solid blue lines are the critical exponent fit to the M-Tcurves. (b) Variation of Anomalous Hall coefficient Rs with the longitudinal resistivity ρxx. (c)Scaling of ρAHE with ρxx for films A, B and C.

line indicates the fit to the behavior using power law when the Tc is fixed at 120 K. Thisfit helps in normalizing the M⊥ to extract the Rs. Since the side jump term (C) cannotexceed beyond 10 A, it has a smaller dependence with increasing ρxx as can be seen in Eq.S2. A reasonable fit is obtained with Eq. S1 (solid blue line) with B being negative andone order higher in magnitude than C, indicating a dominant intrinsic mechanism over sidejump. However, from both the fits, it is clear that there is a significant contribution of skewscattering in the SRO films, although the conductivity σxx (1/ρxx), lies in the moderatemetallic regime, i.e ≤ 104 S/cm. The overall ρAHE variation in films A, B and C is plottedwith ρxx as shown in Fig. S5c. Films B and C do not follow a scaling law of ρAHE withρxx while on the other hand, film B scales with magnetization as shown in Fig. S3b. ρAHE

in film B primarily originates from the intrinsic mechanism.

6

-4 -2 0 2 4

-3

-2

-1

0

1

2

3

OOP

IP

M (

B /

Ru)

0H (T)

C

10 K

FIG. S6. Magnetic hysteresis loops in Film C, measured at 10 K. OOP and IP correspond toout-of-plane and in-plane directions of the applied magnetic field, respectively.

III. MAGNETIZATION STUDIES ON FILM C

Film C was deposited by PLD at a laser fluence of 1.5 J/cm2 and substrate temperatureof 650C. The other growth parameters were kept the same as for Films A and B. The AFMimages before and after deposition are shown in Fig. S2 c. The magnetization dependencewith field for Film C, at 10 K, is shown in Fig. S6 for both the in and out of plane directions.The saturation magnetization (Ms) is around 3 µB/Ru in the OOP direction for Film C.

IV. NUMERICAL SIMULATIONS

The energy of magnetostatic interactions stabilizing inhomogeneous magnetic states isgiven by9,

Ems = 2πhS∑q

[f(qh)|Mz(q)|2 + (1− f(qh))|q · M(q)|2

], (S3)

where S is the film area, q = qq , f(x) = 1−e−x

x and M(q) is the Fourier transformation of

the magnetization,

M(q) =1

S

∫d2x e−iq·x M(x). (S4)

The energy of the magnetic bubble array phase is found by numerical minimization ofthe total energy Eq.(1) with respect to the Fourier harmonics of the magnetization, M(q),where the wave vector q belongs to the reciprocal lattice of the array with the basis vectors,

b∗1 = 2√3a

(1, 0) and b∗2 = 2√3a

(− 12 ,√32 ), a being the lattice constant of the bubble crystal, an

additional parameter with respect to which the energy is minimized. We limit the numberof the Fourier harmonics by q ≤ 4b∗1 (61 harmonics in total, see Fig. S7 a) and do checks forlarger wave vectors sets to ensure that the contribution of the neglected harmonics is small.

7

A similar procedure is applied to study the stripe domain state (Fig. S7 b), in which caseall wave vectors are collinear. The number of harmonics is large enough to simulate spintextures of all competing phases as well as their deformations in a tilted magnetic field.

We also studied effect of the interfacial Dzyaloshinskii-Moriya interaction,

EDM = Dh2∫d2x

[(M ·∇)Mz −Mz(∇ ·M)

]. (S5)

Although relatively weak, it does make skyrmions more resilient to high magnetic fields (seeFigs. S8 c,d), as they acquire a Neel component (Fig. S7 c,d)10–13. In addition, bubbles withskyrmion topology become more stable against the transition into non-topological bubblesin tilted magnetic fields, as shown in Figs. S9 c,d.

To obtain phase diagrams shown in Fig. 4 of the main text, we performed simulationsfor h = 8.9 nm, the saturation magnetization Ms = 3µB per Ru ion (see Fig. 1b, S6) andthe spin stiffness constant, A = VRuM

2s c = 62 meV·A2, taken from neutron scattering data

on bulk SRO14 (VRu is the volume per Ru ion). Distances are measured in units of h andenergy is given in units of VRuM

2s =0.0080 meV. The dimensionless DM constant, D = 1,

used to calculate the spin configurations shown in Fig. S7 c,d and the phase diagramsFig. S8 c,d correspond to the Dzyaloshinskii-Moriya interaction Dnn = 0.8 meV betweennearest-neighbor spins of the uppermost layer, which has to be compared with the nearest-neighbor Heisenberg exchange constant Jnn = 6.2 meV in a discrete spin model that givesthe experimental value of the stiffness constant A.

b∗2

b∗1

-5 0 5

-5

0

5

-1

-0.5

0

0.5

1

-5 0 5

-5

0

5

-1

-0.5

0

0.5

1

-5 0 5

-5

0

5

-1

-0.5

0

0.5

1

Y

Y

X X

(a) (b)

(c) (d)

mz mz

mz

mz mz

FIG. S7. (a) Skyrmion crystal in reciprocal space (large zero harmonic is excluded for clarity).The green dot area is proportional to the magnitude of the Fourier harmonic of the magnetization.Black line encircles the subspace of 61 wave vectors. (b) The stripe domain state. Effect of theinterfacial Dzyaloshinskii-Moriya interaction, D = 1, on (c) skyrmions and (d) domain walls inthe stripe domain state. Both acquire a Neel component. mz is color-coded and the in-planecomponents of the unit vector m are shown with arrows. Distances are given in units of the filmthickness, h.

8

⟨mz⟩

⟨mz⟩

⟨mz⟩

⟨mz⟩

(a) (b)

(c) (d)

USkX

SD

USkX

SD

USkX

SD

U

SkX

SD

FIG. S8. Magnetic field, H, vs quality factor, Q = K14π

, phase diagrams, which include the skyrmioncrystal (SkX), stripe domain (SD) and uniform (U) states. Red color intensity indicates mz in theuniform state with a tilted magnetization. The state with the magnetization normal to the film isshown with white color. These diagrams are calculated for (a) R = K2

4π= 0 and (b) R = 0.4. The

interfacial DM interaction with D = 1 widens the regions occupied by the SkX and SD phases,calculated for (c) R = 0 and (d) R = 0.4.

V. DENSITY FUNCTIONAL THEORY

To study the SrRuO3:SrTiO3 interface system with atomic mixing layer, we construct aSrRuO3:SrTiO3 superlattice system with 4 SrRuO3 (SRO) unit cell layers and 3 SrTiO3

(STO) unit cell layers, between which, a mixing layers with 50% SrRuO3 and 50% SrTiO3

layer is constructed by arranging Ru and Ti atoms in a G-type ordering (see Fig. S10).The atomic positions and the lattice constant along Z-direction are relaxed to reach theground state with lowest energy while the in-plane lattice constant is fixed at 3.905 Aforthe experimental value of SrTiO3. The structural relaxation is carried out by using Viennaab-initio simulation package (VASP)15,16 with GGA-PBE functional17. The cut-off energyis set as 500 eV and Brillouin zone is sampled with a 7×7×1 k-mesh. To better describethe correlation effects of transition metal d electrons, we use the Dudarev’s rotationallyinvariant GGA+U approach18 with U=3.0 eV for Ru-4d and 5.0 eV for Ti-3d. To simulateall possible magnetic orderings, a 2×2 in-plane superlattice is used for our DFT calculations(Fig. S10).

To figure out the magnetic ground state, we study various magnetic configurations(Fig. S11 and Table S2). In our experimental M-H curves, under strong H field an averagemagnetic moment ∼2.5µB/Ru is confirmed (Fig.1a) of main text), which motivates us toconsider the possibility of a high-spin state in the SRO:STO intermixing system. We listall the possible magnetic ground states: ferromagnetism (FM) with both high-spin (HS)and low-spin states (LS), and antiferromagnetic states (AFM-1 to AFM-10, and AFM-ST1and AFM-ST2 as shown in Fig. S11, here ST means stripe-like AFM domain).

We find that the ground state is the AFM-ST1 configuration: it is about 11.22 meV perintermixing Ru energetically lower than that of the bulk-like state: FM-LS (∼2.0µB/Ru).The second stable phase is the AFM-3 state (2.17 meV/Ru higher than AFM-ST1); the

9

-180 -90 0 90 1800

0.2

0.4

0.6

0.8

1

-180 -90 0 90 1800

0.2

0.4

0.6

0.8

1

-180 -90 0 90 1800

0.2

0.4

0.6

0.8

ea

θm

ea

θm

ea

θm

U

SkX NTB

SD

U

SkXNTB

SD

USkX

NTB

SD

U

SkX NTB

SD

(a) (b)

(c) (d)

FIG. S9. Phase diagrams in a tilted magnetic field, θ being the tilt angle, for (a) Q = 0.87 andR = 0, (b) Q = 0.65 and R = 0.40 and (c) Q = 0.65 and R = 0.60, and (d) Q = 0.65, R = 0.60 andD = 1. In addition to the skyrmion crystal (SkX, blue), stripe domain (SD, green) and uniform(U, white) states, these diagrams include an array of non-topological bubbles (NTB, yellow). Theinsets show the dependence of the dimensionless anisotropy energy density, ea, in the uniform stateon the magnetization tilt angle, θm, for the corresponding parameter sets.

TABLE S2. Average magnetic moment (per Ru) and total energy (per intermixing Ru) of thevarious magnetic configurations which have been considered in our calculations. For the results oftotal energy, the energy of the ground state AFM-ST1 has been set as 0.00 meV. The magneticmoment is calculated as the total moment/number of Ru atoms in the supercell. FM-HS indicatesthe ferromagnetic state with high-spin state, and FM-LS indicates the ferromagnetic state withlow-spin state.

FM-HS FM-LS AFM-1 AFM-2 AFM-3 AFM-4 AFM-5E (mev/Ru) 38.22 11.12 85.44 251.98 2.17 84.95 10.16Moment (µB/Ru) 2.50 2.00 1.56 1.78 1.78 1.56 1.78

AFM-6 AFM-7 AFM-8 AFM-9 AFM-10 AFM-ST1 AFM-ST2E (mev/Ru) 105.70 29.89 102.09 20.44 78.95 0.00 41.36Moment (µB/Ru) 1.61 1.34 1.14 1.56 1.14 1.55 1.14

third stable phase is the AFM-5 state, whose energy is 10.16 meV/Ru higher than AFM-ST1 state. Thereafter the next stable phase is the bulk-like FM-LS state, its energy is11.12 meV/Ru higher than AFM-ST1 state. The total energy of other magnetic states areremarkably higher than these 4 states, thus we exclude the possibilities of their existencein the magnetic transitions. Surprisingly, we also found a metastable FM state, in whicha higher average magnetic moment ∼ 2.5µB/Ru is converged in DFT+U calculation. Thisvalue of the magnetic moment is remarkably higher than the low-spin limitation of 4d4

configuration in Ru4+: 2.0µB/Ru, and also higher than the experimental value of 1.1-1.6 µB/Ru19–22 of bulk SRO, DMFT value of 1.3µB/Ru in our previous research23. Theconfirmation of this FM-HS state can explain the anomalous high-moment in the M-H curve,in which a magnetic moment ∼ 2.5µB/Ru is experimentally observed under a magnetic field

10

SrRuTiO

SrRuO3

SrTiO3

Intermixing Layer

Top interface SrTiO3

2nd interface SrTiO3

Top interface SrRuO3

2nd interface SrRuO3

3rd interface SrRuO3

….

….

FIG. S10. The computational model in our DFT+U calculations.

> 1T.The AFM-ST1 state is a very unusual AFM magnetic state, however, our theory predic-

tion of magnetic moment (∼ 1.56µB/Ru) and the moment of our samples under zero field (∼1.5µB/Ru in the M -H curve are consistent and thus our theoretical results are reasonable.Moreover, a similar conclusion about stripe domain in SrRuO3 under zero field had beenreported24, which not only supports our conclusion, but reveals that the atomic mixing candrive SRO-based system to unusual AFM magnetic states. Under weak magnetic field, theAFM-ST1 state should transfer to a state with higher average moment and higher totalenergy, as shown in Fig. S12. From Table S2 one can see that the second stable state isthe AFM-3 state. The average moment of AFM-3 is 1.78 µB/Ru, which is consistent withthe M-H curve at H <1T. Further increasing the external magnetic field should continu-ously enhance the observed moment of the samples, this corresponds to the transition toFM-LS state, whose energy is 8.95 meV/Ru higher than the AFM-3 state. This transitionreaches the state in the M -H curve at H ∼ 1T. Under such a field an average moment of ∼2.0µB/Ru is observed. Please note, the energy of AFM-5 state is 0.96 meV/Ru lower thanthat of FM-LS, i.e., their energies are comparable. Thus, our DFT+U results hint thattwo magnetically different states (AFM-5 and FM-LS) may coexist due to comparable totalenergy and lattice thermodynamics, and structural disorder or crystal defects may possiblypin one solution in one region. The possible transition between AFM-3 to AFM-5 state,can be effectively considered as a diffusion of some movable AFM single domains. Finally,the samples exhibit as FM-HS state (under magnetic field H > 1T), whose total energy is38.22 meV/Ru higher than the ground state AFM-ST1, as discussed in Fig. 5 of the maintext.

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(a) (b) (c) (d) (e)

FM-HS/LS AFM-1 AFM-2 AFM-ST1 AFM-ST2

(f) (g) (h) (i)

(j) (k) (l) (m)

AFM-3 AFM-4 AFM-5 AFM-6

AFM-7 AFM-8 AFM-9 AFM-10

FIG. S11. Various magnetic configurations which have been considered in our DFT+U calculations.The blue dots indicate non-ferromagnetic Ti sites, and the red-up arrows and green-down arrowsindicate spin-up and spin-down Ru sites, respectively.

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FIG. S12. Schematic picture of magnetic transitions under external magnetic field. From ourexperimental M-H curve, the H1 is deduced as ∼1 T. The values of average magnetic moment areobtained from our DFT+U calculations for the three AFM magnetic orderings (AFM-ST1, AFM-3,AFM-5) and FM-LS, FM-HS states.

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