Tailoring exchange couplings in magnetic topological ... · Tailoring Exchange Couplings in Magnetic Topological Insulator/Antiferromagnet Heterostructures Qing Lin He 1†*, Xufeng

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Tailoring Exchange Couplings in Magnetic Topological Insulator/Antiferromagnet

Heterostructures

Qing Lin He1†*, Xufeng Kou1†, Alexander J. Grutter2†, Gen Yin1†, Lei Pan1, Xiaoyu Che1, Yuxiang

Liu1, Tianxiao Nie1, Bin Zhang3, Steven M. Disseler2, Brian J. Kirby2, William Ratcliff II2, Qiming

Shao1, Koichi Murata1, Xiaodan Zhu1, Guoqiang Yu1, Yabin Fan1, Mohammad Montazeri1, Xiaodong

Han3, Julie A. Borchers2, and Kang L. Wang1*

1Department of Electrical Engineering, University of California, Los Angeles, California 90095, USA.

2NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD

20899-6102, USA.

3Beijing Key Lab of Microstructure and Property of Advanced Materials, Beijing University of

Technology, 100124, Beijing, China

*Correspondence to: [email protected]; [email protected].

These authors contributed to this work equally.

Supplementary Information

(a) Structural characterizations of the AFM/MTI structures

We have carried out detailed structural characterizations, including X-ray diffraction (XRD), X-ray

reflectivity (XRR), neutron diffraction, and high-resolution scanning transmission electron microscopy

(HRSTEM) on our samples.

Tailoring exchange couplings in magnetic topological-insulator/antiferromagnet heterostructures

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We begin with a XRD measurements (shown in Fig. S1) on a typical [AFM(4nm)/MTI(7nm)]n=4

superlattice (SL). In addition to the expected GaAs (111) and (222) substrate peaks, we observed a series

of strong diffraction peaks originating in the MTI layers. Specifically, we can index the (003), (006),

(009), (00 15), and (00 18) Cr-doped (Bi,Sb)2Te3 peaks, indicating extremely high crystallographic

quality. On the other hand, peaks associated with interfacial tellurides (CrTe, Cr3Te4, and Cr2Te3) are not

found in the resulting XRD spectrum, as illustrated by the associated powder diffraction files (PDFs).

Crystal symmetry and lattice matching demand that the growth axis of the CrSb layers is (0001).

However, in the PDF of CrSb we can see that the (0001) diffraction peak is forbidden, while the (0002)

and (0004) peaks are present. As expected, we observed clear (0002) and (0004) CrSb peaks and indexed

them correspondingly.

We can also exploit the lower background levels and increased nuclear contrast offered by neutron

diffraction to detect the NiAs-phase CrSb. We observed the CrSb (0002) neutron diffraction peak

displayed in Fig. S9, which confirms the NiAs-phase CrSb in our material system.

We also carried out HRSTEM from the cross-section of our [AFM(4nm)/MTI(7nm)]n=4 SL to

probe the crystalline structure and local interfacial quality, with results shown in Fig. 1a in the main text.

The upper panel of Fig. 1a shows the low-magnification HAADF image of this superlattice. One can

clearly see that the each layer component is well defined even in a large scale, indicating the high

crystalline qualities of both layer components. The lower panel shows the interfacial region in a higher

magnification, in which the crystalline structures of both materials are well distinguished and form their

own independent lattices, confirming high quality epitaxial growth. We have scanned multiple areas of

several different samples and did not observed intermixing or interfacial phases like CrTe, Cr3Te4, or

Cr2Te3, in agreement with the XRD measurements.

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In addition to probing the local interface quality through HRSTEM, we have characterized the

long-range interfacial roughness through XRR measurements. Where HRSTEM provides a picture of the

intermixing over nanoscale areas, XRR allows the estimation of roughness across the entire sample and

will incorporate both short-range sources of interfacial roughness, such as chemical intermixing, and long

range effects such as thickness drift across the sample, conformal roughness (such as waviness in the

film), and terracing of the MTI layers. We find that even when all of these effects are combined across a

surface area of approximately 2 cm2, the superlattices are extremely flat. For example, Fig. S2a shows

fitted XRR result of a typical (AFM/MTI)n=4 SL, along with the associated scattering length density depth

profile as shown in b. Here the long-range roughness of the interfaces is ~0.8 nm, indicating a high-quality,

well-defined superlattice spanning the entire surface.

However, it is not sufficient to simply demonstrate high-quality interfaces. We must demonstrate

that intermixing and interfacial chemical reactions cannot be responsible for the observed magnetic

properties of the (AFM/MTI)n SLs. Therefore, we have also performed XRR measurements on an

effectivity identical sample which incorporates undoped TI, (Bi,Sb)2Te3. The fitted reflectivity and

resulting scattering length density are shown in Figs. S2c and d. Here, we extract a slightly larger

interfacial roughness of ~1.5 nm – higher than the (AFM/MTI)n=4 SL but still indicative of a well-defined,

high-quality superlattice. It is unlikely that the increased roughness is due to a change in sample uniformity

between samples, but it may indicate slightly different chemical intermixing or conformal roughness.

Critically, however, the (AFM/undoped TI)n=4 SL does not exhibit the unusual magnetic properties of the

(AFM/MTI)n=4 SL, strongly suggesting that our observations cannot be accounted for by intermixing or

the formation of intermixing tellurides at the interface.

(b) Magneto-transport signature of exchange bias effect




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Since the composition of the MTI layers are carefully adjusted such that the Fermi level is right at

the bulk band gap, the transport in the MTIs should be dominated by the surface states, which becomes

massive Dirac fermion under magnetization.

To present a comprehensive study on our material system, we have carried out the magneto-transport

experiments on some typical samples at 1.9 K with results summarized in Fig. S5. Figure S5 shows the

anomalous Hall resistance of an AFM(7nm)/MTI(7nm) heterostructure under zero-field- and ±3T-cooled

processes from 300 K to 1.9 K. We observed that a +3T-cooled process shifts the anomalous Hall signals

to the negative field direction by about 3.9 mT while a -3T-cooled process shifts to the positive field

direction by about 8.2 mT, which is reasonably consistent with the exchange bias effect probed by the

SQUID (Fig. 1c in the main text). Interestingly, we also found that after field-cooled process, the shape

of the anomalous Hall resistance evolves from a symmetric square-shape (zero-field cooled) to

asymmetric, as shown in the inset of Fig. S5a. Such an asymmetric square-shaped anomalous Hall

resistance may stem from the multi-channels in the MTI layer, i.e. both the surface and the bulk,

accompanied with the contributions from the adjacent AFM layer. We cannot observe signature of

exchange bias effect from the anomalous Hall resistance of the [AFM(4nm)/MTI(7nm)]n=4 SL as shown

in Fig. S5b. This is consistent with the SQUID measurements as shown in Fig. 1d and Fig. 2 in the main

text.

We have also compared the magneto-transport results among an isolated MTI layer (Fig. S11), the

AFM(7nm)/MTI(7nm) heterostructure (Fig. S5), and an AFM(4nm)/MTI(7nm)/AFM(4nm) trilayer (Fig.

S3), and the [AFM(4nm)/MTI(7nm)]n=4 SL (Fig. S5). We found that significant exchange bias effect can

only be observed in the AFM/MTI heterostructure but not in an isolated MTI layer, AFM/MTI/AFM

trilayer, or the (AFM/MTI)n SL. These results are consistent with those from the SQUID measurements




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(Figs. 1d, g, and Fig. 2), which highlight the different roles played by the interfacial exchange coupling

and effective long-range exchange coupling.

(c) Micromagnetic simulations

To further specify the possible spin textures in both the AFM CrSb and the MTI Crx(Bi,Sb)2-xTe3 in

the trilayer and SL structures, micromagnetic simulations were carried out using the LLG Micromagnetics

Simulator on a Crx(Bi,Sb)2-xTe3/CrSb/Crx(Bi,Sb)2-xTe3 trilayer. The magnetic energies considered in this

model are the Zeeman energy, interfacial and internal exchange energies, dipole energy and anisotropy

energy. Specifically, the effective long-range exchange coupling was modeled by setting the exchange

stiffness at the MTI/AFM interface to be non-zero, while setting the exchange stiffness between sublayers

within CrSb to be negative to respect the neutron diffraction measurement results. Within the AFM, two

sets of interleaving sublayers essentially represented the two sublattices of CrSb, which are

antiferromagnetically coupled. The material magnetic parameters used in the simulation are estimated as

followed: magnetic stiffnesses: AMTI = 0.2 μerg/cm, AAFM = -0.3 μerg/cm, and AIEC = 0.1 μerg/cm,

estimated from Curie, Néel temperatures, and the exchange field.

To compare simulation outputs with experimental data, an M-H hysteresis loop was extracted from

the simulation data, which is shown in Fig. S6a. As shown in this figure, the extracted M-H hysteresis

loop is reasonably consistent to the major features of the measured M-H hysteresis loop as shown in Fig.

1g, both of which exhibit similar double-step switching behaviors. Specific spin textures of both the MTI

and the AFM layers, as shown in Fig. S6b, are found to be significantly modified from their bulk magnetic

structures (particularly the AFM one) in the trilayer. Such spin textures agree well with the PNR and

neutron diffraction observations of a ferromagnetic- or ferrimagentic- like in-plane magnetization with a

dominant out-of-plane antiferromagnetic spin alignment of the AFM layer. The magnetic spins in MTI




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layers near the interfacial regions are found to be canted to lower the interfacial energy. Besides, the

resulting spin texture throughout the whole structure also clearly revealed a Néel-type magnetic domain

wall formed within the AFM layer. It is worth mentioning that setting the MTI/AFM interface exchange

stiffness to be zero results in an absence of the plateau steps in the M-H hysteresis loop, i.e. a single step

switching behavior, which agrees with experimental result when the antiparallel effective long-range

exchange coupling was taken over by a single switching as shown in Fig. 1h.

These simulations assume atomically flat interfaces, highly uniform layer thicknesses, and the growth

of complete CrSb unit cells – all assumptions achievable through molecular beam epitaxy and supported

by the fitting of the neutron reflectivity. The growth of fractional CrSb unit cells or significant thickness

nonuniformity is expected to inhibit the observed antiparallel coupling.

In addition, domain wall width is theoretically proportional to (A/K)1/2, where A and K are exchange

stiffness and anisotropy coefficient, respectively. Therefore, the Néel-type magnetic domain wall may not

be essential as such a spin configuration is strongly dependent on the relative magnitudes of both A and K

of the CrSb.

(d) Magnetic interactions from the perspective of thickness dependences in tri- and bi-layer systems

In the trilayer system, the missing exchange bias and the double switching behavior occur only

when the two AFM surfaces coincide with different AFM sublattices. Specifically speaking, as the two

sublattices of the CrSb AFM spins are arranged layer by layer, when the arrangement is A(BA)n, the top

and the bottom MTI layers experience the same spin. In this case, exchange bias is expected to occur. On

the other hand, when the AFM order is AB(AB)n, the two MTI layers experience an opposite interfacial

exchange coupling, thus leading to the double-switching behavior with a minimized exchange bias. In

most (~70%) MTI/AFM/MTI trilayer samples with various thicknesses, we observed the double-




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switching behavior, and the exchange bias vanishes as expected in the AB(AB)n case. We have

investigated a series of typical MTI/AFM/MTI trilayer samples with different AFM and MTI thickness,

as summarized in Table S1.

In the bilayer system, we observed a significant exchange bias effect as shown in the main text.

Thickness variation was also investigated for the MTI/AFM bilayers and the results are shown in Table

S2. Comparing the tri-layer and bi-layer results, it is clear that the exchange bias in these tri-layer samples

is roughly one order of magnitude smaller than that in the bi-layers, indicating the AB(AB)n type sublattice

is the dominant configuration at the interfaces in these samples.

(e) Dirac fermion spectrum in the (AFM/MTI)n superlattice

The modification of the Dirac fermion spectrum in the (AFM/MTI)n superlattice is mainly induced

by the Cr dopants through a spin-texture modulation due to the interfacial exchange coupling at the

AFM/MTI interface. There are two contributions to the time-reversal-symmetry breaking exchange

coupling: (i) the on-site Hund’s-rule coupling given by the Cr dopants and (ii) the Heisenberg-type

interfacial exchange at the AFM/MTI interface. The Hund’s-rule coupling is between atomic orbitals on

the same site, which is usually at the level of ~eV, while the interfacial Heisenberg exchange is mainly

induced by the nearest-neighbor wave function overlap. Since the wave functions exponentially decay at

the interfaces, the Heisenberg exchange (~meV) is usually orders of magnitude smaller than the on-site

Hund’s rule coupling. Thus, the Hund’s rule coupling given by the Cr dopants dominates the spectrum of

the MTI surface states.

Equally important, it is noted that the magnetization of the doped Cr atoms would tilt when the MTI

layer is adjacent to the AFM layer; in other words, the exchange gap in the MTI layer is also modulated

by the AFM layer indirectly. As shown in Fig. 1f and Fig. S6b, the Néel vectors of the CrSb layer are




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canted to in-plane, while the Cr dopants on the TI side are out-of-plane, aligned with the external field. At

the interface, the Cr spins tilt to the in-plane direction, hence reducing the perpendicular magnetic moment

as well as the exchange gap. Under such circumstances, it is expected to have a smaller anomalous Hall

signal in the AFM/MTI structure as compared with the pure MTI case. Indeed, in our experiments, by

replacing the AFM layers with MTI, the anomalous Hall resistance is found to increase by ~25%, as shown

in Fig. S11c.

(f) Alternative models for fitting the PNR data

One of the most common challenges to the understanding and interpretation of PNR measurements

is the existence of multiple scattering length density profiles which describe the data equally well. In fact,

cases in which only a single unique model fits the data within the measured Q-range are extremely rare.

It is therefore critical to compare multiple fitted models to understand what conclusions may confidently

be drawn from the measurements. To that end, we have attempted to fit the PNR data using a wide variety

of models, and present several below.

First, we optimized the model presented in the main text, which assumes a ferromagnetic moment in

the Cr-doped (Bi,Sb)2Te3 layers as well as the CrSb layers. Additionally, the interfacial magnetization was

allowed to vary independently from either of the two constituent layers. This model resulted in the best

overall fit as shown in Fig. 3(a) in the main text. However, a reasonable fit was also achieved using a

model in which the magnetism within the CrSb was fit as three magnetically independent layers while the

magnetization of the Cr-doped (Bi,Sb)2Te3 at the interfaces was also allowed to vary separately from the

bulk, as shown in Figs. S7g-i. Such a model allows for the possibility of a more complex Néel-type

magnetic domain wall configuration within the CrSb layers. Because the magnetization in the CrSb layer

may vary over extremely short length scales in this model, it is expected that, as long as the average net




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in-plane magnetization within the CrSb layer remains similar, this fit will agree well with the one

presented in the main text throughout the measured Q-range. Thus, it is not surprising that such optimized

model as shown in Figs. S7g-i yields a very similar fit to the one presented in the main text, suggesting

that more complex spin structures within the CrSb cannot be excluded. Although the best fit is achieved

with a complex, oscillatory magnetization depth profile within the CrSb layers, several other CrSb spin

structures are also supportive. These alternative structures include a uniform magnetization in the CrSb

layers, or magnetized CrSb interfaces and purely antiferromagnetic CrSb in the center of the layer. Despite

the variety of possible spin structures supported by the PNR, all models which describe the data are highly

sensitive to the average magnetization in Crx(Bi,Sb)2-xTe3 (40-41 emu/cm3) and CrSb (61-63 emu/cm3)

layers. Although PNR modeling often results in multiple models which well describe the data, in this case

we may explicitly exclude several competing models which provided significantly worse fits to the

observed reflectivity.

In contrast, models in which the magnetization is confined exclusively to the Cr-doped (Bi,Sb)2Te3

layers, shown in Figs. S7a-c, or the CrSb layers, in Figs. S7d-f, clearly fail to describe the measured

reflectometry. For example, in Fig. S7c, we can see that the spin asymmetry is much too large from 0.6-

1.3 nm-1. Furthermore, the spin-splitting is the wrong sign at the second order SL reflection. Similarly, the

model incorporating nonmagnetic (Bi,Sb)2Te3 layers fails quite badly throughout the entire Q-range as

shown in Fig. S7f, with incorrect signal magnitude at both SL peaks as well as insufficient spin splitting

near the critical edge.

All presented models have been optimized within the imposed constraints using the NIST Refl1d

software package29. Fitted thicknesses agree reasonably well with those of the designed SL. For example,

the nominal thickness of each CrSb layer in the SL is 30 Å, while the fitted thickness was 25 Å. It should

additionally be noted that the expected nuclear scattering length densities of Cr-doped (Bi,Sb)2Te3 and




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CrSb are nearly identical, and that the majority of the contrast between layers originates in the changing

magnetic scattering length density. Therefore, the model optimization is far more sensitive to the

magnetization and magnetic layer thicknesses, rather than the structural thicknesses. For the model

presented in the main text, we find in-plane magnetizations of approximately 40 emu/cc in the Cr-doped

(Bi,Sb)2Te3 and 63 emu/cc in the CrSb for an in-plane applied field of 700 mT. In summary, the PNR

measurements clearly indicate a spontaneous net magnetization in both the Cr-doped (Bi,Sb)2Te3 and

CrSb layers. Within the CrSb layers, both a simple model with constant magnetization and a more complex

Néel-type magnetic domain wall configuration describe the data equally well.

For the AFM/MTI/AFM trilayer case, a similar approach was also utilized to confirm the magnetic

ordering in the AFM layers. The model used in main text is constructed based on two minimally

magnetized AFM (5 emu/cc or less) and a magnetized MTI layer, which well describes the PNR data. On

the other hand, models with both magnetized AFMs and a magnetized MTI layer were also used to fit the

data (Fig. S8a-c). However, even after parameter optimization such a model clearly fails to describe the

PNR spectra of the trilayer (particularly in the regions of 0.25-0.5 and 1.0-1.25 nm-1), supporting the

conclusion that the magnetization is confined exclusively within the MTI layer. We further measured the

PNR spectra at 150 K and 265 K, above the TC of the MTI but below the AFM Neel temperature, as shown

in Fig. S8d. These measurements show no net magnetization within any layer of the trilayer structure,

demonstrating that the magnetic susceptibility of the AFM layers is small enough to be completely

neglected and further supporting the assertion that the AFM layers are nonmagnetic in this geometry.

(g) Spontaneous magnetization due to interfacial exchange couplings

The interfacial exchange coupling between CrSb and Crx(Bi,Sb)2-xTe3 in SLs is reflected by the initial

magnetization processes in both longitudinal resistance (Rxx) and Hall resistance (Rxy) measurements.

After ZFC the samples from 300 K to 1.9 K, a perpendicular magnetic field is swept from zero to 3 T for




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an initial magnetization, followed by a major hysteresis loop measurement via cycling the field between

3 T and −3 T. As shown in Fig. S11, the resulting butterfly-like Rxx and square-shaped Rxy mainly come

from the contribution of the Crx(Bi,Sb)2-xTe3 layers, which is illustrated by negligible hysteresis behaviors

of Rxx and Rxy in two control samples, a CrSb (12 nm)/(Bi,Sb)2Te3 (2 nm) bilayer and a [CrSb (4

nm)/(Bi,Sb)2Te3 (7 nm)]n=4 SL. Here, in the CrSb (12 nm)/(Bi,Sb)2Te3 (2 nm) bilayer, the thin (Bi,Sb)2Te3

acts as a buffer layer that provides for the epi-growth of CrSb layer.

Taking the [CrSb/Cr0.16(Bi,Sb)1.84Te3]n=4 SL as an example, the initial magnetization curve of Rxx

(light red curve, Fig. S11b) shows a peak located at a field slightly higher than that in the major loop (red

curve, Fig. S11b), while the corresponding initial magnetization curve of Rxy (light red curve, Fig. S11c)

nearly overlaps with the square-shaped major loop (red curve, Fig. S11c). As these magnetizing features

are obtained after a ZFC process, they are considered a signature of spontaneous magnetization. This is

consistent with the conclusion from PNR of a spontaneous net magnetization in the Cr-doped (Bi,Sb)2Te3

layers. In contrast, in a single MTI layer, the initial magnetization curves in Rxx (light blue curve, Fig.

S11b) starts from a higher resistance state, while the corresponding Rxy initial magnetization curve (light

blue curve, Fig. S11c) deviates from the major loop (blue curve, Fig. S11c) until a ~150 mT field is

applied. These features reveals a multi-domain state in a single MTI layer after ZFC process. To

quantitatively characterize such a spontaneous magnetization behavior, a magnon magnetoresistance

(RMMR) model30, which is widely used to describe a partial magnetization state, is adopted to calculate the

ratio of spontaneous magnetization to the saturated magnetization:

𝑅𝑅𝑀𝑀𝑀𝑀𝑀𝑀 = 𝛼𝛼 𝑀𝑀𝑀𝑀𝑆𝑆

𝐵𝐵

where M is the magnetization along the magnetic anisotropy axis (out-of-plane), MS is the saturated

magnetization, B is the external magnetic field along the magnetic anisotropy axis and 𝛼𝛼 = (𝜕𝜕𝑀𝑀𝜕𝜕𝜕𝜕)𝑠𝑠𝑠𝑠𝑠𝑠 is the

slope of RMMR(B) extracted from a saturated magnetization state as shown in Fig. S11b. Thus, the slope




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of the initial magnetization curve is referred to 𝛼𝛼 𝑀𝑀𝑀𝑀𝑆𝑆

during the initial magnetization process. We then

obtained α ~ 3.04 according to the major Rxx loop. This results in an estimated value of M/MS ~ 0.908

using the slope in initial magnetization curve, which reveals that 90.8% magnetization is spontaneously

established after ZFC from 300 K to 1.9 K in the SL.

Considering that the Néel temperature of CrSb is much higher than the TC of Crx(Bi,Sb)2-xTe3, such

a spontaneous magnetization behavior in Crx(Bi,Sb)2-xTe3 is believed to stem from an induced FM

ordering arisen from the CrSb/Crx(Bi,Sb)2-xTe3 interfaces. While cooling from room temperature to 1.9

K, at the interfacial regions, spins in Crx(Bi,Sb)2-xTe3 start to couple to the adjacent antiferromagnetic

spins in AFM. This results in a spontaneous predetermination of spin direction in the FM to that of the

AFM. In contrast, such a spontaneous magnetization is barely observable in a single layer of

Cr0.16(Bi,Sb)1.84Te3 layer as the corresponding Rxx and Rxy do not exhibit a clear hysteresis signature until

the sample is fully magnetized under a large external field (e.g. 3 T). The above observations clearly

demonstrate that interfacial exchange interaction between Crx(Bi,Sb)2-xTe3 and CrSb results in a strong

spontaneous magnetization in the absence of an external field31.




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Table S1 MTI/AFM/MTI trilayers with different component thicknesses

AFM thickness (nm) MTI thickness (nm) Exchange bias field

(mT) (+2T cooled)

Exchange bias field

(mT) (-2T cooled)

15 7 -1.0±1.5 0.5±1.5

15 14 -1.0±1.0 0±1.0

24 28 -1.5±0.5 1.0±0.5

6 7 -0.5±1.5 1.0±1.5

4 7 0±1.5 0.5±1.5

Table S2 AFM/MTI bilayers with different component thicknesses

AFM thickness (nm) MTI thickness (nm) Exchange bias field (mT)

(-2T cooled)

15 7 11.0±1.0

15 14 11.2±1.0

24 28 10.8±5.0

6 7 10.9±1.5

4 7 11.0±1.0

1 7 0±1.5




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Fig. S1

a, XRD pattern obtained from an [AFM(4nm)/MTI(7nm)]n=4 superlattice. b-e, Standard XRD powder

pattern files (PDFs) of CrSb, CrTe, Cr3Te4, and Cr2Te3, respectively.




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Fig. S2

X-ray reflectivity data obtained from a, an (AFM/MTI)n=4 superlattice and b, the corresponding depth

profiles of real/complex scattering length densities, which best fitted the XRR data. c and d, presenting

the corresponding content for an (TI/MTI)n=4 superlattice.




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Fig. S3

Hall resistance measurements on three AFM/MTI/AFM trilayers. All the AFM layers are fixed to be 4 nm

thick while the intervening MTI thickness is decreased from 21 nm to 14 nm and 7 nm in order to

investigate the origin of the proximity-induced magnetic ordering enhancement. The magnetic ordering

enhancement can be reflected by HC of the Hall resistance, which is found to increase along with the

decrease of the MTI thickness. Such a dependence indicates that the enhancement is dominated by the

interface between the MTI and the AFM layers, implying the important role played by the surface Dirac

fermion from the MTI.




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Fig. S4

a-e, Exchange bias effect and vertical shifts in M-H loops obtained from an AFM/MTI bilayer at different

temperatures. An out-of-plane magnetic field of ± 1.5 T is used in the field-cooled process. f, A summary

of temperature-dependent-exchange bias field.




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Fig. S5 Anomalous Hall resistance measurements obtained from a, an AFM(7nm)/MTI(7nm)

heterostructure and b, an [AFM(4nm)/MTI(7nm)]n=4 SL after zero-field cooled and ±3T cooled processes.

The insets show the low field region of the main panels in finer scales with arrows showing the exchange

bias fields; however, the exchange bias effect is vanishing in the SL. These results are consistent with the

SQUID measurements in Figs. 1 and 2 of the main text.




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Fig. S6

Spin textures obtained from micromagnetic simulations on a MTI/AFM/MTI trilayer. a, Out-of-plane M-

H hysteresis loops extracted from the spin textures in the simulations. Antiparallel effective long-range

exchange coupling, which is signified by a two-step switching behavior (left), can only be observed when

setting the MTI/AFM interface exchange stiffness to be non-zero (like 0.1 uerg/cm), while a zero one can

only result in a single switching M-H hysteresis loop (right). b, Calculated spin textures refer to the three

states as indicated by three black spots in the left M-H hysteresis loop of (a), respectively. The light grey

and yellow regions indicate the MTI and AFM layers, respectively.




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Fig. S7

Three alternative different models used to fit the PNR data. For an example, a shows the structural and

magnetic scattering length densities constructed in the model in which the magnetization is confined

exclusively to the MTI layers. b shows the corresponding fitting results to the PNR data in a while c shows

the corresponding fittings result to the spin asymmetries, respectively. Corresponding PNR data and

fittings are shown in d-f, the magnetization is confined exclusively to the AFM layers, and g-i, a more

complex oscillatory spin structures within the AFM. In the former two models, i.e. a and d, the fittings

clearly fail to describe the measured reflectometries, while in the last model, g, a good fitting to the data

is obtained.




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Fig. S8

A model with a magnetized AFM layer used to fit the PNR data of an AFM/MTI/AFM trilayer. a, PNR

data obtained from such a trilayer along with a fit based on the model demonstrated in (b). b, Structural

and magnetic scattering length densities used in this model, which contains two magnetized AFM and a

magnetized TI layers. c, To carefully check the fitting quality, spin asymmetry of the resulting PNR data

along with the fitting is plotted together. In this model, the fitting clearly fail to describe the measured

reflectometry. d, Spin asymmetry of the AFM/MTI/AFM trilayer measured at 150 K and 265 K, which

statistically shows that the nonmagnetic nature of the AFM layers when the temperature is above TC of

the MTI layer.




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Fig. S9

Growth axis (c-axis) polarized neutron diffraction of the (0002) CrSb peak for a

[CrSb/Cr0.16(Bi,Sb)1.84Te3]n=15 SL. Both spin-flip and non spin-flip channels were collected for neutron

polarization (a) in the film plane and (b) along the scattering vector. In both cases, no significant spin-flip

scattering is observed, demonstrating that there is no magnetic contribution to the observed peak.




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Fig. S10. Arrott plots of Hall resistance

Rxy obtained from various magnetic

structures: a, a MTI single layer, b, an

AFM/MTI bilayer, [AFM/MTI]n SLs

with different period n. c, n=2. d, n=4. e,

n=6. f, n=8. g, n=10.




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Fig. S11

Characterization of spontaneous magnetization obtained through magneto-electrical measurements at 1.9

K with schematic shown in a. The resulting the longitudinal (Rxx) and the Hall resistances (Rxy) are shown

in b and c. Four samples are used in the magneto transport measurements: an AFM (12 nm)/ TI (2 nm)

bilayer, and an [AFM (4 nm)/ TI (7 nm)]n=4 SL, a MTI single layer, and an [AFM (4 nm)/ MTI (7 nm)]n=4

SL, as denoted close to the resulting curves. After a ZFC from 300 K to 1.9 K, in the (AFM/MTI)n=4 SL,

the initial magnetization curve of Rxx (light red curve in b) shows a peak located at a field slightly higher

than that in the major loop (red curve in b), while the corresponding initial magnetization curve of Rxy

(light red curve in c) nearly overlaps with the square-shaped major loop (red curve in c), acting a signature

of spontaneous magnetization. This is consistent with the conclusion from PNR of a spontaneous net

magnetization in the MTI layers. In contrast, in a single MTI layer, the initial magnetization curves in Rxx

(light blue curve in b) starts from a higher resistance state that refers to a multi-domain state, while the

corresponding Rxy initial magnetization curve (light blue curve in c) deviates from the major loop (blue

curve in c) until a ~150 mT field is applied.




25

Fig. S12

Major and minor M-H loops of a [CrSb/Cr0.16(Bi,Sb)1.84Te3]n=2 SL obtained at different temperatures using

SQUID.




26

Fig. S13


SQUID.




27

Fig. S14


SQUID.




28

Fig. S15


SQUID.




29

Fig. S16

Major and minor M-H loops of a [CrSb/Cr0.16(Bi,Sb)1.84Te3]n=10 SL obtained at different temperatures

using SQUID.




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