science.sciencemag.org/content/370/6523/1438/suppl/DC1

Supplementary Materials for

Relativistic kinematics of a magnetic soliton

Lucas Caretta, Se-Hyeok Oh, Takian Fakhrul, Dong-Kyu Lee, Byung Hun Lee, Se Kwon

Kim, Caroline A. Ross, Kyung-Jin Lee, Geoffrey S. D. Beach*

*Corresponding author. Email: gbeach@mit.edu

Published 18 December 2020, Science 370, 1438 (2020)

DOI: 10.1126/science.aba5555

This PDF file includes:

Materials and Methods

Supplementary Text

Figs. S1 to S6

References

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Materials and Methods:

Growth, Structural Characterization and Patterning of Materials. BiYIG films were deposited by

pulsed laser deposition (PLD) on single-side-polished, single crystal Gd3Sc2Ga3O12 (111) (GSGG)

substrates. Replacing Y with Bi in dodecahedral sites of BiYIG in sufficient quantities expands

the lattice parameter and increases the magnitude of the negative magnetostriction coefficient; 5

growth on substituted garnet substrates leads to tensile strain yielding an out-of-plane magnetic

easy axis (25).

The Bi0.8Y2.2Fe5O12 target was prepared from Fe2O3 and Bi2O3 powder by a mixed oxide

sintering method. The chamber was pumped to 5 × 10-6 Torr base pressure prior to introducing

oxygen and depositing the films. PLD growth was performed using a 248 nm KrF excimer laser 10

of fluence ~2 J/cm2 and laser repetition rate of 10 Hz (43). The target-substrate distance was fixed

at 6 cm. During deposition the substrate temperature was 560 ᵒC and the oxygen pressure was

100 mTorr. The films were cooled to room temperature at 10 °C min−1 and 225 Torr oxygen

pressure. Film thicknesses were determined by X-ray reflectivity measurements. Domain wall

dynamics experiments were performed on patterned films with thicknesses 6.9 nm, shown in the 15

Main Text, and 2.4 nm, described in the Supplementary Material below. Brillouin light scattering

experiments were carried out on continuous films with thicknesses 19.0 nm, 18.9 nm, and 4.3 nm.

High-resolution X-ray diffraction (HRXRD) 2θ–ω scans of the (444) reflection for a

similiarly-grown 32.0 nm BiYIG reveal that fully-strained epitaxial growth is maintained at least

up to this thickness, and all films studied here have perpendicular anisotropy as expected for this 20

fully-strained state. Thinner films such as the 6.9nm film do not provide sufficient scattering signal

in the HRXRD, but cross-sectional scanning transmission electron microscopy and electron energy

loss spectroscopy characterization of the film structure and composition has been reported

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elsewhere by some of the present co-authors for thinner BiYIG films (36), indicating fully epitaxial

growth and a sharp, lattice-matched BiYIG/GSGG interface. That analysis revealed a narrow

interfacial region (~1nm) that is Y-rich with intermixed Gd and Ga from the substrate. Previously

reported X-ray magnetic circular dichroism measurements in the same publication (36) on a 2.4nm

TmIG/GGG film grown in a similar manner as the present BiYIG films, revealed only 5

paramagnetic signal from the Gd ions. The measurements were performed at room temperature in

glancing-angle total electron yield mode (penetration depth ~3nm) so the Gd signal is dominated

by the Gd in close proximity to the interface region. We hence do not expect magnetically-ordered

interfacially-substituted Gd to contribute to any compensation effects in the present films.

Pt metallic overlayers were grown using d.c. magnetron sputtering with an Ar sputter gas 10

pressure of 3.5 mTorr and a background base pressure of 1 × 10−7 Torr. Domain wall motion

tracks were patterned using standard photolithography and ion milling, etching through both the

Pt overlayer and the BiYIG. The BiYIG/Pt track had electrical contacts at either end for current

injection and an orthogonal Au strip line for initializing DWs via an Oersted field from a short

current pulse. The initialization line and contact pads [Ta(6 nm)/Au(150 nm)] were patterned using 15

photolithography and lift-off processes. Domain wall measurements were performed on

100 𝜇𝑚 × 40 𝜇𝑚 and 50 𝜇𝑚 × 20 𝜇𝑚 tracks.

Magnetic property characterization. Vibrating sample magnetometry was used to characterize the

magnetic properties of the BiYIG. Three 4 𝑚𝑚 × 4 𝑚𝑚 samples were stacked to improve the

signal-to-noise ratio. For the 6.9 nm BiYIG film used in domain wall dynamics measurements, 20

we extracted 𝑀𝑠 ≈ 140 𝑘𝐴/𝑚 from the easy-axis hysteresis loop in Main Text Fig. 1C. From the

hard-axis (in-plane) hysteresis loop shown in Fig. S1, we measured an anisotropy field 𝐻𝑘

corresponding to an effective anisotropy constant 𝐾𝑢,eff = 1

2𝜇0𝑀𝑆𝐻𝑘 ≈ 21 𝑘𝐽/𝑚3.

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We note that in Main Text Fig. 1C, the coercivity of the sample as measured by VSM is

significantly smaller than the coercivity as measured by MOKE. This can be attributed to two

effects. First, the MOKE hysteresis loop was measured very locally using a focused laser spot of

~8 𝜇𝑚, as opposed to being averaged across a large-area specimen. Second, coercivity depends

on thermally-activated domain nucleation and domain wall motion, and is hence strongly 5

dependent on the measurement frequency (timescale). The MOKE hysteresis loops were acquired

at a field sweep frequency of ~10 Hz, whereas the VSM measurement was acquired over a

measurement timescale of ~hours due to the averaging required for adequate signal-to-noise.

Hence, the measured coercivity is expected to be much lower in the latter measurement.

Magneto-optical Kerr effect (MOKE) measurements. Polar MOKE microscopy measurements 10

were acquired on a custom-built, 3-axis scanning Kerr microscope with independent out-of-plane

and in-plane magnetic field control. The in-plane magnetic field is controlled via a water-cooled,

Fe-cored electromagnet with a magnetic field feedback loop. The out-of-plane magnetic field is

controlled via an air-core electromagnet. The sample sits on a stage that has a high-precision

angular adjustment for eliminating out-of-plane contributions from the in-plane magnet. Residual 15

out-of-plane component is nulled using the out-of-plane air coil. This is accomplished by

identifying and nulling out-of-plane fields detected through domain wall creep motion, which is

exponentially sensitive to the field. Focused MOKE measurements were performed using a

continuous-wave 445 nm diode laser focused with a 10x objective to a spot size of ~8 μm. The

laser is attenuated to ~3 mW to prevent heating on the sample. Wide-field MOKE microscopy 20

images were acquired using a white light emitting diode source and a CCD camera.

Domain wall track current injection. High-bandwidth pulses for domain wall motion experiments

were delivered to the sample from a custom-built nanosecond pulse generator via a waveguide. A

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low duty cycle (kHz) was used to avoid heat build-up and damage. Domain walls were nucleated

using an Oersted field generated by passing short (microsecond) current pulses through an

orthogonally patterned Au stripline. The velocity of the domain wall was determined by dividing

the change in the position of the domain wall (as viewed by MOKE) by the duration of the current

pulse. All velocity data points in this manuscript represent an average of three velocity 5

measurements that correspond to a total domain wall travel distance of at least ~60 μm.

Measurement of spin Hall efficiency, 𝛘 and effective spin Hall angle, 𝜽𝐞𝐟𝐟. The spin Hall efficiency

𝜒 and the effective spin Hall angle 𝜃eff were quantified by measuring the dependence of the domain

wall (DW) depinning field on current density j flowing in a DW track, as done elsewhere (44, 45).

First, a DW is nucleated in a DW track using an Oe field from an orthogonal overlaid Ta(5 10

nm)/Au(150 nm) stripline. The nucleated DW is driven along the length of the track with a slowly

swept easy axis magnetic field 𝐻z that depins the domain wall and drives it along the track and

across the probe laser spot, where it is detected by MOKE (see Fig. S2A). The magnetic field

required to depin the DW is the DW depinning field 𝐻𝑑𝑝. A d.c. current is applied through the Pt

overlayer concurrently simultaneously with 𝐻𝑧. The spin Hall current from the Pt overlayer acts 15

as an easy-axis magnetic field 𝐻eff driving a Néel DW, where 𝐻eff ≡ 𝜒𝑗 = 𝜒0 𝑗 𝑐𝑜𝑠(Φ). Here 𝜒0 =

𝜋

2

ℏ𝜃eff

2𝑒𝜇0𝑀𝑠𝑡, where 𝜃eff is the effective spin Hall angle, 𝑒 is the electron charge, ℏ is the reduced

Planck constant, 𝑀𝑠 is the saturation magnetization, and 𝑡 is the magnetic film thickness. Φ is the

angle between the DW moment and the current flow direction. For Néel DWs, Φ = 𝑛𝜋, and for

Bloch DWs (as is the case of BiYIG with Hx=0), Φ = 𝑛𝜋/2. The slope of the measured depinning 20

field 𝐻𝑑𝑝 versus current density 𝑗 is the spin Hall efficiency 𝜒. Fig. S2B plots 𝜒 as a function of

in-plane field 𝐻𝑥 in the GSGG/BiYIG(6.9 nm)/Pt(4 nm) film. We find 𝜒 = 0 when 𝐻𝑥 = 0,

confirming that the DWs in BiYIG are of Bloch type at equilibrium and hence the Dzyaloshinskii-

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Moriya effective field is negligible. An in-plane magnetic field is needed to orient the DW into a

Néel configuration. In Fig. S2B, the saturation value of 𝜒 reached at large 𝐻𝑥 represents the

maximum 𝜒, when the DW is fully Néel (𝜓 = 𝑛𝜋). From the value at saturation, we determine 𝜒0

and hence find 𝜃eff ≈ 1.8%.

Brillouin light scattering (BLS) measurements and dispersion curve fitting. BLS was used to 5

measure the exchange constant A of our thin-film BiYIG from the perpendicular standing spin

wave (PSSW) mode frequency, which exhibits a substantial exchange energy contribution and

hence provides an accurate means of extracting A (37, 38, 46). The BLS measurements were

performed using a Sandercock-style (3+3) tandem Fabry-Pérot interferometer (47) and 532 nm

continuous-wave laser. We employed the Damon-Eshbach geometry (40), with an external field 10

𝐻𝑥 applied in the plane of the sample and perpendicular to the spin wave propagation direction.

At room temperature, thermally excited spin waves are detected through a frequency-shift of the

backscattered light, collected and analyzed through the interferometer. We obtained BLS spectra

with 𝐻𝑥 > 4500 𝑂𝑒, which exceeds the out-of-plane anisotropy field Hk and hence is sufficient to

saturate the magnetization into the film plane. The incident angle 𝜃 of the laser was varied from 15

10⁰ to 60⁰ by varying the angle of the sample with respect to the incident beam and collection

optics. This range corresponds to in-plane wavevectors 𝑞∥ =4𝜋𝑠𝑖𝑛𝜃

𝜆= 4.1 ~ 20.5 𝑟𝑎𝑑 ∙ 𝜇𝑚−1.

The dispersion relations for the Damon-Eshbach magnetostatic surface spin wave mode

(DE) and PSSW mode are given by (41, 46, 48)

𝑓𝐷𝐸 =

𝜇0𝛾

2𝜋√[𝐻𝑥 +

2𝐴

𝜇0𝑀s

𝑞∥2] [𝐻𝑥 +

2𝐴

𝜇0𝑀s

𝑞∥2 + 𝑀s − 𝐻𝑢] + (

𝑀s

2)

2

(1 − exp (−2𝑞∥𝑡)) S1

and 𝑓𝑃𝑆𝑆𝑊 =

𝜇0𝛾

2𝜋√[𝐻𝑥 +

2𝐴

𝜇0𝑀𝑠

(𝑞∥2 + (

𝑛𝜋

𝑡)

2

)] [𝐻𝑥 +2𝐴

𝜇0𝑀𝑠

(𝑞∥2 + (

𝑛𝜋

𝑡)

2

) + 𝑀𝑠 − 𝐻𝑢] S2

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respectively. Here, 𝑞∥ is the in plane component of the spin-wave vector, Hu is the uniaxial

anisotropy field directed out of the film plane (49), t is the film thickness, n is the mode number

for the standing wave perpendicular to the film plane, 𝜇0 = 4𝜋 × 10−7 𝑇 ∙𝑚

𝐴, and

γ

2π= 28 GHz/T,

and the other parameters are defined in preceding sections. In Eq. S2 we have omitted the term

𝐻𝑥 (𝑀𝑠 𝐻𝑥⁄

𝑛𝜋 𝑡⁄)

2

𝑞∥2 that is negligible compared to Hx. Equations S1 and S2 have a very similar form, 5

but the exchange contribution to the frequency in Eq. S2 is much larger (37, 38), since

(𝜋 𝑡)2/⁄ 𝑞∥2 ≈ 102. This allows us to identify the mode character of the BLS spectral peaks in

Main Text Fig. 3a: the DE modes are nearly independent of thickness, whereas the PSSW mode

depends strongly on thickness (37, 38) and is hence only visible at high frequency for the thickest

film studied (Main Text Fig. 3B). 10

The spin wave frequency peak positions were determined by Lorentzian fits to the BLS

spectra. The 𝑞∥-dependence (Main Text Fig. 3C) and Hx-dependence (Fig. S3) of fDE and fPSSW for

the t = 29 nm sample were fitted to Eqs. S1 and S2, respectively. We used t = 29 nm and set n = 1

since it is the first PSSW mode to appear with increasing t (as a further check, using a larger mode

number gives an unphysical result). Ms and Hu were fixed based on VSM measurements of the 15

saturation magnetization Ms = 1.40x105 A/m and the effective out-of-plane anisotropy field (hard-

axis saturation field) 𝐻k,eff = 𝐻𝑢 − 𝑀𝑠 = 244 𝑘𝐴/𝑚. Linear least-squares fits to fPSSW versus 𝑞∥

and fPSSW versus Hx yielded A = 4.16 pJ/m and A = 4.26 pJ/m, respectively, with a fitting

uncertainty <0.01 pJ/m. The experimental uncertainty on A is hence dominated by that on Ms,

since A/Ms appears as a ratio in Eq. S2. We estimate a 5% uncertainty on the VSM measurement 20

of Ms, and hence report an experimental value 𝐴 = (4.2 ± 0.2)p J/m in the Main Text.

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We note that, as described in detail in Ref. (39), since exchange makes only a minor

contribution to fDE owing to the small value of 𝑞∥ (see Eq. S1), the DE mode cannot be used to

accurately determine A. In our case, the fitting uncertainty on A for the DE mode is ~ 3 pJ/m,

similar to what was reported in Ref. (39). For this reason, we cannot reliably measure A in thinner

films, where the PSSW mode is at frequencies that are not accessible in our BLS setup. 5

Exchange spin-wave dispersion and maximum group velocity calculation. The exchange mode

spin-wave dispersion relation for a two-sublattice ferrimagnet has been reported elsewhere (12),

where it was derived from the equations of motion obtained from the Lagrangian density ℒ and

Rayleigh dissipation ℛ for two-sublattice ferrimagnets,

ω± =

±𝛿𝑠 + √𝛿𝑠2 + 4𝜌(𝐴𝑘2 + 𝐾𝑢)

2𝜌

S3

with k the wavevector and the angular frequency, as also given in Main Text Eq. 3. The two 10

signs correspond to two different modes. Here, 𝛿𝑠 = 𝑆1 − 𝑆2 is the net spin density from the two

opposing sublattices, 𝐴 is the exchange constant, 𝐾𝑢 is the uniaxial anisotropy constant, 𝜌 =

𝑆2𝑑2/𝐴 is the inertia of the dynamics (9), d is the lattice spacing, and 𝑆 = (|𝑆1| + |𝑆2|) is the total

sublattice spin density. To generate the curves in Fig. 3a of the main text, we used the measured

Ku and A, and we extracted 𝛿𝑠 from 𝛾eff determined from the domain wall dynamics measurements 15

as described in the previous subsection. We used a lattice spacing 𝑑 = 0.346 nm, corresponding

to the tetrahedral-octahedral Fe3+ spacing in BiYIG. We computed 𝑆 = 4.78 ×10−6𝑘𝑔

𝑚 𝑠 based on

the number density of tetrahedral and octahedral Fe3+ ions in BiYIG. Note that this quantity is

quite insensitive to deviations from the ideal Fe3+ site occupancy since 𝑆 involves the sum of the

absolute values of the spin densities on the opposing tetrahedral and octahedral sites, not the 20

difference between them.

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The group velocity can be derived from the dispersion relation above,

𝑣𝑔 =

𝑑𝜔

𝑑𝑘=

2𝐴𝑘

√𝛿𝑠2 + 4𝜌(𝐴𝑘2 + 𝐾)

. S4

The group velocity is the same for both spin wave modes. The maximum group velocity comes

as the limit as 𝑘 → ∞, and is given by

𝑣𝑔

𝑚𝑎𝑥 =2𝐴

𝑑𝑆

S5

We note here that A is the only experimental parameter that appears in the expression for vg,max.

We hence arrive at 𝑣𝑔,𝑚𝑎𝑥 ≈ 5000 𝑚/𝑠 for our BiYIG films based on the experimentally-5

measured A.

Effective Scaling Model for Ferrimagnetic Dynamics. In the case of ferrimagnets with strongly

exchanged-coupled, antiparallel sublattices, one can model DW dynamics by adapting the 1D

equation of motion using a well-established effective ferromagnet model with the following

scaling (16–18): 10

𝛼 → 𝛼eff =

𝑆𝛼

𝛿𝑠

S6

𝛾 → 𝛾eff =

𝑀𝑠

𝛿𝑠

S7

Here, 𝑀𝑠 = |∑ 𝑀𝑠,𝑖𝑖 |, 𝛿𝑠 = |∑𝑀𝑠,𝑖

𝛾𝑖𝑖 |, 𝑆𝛼 = 𝛼 ∑

|𝑀𝑠,𝑖|

𝛾𝑖𝑖 are the net magnetization, net angular

momentum (spin density), and Rayleigh dissipation coefficient, respectively, and 𝛼 characterizes

the sublattice Gilbert damping. The subscript 𝑖 denotes the sublattices. Substituting these

expressions into the 1D model for domain walls driven by the spin Hall effect yields 𝑣𝐷𝑊 =15

𝛾eff𝛥

𝛼eff

𝜒0𝑗

√1+[𝐻𝑆𝐻 𝛼eff(⁄ 𝐻𝐷+𝐻𝑥)]2, where Hx is a longitudinal in-plane field, and HD is the DMI effective

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field, which is absent in the present samples. Although the BiYIG system studied here is not

expected to have an accessible (𝑇 > 0 𝐾) angular momentum compensation temperature 𝑇𝐴, this

modification of the 1D model accounts for all compensation effects of the antiferromagnetically-

coupled system, both close to, and far away from 𝑇𝐴, as shown in Ref (17). This simple and

effective scaling model allows for easy adaptation of existing ferromagnet models to strongly 5

exchange-coupled, multi-sublattice ferrimagnets.

For the analytical modelling, 𝜒0 was extracted experimentally from current-assisted DW

depinning measurements as described above. The equilibrium DW width ∆𝟎 was taken as ∆𝟎=

√𝐴

𝐾u,eff = 14 nm, using the measured BiYIG exchange constant and effective uniaxial anisotropy

energy density 𝐾u,eff reported above. We extracted 𝛾eff ≈ 1.54 × 1013 𝑟𝑎𝑑 ⋅ 𝑠−1 ⋅ 𝑇−1 from the 10

linear slope of the DW velocity v versus Hx under high current density in the low-Hx (non-

relativistic) regime, where 𝑣 =𝜋

2𝛾eff𝛥0𝐻𝑥. Finally, we estimated 𝛼eff ≈ 0.85 from the slope of v

versus j at low currents, where 𝑣 =𝛾eff𝛥

𝛼eff𝜒0𝑗.

We note that 𝛼eff and 𝛾eff are enhanced due to partial angular momentum compensation in

the ferrimagnet, but their ratio, which determines the DW mobility, is not. The latter depends not 15

on 𝛼eff but on the factor 𝛼 that determines the dissipation rate, which we estimate as 𝛼 ≈0.0027.

This value is much smaller than 𝛼 in most metallic ferromagnets, contributing to the high current-

driven mobility in these samples.

Numerical Atomistic Modeling. We carry out numerical simulation with the one-dimensional

atomistic Landau-Lifshitz-Gilbert (LLG) equation including damping-like spin-orbit torque (DLT) 20

given by

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𝑑𝑺𝑖

𝑑𝑡= −𝛾𝑖𝑺𝑖 × 𝑩eff,𝑖 + 𝛼𝑖𝑺𝑖 ×

𝑑𝑺𝑖

𝑑𝑡+ 𝝉𝒎 ,

S8

Here, 𝑺i is the normalized spin vector at the lattice site 𝑖, 𝛾𝑖 is the gyromagnetic ratio, 𝑩eff,𝑖 =

−(1 𝜇𝑖⁄ ) 𝜕ℋ 𝜕𝑺𝑖⁄ is the effective field including exchange, uniaxial anisotropy, and external

magnetic fields, 𝜇𝑖 is the magnetic moment per atom, the discrete Hamiltonian is ℋ =

𝐴𝑠𝑖𝑚 ∑ 𝑺𝑖 ∙ 𝑺𝑖+1𝑖 − 𝐾𝑠𝑖𝑚 ∑ (𝑺𝑖 ∙ 𝒛)2𝑖 + 𝑔𝑖𝜇𝐵 ∑ 𝑺𝑖 ∙ 𝑩ext𝑖 , the external magnetic field 𝑩𝑒𝑥𝑡 is

applied along 𝑥 axis, the DLT is 𝝉𝒎 = 𝛾𝑖𝐵𝐷,𝑖𝑺𝑖 × (𝑺𝑖 × 𝒚) where 𝐵𝐷,𝑖 = ℏ𝜃eff𝑗 2𝑒𝑀𝑠,𝑖𝑡⁄ , 𝜃eff is 5

the effective spin-Hall angle, 𝑗 is the current density, 𝑀𝑠 is the saturation magnetization, and 𝑡 is

the thickness.

In the numerical simulations, we assume that there are two alternating sublattices along the

atomic chain. Odd (even) 𝑖 represent Fe3+ ions in tetrahedral (octahedral) sites. We used lattice

spacing 𝑑 = 0.346 nm, corresponding to the tetrahedral-octahedral Fe3+ spacing in BiYIG; 𝐴𝑠𝑖𝑚 =10

9.00 meV, corresponding to the bulk BiYIG exchange constant; 𝐾𝑠𝑖𝑚 = 1.39 μeV, corresponding

to the measured uniaxial anisotropy, and Gilbert damping constants 𝛼1 = 𝛼2 = 0.0027, consistent

with the experimental result. The individual sublattice 𝑀𝑠,𝑖 and 𝑔𝑖 cannot be experimentally

resolved, so to match the experimental net 𝑀𝑠 and 𝛾eff, we used 𝑀𝑠,1 = 421 kA m⁄ , 𝑀𝑠,2 =

281 kA m⁄ and 𝑔1 = 2 and 𝑔2 = 1.34, where the values for sublattice ‘1’ were fixed to the bulk 15

BiYIG tetrahedral site values. The results do not depend on these individual sublattice choices as

long as 𝑀𝑠 and 𝛾eff are fixed. The somewhat low value of 𝑔2 is not completely understood but may

originate from stoichiometry and site-occupancy changes from interdiffusion of Gd (50, 51) and

Ga (52) from the substrate.

𝚫(𝑯𝒙) Domain wall broadening correction. For the analytical modelling, the equilibrium DW 20

width ∆0 is computed as ∆0= √𝐴

𝐾𝑢,eff= 14 𝑛𝑚 using experimental material parameters as

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described above. Under the application of an in-plane field 𝐻𝑥, ∆0 broadens due to the Zeeman

contribution to the DW energy density. Therefore, Δ0 is a function of 𝐻𝑥. This is seen in Fig. S3,

where the fractional expansion of the DW width as a function of Hx extracted from numerical

atomistic simulations is plotted. Within this field range, the variation is linear with Hx. Both the

experimental and numerical data in Main Text Fig. 3B are obtained with an applied 𝐻𝑥. To account 5

for the 𝐻𝑥-induced DW broadening in the experimental and numerical data points, we plot the

ratio 𝛥/∆0(𝐻𝑥) instead of 𝛥 alone in Main Text Fig. 3B. Here, ∆0(𝐻𝑥) for each experimental data

point or numerically-computed data point corresponds to a fixed 𝐻𝑥, which is determined from the

calibration curve in Fig. S4. This way, the variation of 𝛥/∆0 relates only to the Lorentz contraction,

even though different data points correspond to different Hx. 10

Supplementary Text

S1. Mapping of the spin dynamics of uniaxial magnets to the sine-Gordon equation

In this section, we provide continuum equations of motion for low energy spin dynamics (i.e.,

excitations in which the magnetization varies smoothly enough in space that the continuum 15

approximation is valid) in uniaxial antiferromagnets, ferrimagnets, and ferromagnets, and review

the mapping of those dynamics to the sine-Gordon equation. These mappings have been derived

previously in the literature and are collected here to highlight their assumptions and common

features and to provide context to the Main Text discussions. The equations of motion in the case

of driving and damping terms are discussed for the specific case of ferrimagnets, which are the 20

focus of the experiments and simulations presented in the Main Text.

Antiferromagnets with easy-axis anisotropy. The low-energy dynamics of one-

dimensional antiferromagnets with easy-axis anisotropy can be described by the following

Lagrangian (7–9, 53)

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𝐿[𝒏] = 𝑇[𝒏] − 𝑈[𝒏] = ∫𝜌 �̇�𝟐

2𝑑𝑥 − ∫

𝐴(𝒏′)2

− 𝐾𝑢𝑛𝑧2

2𝑑𝑥

S9

where 𝑇[𝒏] and 𝑈[𝒏] are the kinetic and the potential energies of the magnet, respectively, 𝐧 is

the unit vector along the staggered magnetization (i.e., the Néel order parameter), 𝜌 is the inertia

for the dynamics, which may be written 𝜌 = 𝑆2𝑑2/𝐴 (9) in terms of the lattice spacing d and

sublattice spin density S, 𝐴 is the exchange coefficient, and 𝐾𝑢 is the easy-axis uniaxial anisotropy

constant. The “dot” symbol denotes a time derivative, and the “prime” symbol (′ ) denotes a 5

spatial derivative in the 𝑥 direction. As noted explicitly in Ref. (7), this is the Lorentz invariant

O(3) nonlinear sigma model with an additional easy-axis anisotropy term.

When 𝒏 is expressed in terms of the usual polar angle 𝜗 and azimuthal angle 𝜙, with

Cartesian coordinates given by 𝒏 = (sin 𝜗 cos 𝜙 , sin 𝜗 sin 𝜙 , cos 𝜗), the Lagrangian is written as

𝐿[𝜗, 𝜙]

= ∫𝜌 (�̇�2 + sin2 𝜗 �̇�2) − 𝐴 ((𝜗′)

2+ sin2 𝜗 (𝜙′)

2) + 𝐾𝑢 cos2 𝜗

2𝑑𝑥 .

S10

The Euler-Lagrange equation corresponding to the azimuthal angle 𝜙(𝑥, 𝑡) is given by 10

𝜌 (

𝜕(sin2 𝜗 �̇�)

𝜕𝑡) = 𝐴 (

𝜕(sin2 𝜗 𝜙′)

𝜕𝑥) .

S11

It is solved by the constant uniform angle 𝜙(𝑥, 𝑡) ≡ 𝜙0, which is the minimizer of the potential

energy 𝑈[𝜗, 𝜙]. By plugging in this solution for 𝜙, we obtain the Lagrangian only in terms of the

polar angle:

𝐿[𝜗] = ∫𝜌 (�̇�2) − 𝐴(𝜗′)

2+ 𝐾𝑢 cos2 𝜗

2𝑑𝑥 .

S12

The resultant Euler-Lagrange equation is given by

Submitted Manuscript: Confidential

14

𝜌�̈� − 𝐴𝜗′′ + 𝐾𝑢 cos 𝜗 sin 𝜗

= 0 .

S13

This equation takes the form of the sine-Gordon equation upon the change of variable 𝜃 = 2𝜗:

𝜕2𝜃

𝜕𝑡2− 𝑐𝑚

2𝜕2𝜃

𝜕𝑥2+

𝑐𝑚2

Δ02 sin 𝜃 = 0 ,

S14

where 𝑐𝑚 = √𝐴/𝜌 is the maximum magnon group velocity and Δ0 = √𝐴/𝐾𝑢 is the equilibrium

domain wall width. This equation is invariant under the following transformation:

𝑥 ↦ 𝑥′ =

𝑥 − 𝑣𝑡

√1 − 𝑣2/𝑐𝑚2

,

S15

𝑡 ↦ 𝑡′ =𝑡 − 𝑣𝑥/𝑐𝑚

2

√1 − 𝑣2/𝑐𝑚2

,

with an arbitrary velocity −𝑐𝑚 < 𝑣 < 𝑐𝑚. This transformation resembles the Lorentz 5

transformation in the theory of special relativity: the maximum magnon group velocity 𝑐𝑚 in the

former plays an analogous role to the speed of light in the latter. The solution to the sine-Gordon

equation is (4, 5)

𝜃(𝑥, 𝑡) = 4 arctan [exp (𝑥 − 𝑣𝑡

Δ)] S16

with the parameter Δ given by

Δ = Δ0√1 −𝑣2

𝑐𝑚2

.

S17

This solution takes the form of the familiar Walker ansatz (15) for a magnetic domain wall, but 10

with the static domain-wall width Δ0 scaled by the inverse of the Lorentz factor 𝛾′ =

1/√1 − 𝑣2/𝑐m2 . In the limit of zero dissipation (damping), domain walls are freely-moving

particle-like solutions, and Lorentz boosts between inertial frames follow in a manner that is fully

Submitted Manuscript: Confidential

15

analogous to those in special relativity, here manifesting as Lorentz contractions of the domain

wall width (5).

Antiferromagnetically-coupled ferrimagnets with easy-axis anisotropy. We next consider

ferrimagnets consisting of two inequivalent antiferromagnetically-coupled sublattices with an

easy-axis anisotropy along the 𝑧 axis. We include an external magnetic field Hx along the x 5

direction, for reasons that can be understood below. We note that this corresponds to the

experimental conditions in the experiments in the Main Text.

The equation of motion for the dynamics of a ferrimagnet can be described by the following

Lagrangian (12, 54)

𝐿[𝒏] = 𝑇[𝒏] − 𝑈[𝒏] = ∫ [𝜌 �̇�𝟐

2− 𝛿𝑠𝒂(𝒏) ⋅ �̇�] 𝑑𝑥 − ∫ [

𝐴(𝒏′)2

− 𝐾𝑢𝑛𝑧2 − 𝐻𝑥𝑀𝑠𝑛𝑥

2] 𝑑𝑥, S18

where 𝐧 is the unit vector along the local spin direction, 𝛿𝑠 = 𝑆1 − 𝑆2 is the net spin density (i.e.,

the uncompensated spin density in the direction of 𝒏), 𝒂(𝒏) is the vector potential of a magnetic

monopole satisfying 𝛁𝒏 × 𝒂(𝒏) = 𝒏, 𝐻𝑥 is the external field in the 𝑥 direction, and 𝑀𝑠 is the

saturation magnetization of the ferrimagnet. Note that there are two terms involving the time

derivative of the order parameter 𝒏. The first term, proportional to 𝜌, is the same kinetic term as

appeared in the antiferromagnet case discussed above. The second term is a spin Berry phase term

(55) that appears due to the finite net spin density 𝛿𝑠, which is absent in pure antiferromagnets and

is responsible for precessional dynamics. Due to significant compensation of the two spin

densities, the net spin density is typically small in ferrimagnets and thus their magnetization

dynamics is generally dominated by the first term in the Lagrangian.

In terms of polar angle 𝜗 and azimuthal angle 𝜙, the spin Berry phase term takes the form

𝑠(cos 𝜗 − 1)�̇�. The Euler-Lagrange equation corresponding to the azimuthal angle 𝜙 is given by

Submitted Manuscript: Confidential

16

−𝛿𝑠 sin 𝜗 �̇� +𝜕

𝜕𝑡(sin2 𝜗

𝜕𝜙

𝜕𝑡) − 𝐴

𝜕

𝜕𝑥(sin2 𝜗

𝜕𝜙

𝜕𝑥) + 𝐻𝑥𝑀𝑠 sin 𝜗 sin 𝜙 = 0 . S19

If the net spin density vanishes, i.e., 𝛿𝑠 = 0, the equation is solved by a constant, uniform azimuthal

angle 𝜙(𝑥, 𝑡) ≡ 0 (See the discussion around Eq. S11). For small but finite 𝛿𝑠, the azimuthal-angle

solution to the equation is finite and is given approximately by

𝜙 = arcsin (𝛿𝑠�̇�

𝐻𝑥𝑀𝑠) ≃

𝛿𝑠�̇�

𝐻𝑥𝑀𝑠 to linear order in the time derivative and the spatial derivative of the

fields (assuming sufficiently long-wavelength low-energy dynamics). This is the situation for

stationary-state domain wall solutions, in which 𝜙 is uniform and stationary in time. We note that,

as can be seen in the equation above, a finite field Hx is required for a nontrivial stationary-state

solution to exist.

Plugging this solution for 𝜙 into Eq. S11 yields the Lagrangian only in terms of the polar

angle 𝜗 and it is given by Eq. S12 when we neglect the term (𝛿𝑠2/𝐻𝑥𝑀𝑠) sin 𝜗 �̇�2 from the spin

Berry phase. This approximation is valid when 𝛿𝑠 is sufficiently small and 𝐻𝑥𝑀𝑠 is sufficiently

large so that 𝐻𝑥𝑀𝑠 ≫ 𝛿𝑠2/𝜌, which is satisfied for our experiments and is also verified by the

agreement of this model with our atomistic simulations. With this approximation, all the

discussions below Eq. S12 for antiferromagnets can be applied to ferrimagnets. In particular, the

equation of motion for 𝜃 = 𝜗/2 is given by the sine-Gordon equation, Eq. S14.

Damping and driving terms can be added to the sine-Gordon equation in a straightforward

manner, as has been shown in the literature (5, 6). The damping term can be added through the

Rayleigh dissipation function (12, 16, 56)

𝑅 = (𝛼𝑆/2) ∫ �̇�𝟐𝑑𝑥 = (𝛼𝑆/2) ∫(�̇�𝟐 + sin2 𝜗 �̇�𝟐)𝑑𝑥

S20

Submitted Manuscript: Confidential

17

where 𝛼 is the dimensionless Gilbert damping parameter and 𝑆 = (|𝑆1| + |𝑆2|) is the total

sublattice spin density. Upon the inclusion of this damping term, the Euler-Lagrange equation for

𝜗 (Eq. S13) acquires 𝛼𝑆�̇� on the right-hand side:

𝜌�̈� − 𝐴𝜗′′ + 𝐾 cos 𝜗 sin 𝜗

= −𝛼𝑆 �̇� .

S20

For a driving force, let us consider an effective magnetic field along the 𝑧 direction, 𝐻𝑧�̂�,

which enters the Lagrangian as the potential-energy term −𝐻𝑧𝑀𝑠𝑛𝑧 = −𝐻𝑧𝑀𝑠 cos 𝜗. With this

additional term and also with the damping term, the equation of motion for 𝜗 is given by

�̈� − 𝐴𝜗′′ + 𝐾𝑢 cos 𝜗 sin 𝜗 = −𝛼𝑆 �̇� + 𝐻𝑧𝑀𝑠 sin 𝜗 . S22

Upon the change of variable 𝜃 = 𝜗/2, the equation of motion is transformed into

𝜕2𝜃

𝜕𝑡2− 𝑐𝑚

2𝜕2𝜃

𝜕𝑥2+

𝑐𝑚2

Δ02 sin 𝜃 = −𝜏−1

𝜕𝜃

𝜕𝑡+ 𝑓 sin

𝜃

2

S23

where 𝜏 ≡ 𝜌/𝛼𝑠𝑡 is the relaxation time of the dynamics of the magnet and 𝑓 ≡ 2𝐻𝑧𝑀𝑠/𝜌 5

represents the magnitude of the driving force. Although the damping term ∝ τ−1 breaks the

Lorentz invariance of the equation, it has been shown numerically and analytically (5, 6, 9–12)

that the modified Walker ansatz 𝜃(𝑥, 𝑡) = 4 arctan [exp (𝑥−𝑣𝑡

Δ)], remains an exact solution with

Δ = Δ0/√1 − 𝑣2/𝑐𝑚2 , and with the velocity given by

=

𝐻𝑧𝑀𝑠Δ

𝛼𝑆=

𝐻𝑧𝑀𝑠Δ0

𝛼𝑆

√1 + (𝐻𝑧𝑀Δ0

𝛼𝑆𝑐𝑚)

2

. S24

Submitted Manuscript: Confidential

18

Using effective parameters 𝛼eff =𝛼𝑆

𝛿𝑠 and 𝛾eff =

𝑀𝑠

𝛿𝑠 as defined above, and defining

𝑣0 = (𝛾eff∆0 /𝛼eff)𝐻𝑧, which corresponds to Walker’s classical stationary-state solution (15), we

can rewrite the expression above to read

𝑣 =𝛾effΔ

𝛼eff

𝐻𝑧 =𝑣0

√1 + (𝑣

𝑐𝑚)

2, S25

as was stated in the Main Text. In the case of a spin Hall effective field, Hz in the equation above

is replaced by HSHE, which takes the form of a z-axis effective field as described in the Main Text. 5

We make several important observations. First, in the case of a damped, driven system, the

domain wall is still seen to exhibit a Lorentz contraction governed by the factor √1 − 𝑣2/𝑐𝑚2 that

holds for the undamped Lorentz-invariant sine-Gordon Equation. Hence, we see that the domain

wall cannot exceed the maximum spin-wave group velocity 𝑐𝑚, which resembles the fundamental

upper limit on the particle velocity given by the speed of light. Second, although in the case of 10

zero dissipation (damping), all frames are indistinguishable inertial frames, with finite dissipation,

the only true inertial frame is the frame in which the domain wall is at rest with respect to the

magnet, since in all other frames, a force is required in order to maintain a constant velocity.

Although the expressions above are derived by treating the magnet as an infinite continuum, with

no explicit reference to a lattice and hence no well-defined way to speak of motion relative to the 15

magnet, damping is implicitly tied to the lattice and its presence amounts to something akin to an

“ether.” It is therefore to be understood that the relativistic transformations describe here, in the

case of a real magnet, must be viewed from the magnet rest frame, although this imposes no

practical loss of generality. Finally, it can be emphasized that the validity of this model and the

assumptions therein is verified by the excellent agreement between our atomistic simulations and 20

the analytical treatment based on this model, seen in Main Text Fig. 4.

Submitted Manuscript: Confidential

19

Ferromagnets with easy-axis anisotropy. For completeness, we consider the case of a

ferromagnet, which in the context of the ferrimagnet treatment discussed above, corresponds to

the limit in which the spin Berry phase term dominates. We treat the low-energy dynamics of an

easy-axis ferromagnet described by the following Lagrangian:

𝐿[𝒏] = 𝑇[𝒏] − 𝑈[𝒏] = −𝑆 ∫ 𝒂(𝒏) ⋅ �̇� 𝑑𝑥 − ∫ [ 𝐴(𝒏′)

2− 𝐾𝑢𝑛𝑧

2 + 4𝜋𝑀𝑠2𝑛𝑥

2

2] 𝑑𝑥 . S26 5

Here, 𝐧 is the unit vector along the local spin direction, 𝑆 is the spin density (corresponding to 𝛿𝑠

in the ferrimagnet case), 𝒂(𝒏) is the vector potential of a magnetic monopole satisfying

𝛁𝒏 × 𝒂(𝒏) = 𝒏, 𝑀𝑠 is the saturation magnetization, and 𝐾𝑢 is the easy-axis anisotropy constant.

The term ∝ 4𝜋𝑀𝑠2 is the magnetostatic energy capturing the effect of the demagnetizing field (15,

57), which plays a role akin to the field Hx in the ferrimagnet case treated above. U. Enz (57) first 10

derived the mapping of the equations of motion of ferromagnets to the sine-Gordon equation under

the condition that the magnetostatic energy is stronger than the easy-axis anisotropy, and the result

has been invoked in subsequent publications on a ferromagnetic domain-wall motion, e.g., in (4–

6, 58). Below, we discuss the mapping presented in (57), which is applicable to easy-plane

ferromagnets, i.e., in-plane magnetized films with a weak in-plane uniaxial anisotropy 𝐾𝑢 and a 15

large out-of-plane demagnetizing field.

When expressing the magnetization vector 𝒏 in terms of angular coordinates 𝜗 and 𝜙 such

that 𝒏 = (− sin 𝜙 , cos 𝜙 sin 𝜗 , cos 𝜙 cos 𝜗) (note the unconventional use of spherical angles), the

Lagrangian is given by

Submitted Manuscript: Confidential

20

𝐿[𝜗, 𝜙] = ∫ [−𝑠 sin 𝜙 �̇�

−𝐴 ((𝜙′)

2+ cos2 𝜙 (𝜗′)2) − 𝐾𝑢 cos2 𝜙 cos2 𝜗 + 4𝜋𝑀𝑠

2 sin2 𝜙

2] 𝑑𝑥. S27

Note that the potential energy is minimized by 𝜙 ≡ 0, i.e., when the magnetization is in the 𝑦𝑧

plane. Domains align along the z-axis and the demagnetizing field is along the out-of-plane x-

axis. 5

The Euler-Lagrange equation of motion for 𝜙 is given by

𝑆 cos 𝜙 �̇� = 𝐴𝜙′′ + 𝐴 sin 𝜙 cos 𝜙 (𝜗′)2 − 𝐾𝑢 sin 𝜙 cos 𝜙 cos2 𝜗 − 4𝜋𝑀𝑠2 cos 𝜙 sin 𝜙 . S28

By assuming 4𝜋𝑀𝑠2 ≫ 𝐾𝑢, a small 𝑛𝑥 component (|𝜙| ≪ 1) (due to the strong magnetostatic

energy), and small spatial gradients of the angles (|𝜙′′|, |𝜗′|2 ≪ 4𝜋𝑀𝑠2/𝐴), the equation can be

approximated by 10

𝜙 = − (𝑆

4𝜋𝑀𝑠2

) �̇� . S29

Upon the replacement of 𝜙 with the above expression in the Lagrangian, we obtain

𝐿[𝜗] = ∫

𝜌 (�̇�2) − 𝐴(𝜗′)2 + 𝐾𝑢 cos2 𝜗

2𝑑𝑥 ,

S30

where 𝜌 ≡ 𝑆2/(4 𝜋𝑀𝑠2) is the inertia of the dynamics of 𝜗, to quadratic order in 𝜙 and in the

spatial and the temporal gradients of 𝜗 (57). Then, the discussions for ferrimagnets after Eq. S21

can be directly applied to ferromagnets. Explicitly, an external magnetic field along the 𝑧-axis,

𝐻𝑧�̂�, enters the Lagrangian as the potential-energy term −𝐻𝑧𝑀𝑠𝑛𝑧 = −𝐻𝑧𝑀𝑠 cos 𝜗 (to linear order

in 𝜙). With this force term as well as the damping term, the equation of motion for 𝜗 is given by

Submitted Manuscript: Confidential

21

𝜌�̈� − 𝐴𝜗′′ + 𝐾𝑢 cos 𝜗 sin 𝜗

= −𝛼𝑠�̇� + 𝐻𝑧𝑀𝑠 sin 𝜗 .

S31

Upon the change of variable, 𝜃 = 𝜗/2, the equation of motion is given by

𝜕2𝜃

𝜕𝑡2− 𝑐𝑚

2𝜕2𝜃

𝜕𝑥2+

𝑐𝑚2

Δ02 sin 𝜃 = −𝜏−1

𝜕𝜃

𝜕𝑡+ 𝑓 sin

𝜃

2 ,

S32

Where 𝜏 ≡ 𝜌/𝛼𝑆 is the relaxation time and 𝑓 ≡ 2𝐻𝑀𝑠/𝜌 represents the magnitude of the driving

force. The exact solution for a moving domain wall is given by the same modified Walker ansatz

as before, 𝜃(𝑥, 𝑡) = 4 arctan [exp (𝑦−𝑣𝑡

Δ)], with the Lorenz-contracted domain wall width Δ =

Δ0/√1 − 𝑣2/𝑐𝑚2 . The velocity is given by the expressions in Eqs. S24, 25, with 𝛾eff → 𝛾 =

𝑀𝑠

𝑆 5

and 𝛼eff → 𝛼. As can be seen from Fig. 4 in Ref. (6) and the associated discussion, Walker

instabilities typically set in at velocities well below the relativistic limit cm, which makes

observation of relativistic-like phenomena in ferromagnets unlikely.

S2. Displacement as a function of pulse width. In this work, DW velocities were measured quasi-

statically, where the velocity was calculated by dividing the total distance the DW was displaced

by the cumulative length of the driving current pulses. In order to ensure negligible inertial effects

and/or DW displacement after the current pulse is turned off, we have measured the displacement

of the domain wall as a function of pulse duration for various in-plane fields 𝐻𝑥 (see Fig. S4). For

pulse widths ranging between ~1 ns and 20 ns, the measured displacement scales linearly with the

pulse width, and extrapolates to zero at zero pulse length. This result implies that displacements

only occur while the current is driving the motion, and there is negligible inertial displacement,

which would manifest as a vertical offset in Fig. S5.

Submitted Manuscript: Confidential

22

S3. Domain wall dynamics in BiYIG(2.4 nm)/Pt(4 nm). To prove reproducibility of the

phenomena observed here and to acquire thickness-dependent information, we have performed

current-driven DW motion experiments on a thinner, fully-strained

GSGG/BiYIG(2.4 nm)/Pt(4.0 nm) sample (Fig. S6). Velocity versus current density was measured

as described in the Main Text, with the velocity exhibiting a high-current saturation that scaled

with linearly with Hx at low Hx and then reached a global velocity plateau at higher Hx. The

saturation velocity for the 2.4 nm sample is plotted together with the data for the 6.9 nm sample

that is shown also in Main Text Fig. 2E. We see the same signature of global velocity saturation,

with the limiting velocity being similar for the 2.4 nm film. This implies the result is a general

feature of the dynamics in these materials, and since 𝑣𝑔𝑚𝑎𝑥 is proportional to the exchange constant

A, this result suggests that 𝐴 shows little scaling with thickness in this regime, remaining close to

its bulk value, which coincides with the experimental value measured for the 29nm BiYIG sample

by BLS. The low-Hx slope is proportional to the domain wall width, and the small variation in

slopes between the two thicknesses can likely be accounted for by a difference in the magnetic

anisotropy in the thinner film, which would not be unexpected. We note that the maximum in-

plane field that can be applied during domain wall motion experiments is limited by the spin orbit

torque switching threshold, which is smaller in the thinner BiYIG film due to its smaller coercivity.

At larger values of Hx than are shown in Fig. S6, domain nucleation and switching occurs randomly

throughout the specimen when current pulses are injected, which precludes unambiguous tracking

of the current-driven motion of individual magnetic domain walls.

S4. Comparisons with high speed domain wall dynamics in other systems. As discussed in the

Main Text, saturation of the DW velocity as a function of driving field (or current-induced

effective field) can have several origins. What distinguishes the present observations from past

Submitted Manuscript: Confidential

23

works is the observation of a saturation in the velocity v(j) that can be attributed to relativistic-like

effects rooted in the sine-Gordon equation. Conventional DW dynamics models predict a

maximum steady-state velocity (the Walker velocity) that scales in proportion to the maximum

restoring torque in the domain wall (in our case, due to the field Hx that tends to pin the domain

wall in a Neel configuration). This maximum velocity for steady-state motion exists regardless of 5

the nature of the driving effective field, and in the specific case of damping-like spin orbit torque,

it is approached only asymptotically (34, 35), as is well-known (29). One can readily distinguish

between the conventional (Walker) limit and the magnonic limit that is central to the present work

by determining whether an observed saturation in the velocity versus driving field is sensitive to

the DW restoring torque. This is exemplified by Main Text Figs. 2 E,F, which show a transition 10

between domain wall stiffness-limited velocity (where the saturation scales in proportion to Hx),

and the magnonic limit, where the maximum velocity no longer depends on the strength of the

restoring torque.

Other instances of a velocity saturation have also been reported in the literature, such as

the saturation of the velocity of field-driven domain walls reported in Ref. (59), which is contrary 15

to the usual drop in velocity for fields above the Walker field. One can see in all such cases that

the saturation in the velocity is related to the precessional limit and not the magnonic limit. In Ref.

(59), Yoshimura et al., reported evidence of a terminal DW velocity in field-driven soliton-like

magnetic DW motion in Pt/Co/Ni layers. They explained the high-speed dynamics through the

addition of an additional energy dissipation mechanism through Bloch lines nucleated within the 20

DW. In the experiments, Yoshimura, et al., measured the DW velocity v(Hz) for DWs with and

without significant DMI, driven by an easy axis driving field Hz. In the latter case, they observed

that the DW velocity increased and then saturated (see Fig. 2c and 2f in (59)), which could not be

Submitted Manuscript: Confidential

24

reproduced by the one-dimensional DW model. Full two-dimensional micromagnetic simulations

were performed to explain the results (2f in (59) for the zero-pinning case). They found that the

Walker velocity increased linearly with the DMI strength as expected, but interestingly, the DMI

also tended to suppress the usual velocity drop-off above Walker breakdown.

In the 1D model, the (average) domain wall velocity in the precessional regime (i.e., above 5

the Walker threshold) is given by 𝑣 = 𝛾Δ𝛼𝐻𝑧, which scales directly in proportion to the damping.

In real domain walls, instead of rigid precession of the DW moment, Bloch lines periodically form

and travel along the wall, causing its orientation to reverse periodically (i.e., the DW azimuthal

angle Φ still changes periodically, but not uniformly). The simulations of Yoshimura, et al, showed

that the DMI tended to cause pairs of Bloch lines to annihilate before they could run along the 10

domain wall. They proposed that the resulting spin wave generation opened an additional

dissipation channel that increased the velocity in the precessional regime, since the velocity is

proportional to dissipation. Similar behavior has been reported elsewhere; e.g., it was shown that

edge roughness (60) or a hard-axis applied field (61) can have essentially the same effect.

It is straightforward to see why the velocity plateau in Yoshimura, et al., is distinct from 15

and mechanistically unrelated to the plateau in velocity that we ascribe to relativistic dynamics:

By definition, 2D dynamical processes related to nucleation and motion of Bloch lines can only

occur when the DW reaches the Walker velocity, 𝑣𝑊 =𝜋

2𝛾Δ𝐻𝐷𝑀𝐼. As shown by Yoshimura, et al.,

the Walker threshold and the velocity plateau beyond this threshold scale in proportion to the DW

“stiffness” (DMI, analogous to Hx in our experiments). Any such precessional behavior, whether 20

1D or 2D, would necessarily be sensitive to this parameter. The insensitivity of the velocity

saturation to Hx in our case (see Main Text Fig. 2, E and F) exclude precessional effects as the

Submitted Manuscript: Confidential

25

origin (which, as is well-known, cannot in any case be driven by damping-like spin orbit torque

(34)).

In the present work, the onset of relativistic dynamics occurs at speeds in the range of

several km/s. Similar velocities have been reported in ferrimagnetic CoGd and CoFeGd rare earth

transition metal alloys near angular momentum compensation (17, 21, 62) but in contrast with the 5

observations here, there has been no report of relativistic effects impacting their dynamics at those

speeds. This can be understood from the fact that the exchange constant, A, which sets the

maximum magnon group velocity, is significantly larger in those metallic systems as compared to

the garnets studied here. As given in Eq. (S5), the maximum magnon group velocity for a

ferrimagnet may be written 𝑣𝑔𝑚𝑎𝑥 =

2𝐴

𝑑𝑆. GdCo alloys have an exchange constant ranging between 10

~2.0 − 8.0 × 10−11 J/m (63–65), which is significantly larger than in BiYIG (𝐴 = 4.15 ×

10−12 J/m). Hence, 𝑣𝑔𝑚𝑎𝑥 is expected to be much larger for CoGd than for BiYIG. Using sublattice

saturation magnetizations 𝑀𝑠𝐶𝑜 = 7.0 × 105 A m−1, 𝑀𝑠

𝐺𝑑 = 6.36 × 105 A m−1, Landé g-factors

gCo = 2.2, gGd = 2.0, and d = 0.4 nm taken from Supplementary Note 9 of Ref. (62), and using a

conservative estimate 𝐴 = 2.0 × 10−11 J/m, we estimate a lower bound 𝑣gmax~14 km/s. This 15

limit is ~3 times larger than the upper bound of the domain wall speed estimated in Ref (60), based

on the magnetization switching speed measured beneath a single focused magneto-optical laser

probe spot. Relativistic effects are therefore not expected to manifest in these materials under the

experimental conditions examined so far in the literature.

20

Submitted Manuscript: Confidential

26

Supplementary Figures

Figure S1: Hard-axis hysteresis loop of GSGG/BiYIG(6.9 nm). Best fit lines of the linear regime

and saturation regime are shown in blue and red, respectively. The intersection is used to estimate 5

the anisotropy field, Hk.

Submitted Manuscript: Confidential

27

Figure S2: (A) Schematic of the domain wall track. Red (blue) regions indicate down (up) net

magnetisation in BiYIG. Yellow regions represent the domain wall nucleation line. Magneto-

optical Kerr effect laser spot indicated by a turquoise circle. (B) spin Hall efficiency 𝝌 as a function 5

of in-plane field 𝑯𝒙.

Submitted Manuscript: Confidential

28

Figure S3: (A) BLS spectra for several in-plane fields Hx for the 29nm BiYIG sample

corresponding measured at an incident angle of = 45o, corresponding to wavevector q||=16.7x106

rad/m. (B) Fits to the Damon-Eshbach (DE) and perpendicular standing spin wave (PSSW) mode

frequencies versus field dependence, using expressions S3, S4 in Materials and Methods. 5

Submitted Manuscript: Confidential

29

Figure S4: Calibration curve to correct the experimental data for 𝑯𝒙-induced DW broadening.

Submitted Manuscript: Confidential

30

Figure S5: Domain wall displacement as a function of pulse width for various in-plane fields

𝐻𝑥 at a current density 𝑗 = 1.2 × 1012𝐴/𝑚2. Solid lines are linear fits to the data points.

0 10 200

10

20

30

40 Hx (Oe)

90

60

50

25

dis

pla

ce

men

t (

m)

pulse width (ns)

Submitted Manuscript: Confidential

31

Figure S6: Saturation velocity 𝑣sat taken from velocity versus current density curves as a

function of in-plane field 𝐻𝑥 for up-down and down-up domain walls in

GSGG/BiYIG(t)/Pt(4.0 nm) films, where 𝑡 = 6.9 𝑛𝑚 and 2.4 𝑛𝑚.

5

10

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