electric-field-induced spin disorder-to-order transition ... · ko, kwang-eun kim, yong-jin kim,...

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Supplementary Information Electric-field-induced spin disorder to order transition near a multiferroic triple phase point Byung-Kweon Jang, Jin Hong Lee, Kanghyun Chu, Pankaj Sharma, Gi-Yeop Kim, Kyung-Tae Ko, Kwang-Eun Kim, Yong-Jin Kim, Kyungrok Kang, Han-Byul Jang, Hoyoung Jang, Min Hwa Jung, Kyung Song, Tae Yeong Koo, Si-Young Choi, Jan Seidel, Yoon Hee Jeong, Hendrik Ohldag, Jun-Sik Lee, and Chan-Ho Yang*

* E-mail: [email protected]

Electric-field-induced spin disorder-to-ordertransition near a multiferroic triple phase point

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS3902

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http://dx.doi.org/10.1038/nphys3902


The fitting function of temperature dependent c-axis lattice parameters

The anomaly of c-axis lattice parameter at TN was interpreted using the exchange striction

effect. The length contraction occurs in proportional to the square of magnetic order parameter,

2/l l m S1. Provided the magnetic order parameter follows a power law with an critical

exponent (c) with respect to reduced temperature below TN, we can write the length

contraction due to exchange striction for T < TN:

22 N/ (1 / ) cl l m T T . (S1)

Considering the overall linear temperature background, we can write the c-axis lattice

parameter as a function of T at an Néel temperature T':

20 1 2

0 1

(1 / ) for( , )

for

cc c T c T T T Tc T T

c c T T T

. (S2)

In our model, we took into account spatial inhomogeneity arising from compositional

fluctuation. Thus, the temperature-dependent curve should be convoluted by the following

Gaussian distribution of Néel temperatures centered at TN with a standard width (w):

2N

2

( )1( ) exp[ ]22

T Tg Tww

. (S3)

Accordingly, the final form of our model can be written as:

*( ) ( , ) ( )c T c T T g T dT

. (S4)

The critical exponent c was set to be 0.25 which is close to the middle value between that (1/8)

of two-dimensional Ising model and that (0.365) of three-dimensional isotropic Heisenberg

modelS2 considering the highly elongated anisotropic structure and the easy-axis of spin within

in-plane. The other parameters 0 1 2 N, , , ,c c c T w were determined by fitting the data to the

model and the determined fit parameters were summarized in Table S1.

S1. Chikazumi, S. & Charap, S. H. Physics of Magnetism (Krieger Publishing, Malabar, Florida, 1978).

S2. Guillou, J. C. & Zinn-Justin, J. Phys. Rev. B 21, 3976-3998 (1980).

La doping ratio 0% 5% 7.5% 10% 12.5% 15% 20%

c0 (Å) 4.692 4.686 4.679 4.681 4.669 4.673 4.666

c1 (10-5Å/K) -5.50 -8.16 -4.62 -5.12 -3.13 -3.85 -5.04

c2 (10-2Å) -5.73 -5.07 -3.48 -3.82 -2.52 -2.34 -3.33

TN (K) 357.8 314.5 303.0 272.1 303.4 318.7 324.5

w (K) 2.0 2.2 3.0 4.5 20.0 25.2 35.3

Table S1: The fit parameters of temperature-dependent c-axis lattice parameters.

2 NATURE PHYSICS | www.nature.com/naturephysics

SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS3902



The fitting function of temperature dependent c-axis lattice parameters

The anomaly of c-axis lattice parameter at TN was interpreted using the exchange striction

effect. The length contraction occurs in proportional to the square of magnetic order parameter,

2/l l m S1. Provided the magnetic order parameter follows a power law with an critical

exponent (c) with respect to reduced temperature below TN, we can write the length

contraction due to exchange striction for T < TN:

22 N/ (1 / ) cl l m T T . (S1)

Considering the overall linear temperature background, we can write the c-axis lattice

parameter as a function of T at an Néel temperature T':

20 1 2

0 1

(1 / ) for( , )

for

cc c T c T T T Tc T T

c c T T T

. (S2)

In our model, we took into account spatial inhomogeneity arising from compositional

fluctuation. Thus, the temperature-dependent curve should be convoluted by the following

Gaussian distribution of Néel temperatures centered at TN with a standard width (w):

2N

2

( )1( ) exp[ ]22

T Tg Tww

. (S3)

Accordingly, the final form of our model can be written as:

*( ) ( , ) ( )c T c T T g T dT

. (S4)

The critical exponent c was set to be 0.25 which is close to the middle value between that (1/8)

of two-dimensional Ising model and that (0.365) of three-dimensional isotropic Heisenberg

modelS2 considering the highly elongated anisotropic structure and the easy-axis of spin within

in-plane. The other parameters 0 1 2 N, , , ,c c c T w were determined by fitting the data to the

model and the determined fit parameters were summarized in Table S1.

S1. Chikazumi, S. & Charap, S. H. Physics of Magnetism (Krieger Publishing, Malabar, Florida, 1978).

S2. Guillou, J. C. & Zinn-Justin, J. Phys. Rev. B 21, 3976-3998 (1980).

La doping ratio 0% 5% 7.5% 10% 12.5% 15% 20%

c0 (Å) 4.692 4.686 4.679 4.681 4.669 4.673 4.666

c1 (10-5Å/K) -5.50 -8.16 -4.62 -5.12 -3.13 -3.85 -5.04

c2 (10-2Å) -5.73 -5.07 -3.48 -3.82 -2.52 -2.34 -3.33

TN (K) 357.8 314.5 303.0 272.1 303.4 318.7 324.5

w (K) 2.0 2.2 3.0 4.5 20.0 25.2 35.3

Table S1: The fit parameters of temperature-dependent c-axis lattice parameters.





Landau phenomenological theory

We built up the phenomenological Landau free energy considering competition between the

structural/ferroelectric, magnetic ordering, and spin-lattice coupling energiesS3. The

monoclinic tilt angle was experimentally determined with x-ray reciprocal space maps (RSM)

as a function of La substitution ratio x, as explained in the Figs. S2-4. Although the should

be an order parameter in a strict sense, the larger energy scale of the terms directly related to

compels it to its pristine value preferentially and it is less modified by the other structural and

magnetic transitions. In the context, we note that temperature-dependent monoclinic to

tetragonal transition in pure BFO occurs at the higher temperature of 700 K with a quick

decrease of near the transition temperature and that does not vary severely with the

temperature in the range of measurementS4. Even the MA to MC structural transition across

~370 K only changes the value of from 2.07° to 1.99°20. At x = 0.1, we also checked the value

of at 10 K was nearly the same as the value at 400 K within an error of 10 %, which justified

our assumption. However, we must be cautious when considering the samples near x ~ 0.2 in

the regime of a quantum phase transition, because all of the energy scales that include the -

related terms are significantly smaller and are more likely to be comparable with one another.

In the model calculation, the bare ferroelectric reorientation transition temperature between the

MC and MA ferroelectric phases was expected to be coupled to the monoclinic tilt angle and

was written as R K( ) 182.3 ( )T x x approximately up to the first order term in

expansions. Besides, the bare antiferromagnetic transition temperature TN was fixed to 360 K

because the A-site trivalent ion substitution is expected to not severely change the TN. B-site

random substitutions with other magnetic/non-magnetic ions tend to prominently increase the

magnetic frustration, leading to the suppression of magnetic long-range orders and the

appearance of spin/cluster glasses. However, A-site isovalent substitutions without doping

carriers are expected to influence on the magnetic exchange interaction indirectly by

controlling the Fe3+-O2--Fe3+ bond anglesS5. In the cases, strong antiferromagnetic

superexchange interaction through Fe3+-O2--Fe3+ is robust to A-site substitutions.

For the time-dependent simulation at x = 0.1 and T = 300 K near the triple phase point,

we tracked the order-parameters (, L) over time in a quasi-static manner, changing the external

electric field sinusoidally. The hysteresis curves were simulated through sequential iterations

at the following rates:

0, , cos ,d F dL F dE E tdt dt L dt

(S5)

where the kinetic coefficient = 0.5 and a time step t = 0.01. The electric field sweeping rate

= 2/150 was thus slow enough to find the instantaneous local minimum. After the

application of an electric field along [100], the remnant state became an antiferromagnetic

phase as a result of switching from the original paramagnetic phase, indicating the existence of

multiple competing phases near the triple phase point. Competition between various magnetic,

ferroelectric, structural phases will show versatile phenomena such as phase separation,

hysteresis, and large susceptibilities. To get a hint on the underlying states near the triple phase

point, we calculated the free energies in the order parameter space in the environments of zero

or finite electric fields as plotted in Figs. S6 and S7. The following free energy expressed as a

function of Px and Py is equivalent to the free energy expanded using and in the main text:





4 4 2 2 8 8 6 2 2 6 4 40 10

2 40 1

2 2 2 2 4 4 2 2

( ( ))( 6 ) ( 12 12 38 )4 8

( )2 4

( ) ( 6 )

.

R x y x y x y x y x y x y

N

x y x y x y

x x y y

a aF F T T x P P P P P P P P P P P P

b bT T L L

L P P L P P P PE P E P

S3. Tolédano, J.-C. & Tolédano, P. The Landau Theory of Phase Transitions: Applications to Structural, Incommensurate, Magnetic, and Liquid Crystal Systems (World Scientific, Singapore, 1987).

S4. Beekman, C. et al. Phase transitions, phase coexistence, and piezoelectric switching behavior in highly strained BiFeO3 films. Adv. Mater. 25, 5561-5567 (2013).

S5. Yang, C.-H., Kan, D., Takeuchi, I., Nagarajan, V. & Seidel, J. Doping BiFeO3: approaches and enhanced functionality. Phys. Chem. Chem. Phys. 14, 15953-15962 (2012).

The relation between the effective in-plane electric field and the resultant

piezoresponse vector explored by angle-resolved PFM technique

The electrical poling and the subsequent vector piezoresponse measurements S6, S7 were

performed employing a commercial atomic force microscope AFM (AIST-NT Smart SPM

1000) at room temperature under ambient conditions for a 13-nm-thick La10%-doped BFO

thin film (Fig. S12). Conductive diamond-coated silicon cantilevers (force constant = 48

N/m, radius of curvature = 35 nm, resonance frequency = 420 kHz) were used to

write and image the resulting ferroelectric domain patterns. For electrical poling (axis

of the cantilever is parallel to [010] crystallographic direction of the sample), a set of four

concentric square boxes were written via applying appropriate electrical biases (i.e. -5V for

electrical writing of outer square box comprising of 1.2 x 1.2 μm2 area, and +4V for electrical

writing of inner square box comprising of 0.8 x 0.8 μm2 area) to the conductive AFM tip.

During electrical writing of the inner square boxes, the tip scanning angle was varied from one

set of concentric inner square frame to the other so as to align the effective in-plane trailing

field of the AFM tip (i.e. determined by slow scan direction45) along directions making an angle

of 45°, 60°, 75° and 90° respectively. Typical scanning parameters used during electrical

writing are as follows: scan rate = 1.1 Hz, scan point and scan lines: 256 x 256 points, and

nominal loading force 50-80 nN.

After electrical writing of the four concentric square frames with inner boxes prepared

by effective in-plane electric fields pointing along directions of 45°, 60°, 75°, and 90°

respectively, vector PFM measurements were performed in the resonance enhanced modeS8.

For each electrically prepared pattern (i.e. concentric square box), both out-of-plane and in-

plane piezoresponse images were acquired using the same tip at eight distinct orientations (i.e.

0°, 41°, 88°, 141°, 176°, 231°, 270° and 315°) between the axis of the AFM cantilever and





[010] crystallographic direction of the sample. These different imaging geometries (i.e.

orientations between the cantilever axis and [010] direction of the samples) were achieved via

physical rotation of the sample with respect to the axis of the cantilever. For PFM imaging, the

typical frequency range for the applied ac imaging voltage in the out-of-plane and in-plane

PFM mode was in the range of 1100-1400 kHz and 1500-1800 kHz respectively, with an

amplitude of 1.0 V (peak-to-peak). During piezoresponse imaging, scanning parameters were

the same as mentioned previously except for the scan rate, which was 0.9 Hz.

The angle-resolved analyses for the measured in-plane PFM images were performed

to visualize local in-plane piezoresponse vectors S7 (Fig. S13). Each set of eight in-plane PFM

images were aligned to correct pixel misalignment arising from tip-drift issues during the

measurements. Several particular positions including the step terraces and the poling box edges

were selected as reference points to find the coordinate conversion matrices amongst the

images. The angle-resolved PFM signals per site were fit to a sinusoidal curve with respect to

the tip orientation angle, thereby determining the magnitude and direction of local in-plane

piezoresponse vector. The in-plane PFM signal per site was the average value of 3-by-3

neighboring pixels (the physical size corresponds to 23.4nm-by-23.4nm) to improve statistics

in the sinusoidal fitting. Fitting reliability can be inspected in the representative fitting curves

chosen from the 5 by 5 points of each vector map.

Fig. S14 shows the relation between the effective in-plane electric field and the

resultant in-plane piezoresponse vector. The resultant in-plane PFM vectors were subtlely

influenced by surface damages and the poling boundary effect whereby effective in-plane field

was poorly defined in such small poling boxes. Spatial fluctuation can be severe in this

susceptible material closer to the tetragonal structure as compared to pure BFO. We defined

polygon areas where the effects were minimized and the resultant piezoresponse vectors were

relatively uniform, as displayed in the vector PFM maps. Only the data inside the polygon were

used in estimating the net in-plane piezoresponse vector.

S6. Kalinin, S. V. et al. Vector piezoresponse force microscopy. Microsc. Microanal.12, 206–220 (2006).

S7. Chu, K. et al. Enhancement of the anisotropic photocurrent in ferroelectric oxides by strain gradients. Nature Nanotechnol. 10, 972-979 (2015).

S8. Rodriguez, B. J., Callahan, C., Kalinin, S. V. & Proksch, R. Dual-frequency resonance-tracking atomic force microscopy. Nanotechnology 18, 475504 (2007).





Consideration of possible involvement of oxygen vacancy migration during

electrical poling

Defect chemistry: We address what kinds of defects can exist and how they can be related to

electronic carrier concentration. The ionized oxygen vacancy ( OV 2e ) is regarded as a double

electron donor and the existence of oxygen vacancies results in electron doping into the system

leading to decrease of the valence state of Fe ions (e.g. partially change Fe3+ to Fe2+). Besides,

fully ionized vacancies of three-valent cations (Bi3+, La3+, Fe3+ in our case) play a role of

acceptors creating three holes per defect (e.g. BiV 3h ). The balanced coexistence of these

donor/acceptor defects can cancel the role of each dopant maintaining the valence state of Fe3+.

If we assume the oxygen vacancy is mobile in an electric field, the electrically-formed region

is subject to a non-equilibrium distribution of oxygen vacancies leading to locally p-type or n-

type doping states depending on local oxygen vacancy concentration relative to the cation

vacancies concentration.

The quantitative thermodynamic analysis of defect formation in BFO was reported

based on the Kröger method of quasi-chemical reactions S9. According to the defect chemistry

analysis, bismuth vacancies are a majority of defects in BFO and they are compensated by

spontaneous formation of ionized oxygen vacancies in oxygen partial pressures (less than a

few atm). This self-compensation tendency becomes more favorable in nanostructures like

films and wires. The theoretical work has also estimated that the concentrations of bismuth

vacancies and oxygen vacancies are less than 1016 cm-3 for BFO in thermodynamic equilibrium

at an oxygen partial pressure of 1 atm. We know the value of concentration is a pretty small

number taking into account that stoichiometric BFO contains oxygen ions as much as 4.8 x

1022 cm-3.

The likely mechanism for the electroresistive switching: The observed conduction enhancement

in the positively sample-biased region (Fig. S18) can be explained in the context of the oxygen

vacancy migration scenario. Although it is challenging to experimentally quantify the oxygen

vacancy concentration in the as-grown state, we may get a sense of it by comparing with a

better-known system such as Ca-doped BiFeO3, which has shown similar resistive switching

behavior associated with oxygen vacancy migration. Moreover, the defect concentration can

be estimated quite accurately. The chemical composition in the as-grown state is Bi1-xCaxFeO3-

( =x/2). To keep the valence state of Fe3+, oxygen vacancies are spontaneously formed and

the vacancy content () corresponds to half of the Ca-substitution ratio (x/2) for this aliovalent

Ca2+ substitution into Bi3+. For example, the Ca10%-doped sample contains a lot of oxygen

vacancies as much as 8 x 1020 cm-315, 48. The creation of oxygen vacancies in the system are

thermodynamically stable and their concentration is hardly altered (can be reduced by only less

than 10%) even by thermal annealing in a high-pressure oxygen environment at ~125 bar S10.

On the assumption that the electronic carrier’s mobility of La-doped BFO samples is

comparable with that of Ca-doped samples, the conductive AFM results (compare Fig. S18b

with the Fig. 2a of Ref. 15) give a rough estimation of the modulation of electronic carrier

density as well as oxygen vacancy concentration. According to the previous conductive AFM

measurement on Ca10%-doped BFO (~300 nm in film thickness), the upward poling resulted

in a large enhancement of current ~300 pA at a tip bias of -2.5 V15. The conduction modulation

(the difference between the high conduction and the as-grown state conduction) is proportional

to the oxygen-vacancy-concentration modulation, and the amount of the vacancy modulation

is comparable to the initial oxygen vacancy concentration in the as-grown state48. As compared

with the Ca10%-doped sample having ~0.05, i.e. Bi0.9Ca0.1FeO2.95, a two orders of magnitude

smaller conduction modulation is observed in La10%-doped BFO gives ~0.05/100, i.e.,





Bi0.9La0.1FeO2.9995. We would like to mention that such a small oxygen vacancy content is

conservatively evaluated in terms of the fact that the sample thickness of La-doped BFO is 20

times thinner than that of the Ca-doped one regardless of the use of a similar measuring voltage

and in addition the poling electric field strength used for La-doped BFO is higher. Although

we have barely detected the conduction modulation by using a highly sensitive current

amplifier at a few pA level, the deduced modulation of oxygen vacancies is too tiny to explain

the observed dichroism. Furthermore, we came to know that the La-doped samples are more

resistive than the pure BFO sample, further supporting our above argument (see Fig. S20). It

is likely due to the fact that the isovalent substitution of La ions can partially suppress the

creation of bismuth vacancies as well as oxygen vacancies S11.

Symmetry considerations for the observed dichroism: Even if the oxygen vacancy migration

scenario can successfully explain the resistive switching behavior, the oxygen vacancy

migration is not compatible with the observed dichroism according to symmetry considerations.

The L–edge absorption peak of Fe ions undergoes a blue shift when holes are doped (or the

number of electrons are reduced), while it is shifted toward the left when electrons are doped

(or the number of holes are reduced) S12. If the oxygen vacancy migration were the main origin,

the dichroism spectrum (PP-UP) in the positively-poled (PP) area relative to the absorption

acquired at the unpoled (UP) area should have been opposite to the dichroism (NP-UP) in the

negatively-poled (NP) area rather than the observed same polarity in Fig. 3c.

Charged point defects like oxygen vacancies show odd (with respect to symmetry)

responses to the direction of electric field. For example, in the resistive switching behavior

observed in doped BFOs, the positive sample bias induces a more conductive state but the

negative bias works in the opposite way. On the other hand, our observation in the dichroism

has the even character in response to an applied electric field (so called axial dependence). The

axis of electric field is the main control parameter but the sign of direction is not important. To

create the magnetic ordering, the in-plane electric field should be parallel to the crystal

axes; it is clear that whether the direction is [100] or [-100] doesn’t matter considering the four-

fold rotational symmetry around the film normal. Similarly, the written effect can be erased,

provided the in-plane field satisfies an axial condition i.e. parallel to one of the axes.

Because of this, the second electric polings by the same out-of-plane electric field over

the regions with a magnetic ordering lead to completely different results depending on the axis

of in-plane electric field (compare Fig. 3b with Fig. 4d). The electric poling made with a slow-

scan-axis of [010] maintained the magnetic ordering (Fig. 3b), while the poling along a slow-

scan-axis of [1-10] erased the magnetic ordering with returning the state back to the original

as-grown state (Fig. 4d). This dramatic slow-scan-axis dependence regardless of the same

initial state of sample and the same out-of-plane electric field direction leads us to the physical

origin associated with the structural phase competition.

With this in mind, although the oxygen vacancy migration can be partially involved in

explaining a secondary difference between the PP-UP and NP-UP dichroisms and may

originate in a weak (a few pA level) resistive switching behavior, it is not the primary

mechanism to induce the observed dichroism.

S9. Tchelidze, T., Gagnidze, T. & Shengelaya, A. Thermodynamic analysis of defect formation in BiFeO3. Phys. Status Solidi C 12, 117-119 (2015).

S10. Masó, N. & West, A. R. Electrical properties of Ca-doped BiFeO3 ceramics: From p-type semiconduction to oxide-ion conduction. Chem. Mater. 24, 2127-2132 (2012).

S11. Simões, A. Z., Garcia, F. G., & dos Santos Riccardi, C. Rietveld analysis and electrical properties of lanthanum doped BiFeO3 ceramics. Mater. Chem. Phys. 116, 305-309 (2009).

S12. Tan, H., Verbeeck, J., Abakumov, A. & Van Tendeloo, G. Oxidation state and chemical shift investigation in transition metal oxides by EELS. Ultramicroscopy 116, 24-33 (2012).





Supplementary Figures

Figure S1: The dielectric anomaly in BLFO. a, Temperature dependence of the capacitances measured at 100 kHz. All of the measurements were performed using the interdigitated electrodes described in the inset upon cooling. The polynomial backgrounds were presented by dashed lines. b, The dielectric loss tangents. The data were vertically shifted to avoid overlapping. Provided that the effective area of the interdigitated electrode capacitor is assumed to be the product of total length of electrode and film thickness, 1 pF in capacitance corresponds to a relative permittivity of 347.

0 100 200 300 40026

28

30

32

34

36

C

apac

itanc

e (p

F)

Temperature (K)

N = 40×2

2.7 mm

20 μm

• • • • • •

0 100 200 300 4000.00

0.01

0.02

0.03

Tan

(δ)

Temperature (K)

20% 15% 12.5% 10% 7.5% 5% 0% LAO

ba

Figure S2: X-ray diffraction studies for a series of BLFO thin films. Distorted pseudocubic unitcells of the films were clarified with RSMs for (103) and (113) reflections at ambient conditions, using X’Pert PRO MRD (PANalytical) with Cu K1 radiation. a, X-ray -2 scans around LAO (002). Dashed lines indicate the (002) peak positions of the tetragonal-like BLFO phase. b, A schematic of the pseudocubic unitcell of BLFO. c, The RSMs around the (103) peaks of BLFO thin films. d, The RSMs around the (113) peaks of BLFO thin films. The reciprocal lattice unit (r.l.u.) is defined as 2/3.789 Å-1, the reciprocal value of the pseudocubic lattice parameter of the LAO substrate. The horizontal dashed lines in the RSMs indicate the pseudotetragonal L positions presumed from the peak positions in the -2 scans.

b

c

d

a

0.98 1.00 1.022.362.382.402.422.442.462.482.50

[00L

] (r.l

.u.)

0.98 1.00 1.02

0.98 1.00 1.02[HH0] (r.l.u.)

0.98 1.00 1.02 0.98 1.00 1.02

0.981.001.02

2.362.382.402.422.442.462.482.50

[00L

] (r.l

.u.)

0.981.001.02

0.981.001.02

0.981.001.02

0.981.001.02

[H00] (r.l.u.)

( )( )

( )

( )

( )

( )( )

( )

( )

( )

( )

( )

( )

( )

( )( )

( )( )

( )( )

( )( )

( )

( )

( )

( )

( )

( )

0% 5% 10% 15% 20%

0% 5% 10% 15% 20%

β

δx

y

z

a

c

b36 38 40 42 44 46 48 50

100103106109

1012

Inte

nsity

(cps

)

2 (degree)

LAO ( 2)T-BFO ( 2)

La 0%La 5%La 10%La 15%La 20%





Supplementary Figures

Figure S1: The dielectric anomaly in BLFO. a, Temperature dependence of the capacitances measured at 100 kHz. All of the measurements were performed using the interdigitated electrodes described in the inset upon cooling. The polynomial backgrounds were presented by dashed lines. b, The dielectric loss tangents. The data were vertically shifted to avoid overlapping. Provided that the effective area of the interdigitated electrode capacitor is assumed to be the product of total length of electrode and film thickness, 1 pF in capacitance corresponds to a relative permittivity of 347.

0 100 200 300 40026

28

30

32

34

36

Cap

acita

nce

(pF)

Temperature (K)

N = 40×2

2.7 mm

20 μm

• • • • • •

0 100 200 300 4000.00

0.01

0.02

0.03

Tan

(δ)

Temperature (K)

20% 15% 12.5% 10% 7.5% 5% 0% LAO

ba

Figure S2: X-ray diffraction studies for a series of BLFO thin films. Distorted pseudocubic unitcells of the films were clarified with RSMs for (103) and (113) reflections at ambient conditions, using X’Pert PRO MRD (PANalytical) with Cu K1 radiation. a, X-ray -2 scans around LAO (002). Dashed lines indicate the (002) peak positions of the tetragonal-like BLFO phase. b, A schematic of the pseudocubic unitcell of BLFO. c, The RSMs around the (103) peaks of BLFO thin films. d, The RSMs around the (113) peaks of BLFO thin films. The reciprocal lattice unit (r.l.u.) is defined as 2/3.789 Å-1, the reciprocal value of the pseudocubic lattice parameter of the LAO substrate. The horizontal dashed lines in the RSMs indicate the pseudotetragonal L positions presumed from the peak positions in the -2 scans.

b

c

d

a

0.98 1.00 1.022.362.382.402.422.442.462.482.50

[00L

] (r.l

.u.)

0.98 1.00 1.02

0.98 1.00 1.02[HH0] (r.l.u.)

0.98 1.00 1.02 0.98 1.00 1.02

0.981.001.02

2.362.382.402.422.442.462.482.50

[00L

] (r.l

.u.)

0.981.001.02

0.981.001.02

0.981.001.02

0.981.001.02

[H00] (r.l.u.)

( )( )

( )

( )

( )

( )( )

( )

( )

( )

( )

( )

( )

( )

( )( )

( )( )

( )( )

( )( )

( )

( )

( )

( )

( )

( )

0% 5% 10% 15% 20%

0% 5% 10% 15% 20%

β

δx

y

z

a

c

b36 38 40 42 44 46 48 50

100103106109

1012

Inte

nsity

(cps

)

2 (degree)

LAO ( 2)T-BFO ( 2)

La 0%La 5%La 10%La 15%La 20%





Figure S3: A structural model to interpret the RSMs. a, The expected peak positions in the (H0L) reflections. b, The expected peak positions in the (HHL) reflections. Introducing the azimuthal distortion angle induces additional peak splits, as compared to the ideal MC- or MA-type peak patterns. The blue and green arrows express peak shifts of as much as 2π tan cosa and

2π tan sinb , respectively. For the interpretation of RSMs, we defined

the primitive vectors of the direct ( ia ) and reciprocal ( ib ) spaces as follows:

1 2 3

1 2 3

( ,0,0), (0, ,0), ( sin cos , sin sin , cos );2π 2π tan cos 2π 2π tan sin 2πsec( ,0, ), (0, , ), (0,0, ).

a a a b a c c c

b b ba a b b c

Figure S4: Crystalline structural information extracted from the x-ray diffraction studies. a, The structural information of the series of BLFO samples is summarized. b, The monoclinic tilt angle () gradually decreases at higher La doping and diminishes near x = 0.2, indicating a doping-driven structural transition from a monoclinic to tetragonal structure. The decreasing tendency follows a power law with an exponent of 0.5.

[H00] [HH0]

[00L

]

2

2

2

2

( )

( )

( )

( )

( )

( )

( )

[00L

]

( )a b

0.00 0.05 0.10 0.15 0.200.0

0.5

1.0

1.5

2.0 Experiment Fitted curve

(

)

La doping ratio, x

x - 0.201|0.5

La doping 0% 5% 10% 15% 20%

Structure MC MC MA-like MA-like MA-like

a (Å) 3.84(7) 3.80(3) 3.78(8) 3.78(8) 3.79(3)b (Å) 3.76(3) 3.76(9) 3.77(0) 3.77(7) 3.78(9)c (Å) 4.64(1) 4.67(3) 4.67(7) 4.65(4) 4.65(2)β (°) 2.0(3) 1.7(5) 1.3(8) 1.0(0) 0.1(5)

δ (°) 0.0(0) 0.0(0) 2(7) 3(6) (23)

V (Å3) 67.15 66.96 66.78 66.59 66.85t (nm) 33 47 42 37 42

a b

Figure S5: Calculated susceptibilities according to phenomenological theory. a, Electric susceptibility ( ) with respect to . b, Magnetic susceptibility ( L ) with respect to the order parameter L of the antiferromagnetic sublattice. Each can be obtained by calculating

21

2

F

or

21

2LF

L

at the global minimum. They are plotted on logarithmic scales.

0 5 10 15 200

100

200

300

400

Tem

pera

ture

(K)

La doping ratio (%)

0 5 10 15 200

100

200

300

400

Tem

pera

ture

(K)

La doping ratio (%)

2

2

a

b





Figure S6: Free energy map in the order parameters space for a Bi0.9La0.1FeO3 sample. a, Free energy map at T = 250 K. b, Free energy map at T = 300 K. The radial axis indicates the antiferromagnetic order and the azimuthal angle represents the crystal distortion angle () with four-fold rotational symmetry at no external electric field. The red squares show global minima and the white circles show meta-stable points. The temperature change at x = 0.1 induces a multiferroic transition between an MC-type ferroelectric state with an antiferromagnetic order and an MA-type ferroelectric state without any spin order.

Figure S7: Electric field dependence of the free energy map at x = 0.1 and T = 300 K. a, Free energy map at E = 600 parallel to [100]. b, Free energy map at E = 600 parallel to [110]. The red square represents the global minimum. Notably, the application of an electric field along [100] gives rise to antiferromagnetic ordering and ferroelectric switching from the MA to the MC-type.

0

45

90

135

180

225

270

315

1.5

1.0

0.5

0.0

L (a.u.) 0

45

90

135

180

225

270

315

MC

MA MA

Free energy (a.u.)

-48

55 Free energy (a.u.)

-63

87MCa b

0

45

90

135

180

225

270

315

1.5

1.0

0.5

0.0

L (a.u.) 0

45

90

135

180

225

270

315

MC

MA MA

Free energy (a.u.)-835

1,330 Free energy (a.u.)-916

1,380MCa b

Figure S8: Scanning x-ray absorption spectroscopy for the electrically poled areas. a-j, Spatially resolved x-ray absorption measurements over the electrically poled areas were performed at the selected photon energies in the range of the Fe L3 edge using a focused soft x-ray source with a linear light polarization along an in-plane axis of [100]. The contrast indicates the absorption change relative to that of the un-poled area. The scale bar represents 5 m. k, The XAS of an un-poled area. The absorptions expressed by the symbols were obtained using microspectroscopy, by integrating the intensities from the un-poled area in a stack of images. In comparison, an XAS curve was measured by sweeping photon energy from a focused x-ray beam over an un-poled area. l, We extracted only the XMLD and removed any

706 708 710 712

-0.02

0.00

0.02

0.04

0.4

0.6

0.8

1.0

Dic

hroi

sm

Photon energy (eV)

XA

S

707.9 eV

708.09 eV

708.76 eV

709.05 eV

709.12 eV

709.15 eV

709.2 eV

709.3 eV

709.4 eV

707.76 eV

Eph→ 8

0

Absorption rate (%

)

PP-UP

(NP-UP)×-1

XAS curveMicro-spectroscopy

a

b

c

d

e

f

g

h

i

j

k

l





Figure S6: Free energy map in the order parameters space for a Bi0.9La0.1FeO3 sample. a, Free energy map at T = 250 K. b, Free energy map at T = 300 K. The radial axis indicates the antiferromagnetic order and the azimuthal angle represents the crystal distortion angle () with four-fold rotational symmetry at no external electric field. The red squares show global minima and the white circles show meta-stable points. The temperature change at x = 0.1 induces a multiferroic transition between an MC-type ferroelectric state with an antiferromagnetic order and an MA-type ferroelectric state without any spin order.

Figure S7: Electric field dependence of the free energy map at x = 0.1 and T = 300 K. a, Free energy map at E = 600 parallel to [100]. b, Free energy map at E = 600 parallel to [110]. The red square represents the global minimum. Notably, the application of an electric field along [100] gives rise to antiferromagnetic ordering and ferroelectric switching from the MA to the MC-type.

0

45

90

135

180

225

270

315

1.5

1.0

0.5

0.0

L (a.u.) 0

45

90

135

180

225

270

315

MC

MA MA

Free energy (a.u.)

-48

55 Free energy (a.u.)

-63

87MCa b

0

45

90

135

180

225

270

315

1.5

1.0

0.5

0.0

L (a.u.) 0

45

90

135

180

225

270

315

MC

MA MA

Free energy (a.u.)-835

1,330 Free energy (a.u.)-916

1,380MCa b

Figure S8: Scanning x-ray absorption spectroscopy for the electrically poled areas. a-j, Spatially resolved x-ray absorption measurements over the electrically poled areas were performed at the selected photon energies in the range of the Fe L3 edge using a focused soft x-ray source with a linear light polarization along an in-plane axis of [100]. The contrast indicates the absorption change relative to that of the un-poled area. The scale bar represents 5 m. k, The XAS of an un-poled area. The absorptions expressed by the symbols were obtained using microspectroscopy, by integrating the intensities from the un-poled area in a stack of images. In comparison, an XAS curve was measured by sweeping photon energy from a focused x-ray beam over an un-poled area. l, We extracted only the XMLD and removed any

706 708 710 712

-0.02

0.00

0.02

0.04

0.4

0.6

0.8

1.0D

ichr

oism

Photon energy (eV)

XA

S

707.9 eV

708.09 eV

708.76 eV

709.05 eV

709.12 eV

709.15 eV

709.2 eV

709.3 eV

709.4 eV

707.76 eV

Eph→ 8

0

Absorption rate (%

)

PP-UP

(NP-UP)×-1

XAS curveMicro-spectroscopy

a

b

c

d

e

f

g

h

i

j

k

l





structural contributions, by subtracting the local absorption of un-poled (UP) area from that of electrically poled areas (PP or NP). The magnetic dichroisms obtained with the microspectroscopic approach match reasonably well with those of the corresponding curves from the focused-beam spectroscopy, showing a major dip at 709 eV. All of the investigations shown here were conducted on a paramagnetic Bi0.9La0.1FeO3 sample at room temperature. We found that this behavior was reproducible in other areas and different samples.

Figure S9: Simulated XAS and dichroisms between the MC phase with antiferromagnetism and the MA phase with paramagnetism fixing light polarization. a, The schematic of unitcell models used in the simulation. The MC phase with a magnetic order has an antiferromagnetic axis along [100] with an in-plane polarziation parallel to [010], which corresponds to the electrically poled area made in Bi0.9La0.1FeO3 by tip scanning with a slow scan axis along [010]. b, Simulation result for XAS and dichroisms when light polarization (Eph) is parallel to [100]. A calcualted dichroism (shown in dark green) between XAS of the competing phases (i.e. MC phase with antiferromagnetism and MA phase with paramagnetism) has a good agreement with the measured spectrum obtained at the same light polization. The difference is primarilly attributed to the presence or absence of magnetic order because MC and MA structural phases that have identically no magnetic order yields more than an order of magnitude smaller dichroism (shown in purple). c, Similar simulations were carried out for light polarizations along [010].

705 710 715 720 725

0.0

0.5

1.0

550

Eph

// [100]

XA

S

Photon energy (eV)

MCAFM

MCPM

MAPM

MCAFM - MAPM

MCPM - MAPM

L2

L3b

705 710 715 720 725

0.0

0.5

1.0

Eph

// [010]

XA

S

Photon energy (eV)

505

MCAFM

MCPM

MAPM

MCAFM - MAPM

MCPM - MAPM

L2

L3c

MA(= 0°)

MC(= 45°)

MCAFM MAPMMCPM

[100]

[010

]

⊙[001]

PL

[010]

[100]

[001]a





Figure S10: X-ray absorption spectroscopy for the detection of antiferromagnetic order. a, The XAS of a pure BFO thin film was measured at T = 300 K across the Fe L2,3-edge with two and linear light polarizations, with an electric field (Eph) dominant along the [100] and [001] axes, respectively. The XLD spectrum was obtained by subtracting the XAS of Eph//[001] from that of Eph//[100]. b, The schematic shows the experimental geometry for the XAS measurements. c, The temperature-dependent XLDs at the L2-edge were obtained for various La doping ratios at warming temperatures from 250 K to 450 K, separated by 50 K increments. Each spectrum was normalized by the maximum amplitude of the XLD at 250 K.

717 721 725 717 721 725 717 721 725 717 721 725 717 721 725

Nor

mal

ized

XLD

Photon energy (eV)

705 710 715 720 725 730-0.5

0.0

0.5

1.0

XA

S

Photon energy (eV)

Eph

// [100] E

ph // ~[001]

XLD × 2L2

L3

250 K300 K350 K400 K450 K

0% 5% 10% 15% 20%

[100]

[010]

[001]

σ-polarization

π-polarization

75°

c

a b

Figure S11: Calculation of XMLD. a, Dichroisms for light polarizations [100] and [001]. b, Dichroisms for [010] and [001]. The three phases used in the simulation are described in Fig. S9a. c, For each antiferromagnetic MC-type phase and paramagnetic MA-type phase, averaged linear dichroism of XAS[100] – XAS[001] and XAS[010] – XAS[001] was used to consider twin averaging effect within a macroscopic beam spot. The difference (dashed purple line) between the dichroisms shows a typical shape of magnetic dichroism, i.e. a positive left half and a negative right half.

718 720 722 724 726-0.06

-0.04

-0.02

0.00

0.02

0.04

Dic

hroi

sm

Photon energy (eV)

705 710 715 720 725

-0.2

-0.1

0.0

0.1

Line

ar d

ichr

oism

Photon energy (eV)

XAS[100] - XAS[001]

705 710 715 720 725

-0.2

-0.1

0.0

0.1

XAS[010] - XAS[001]

Line

ar d

ichr

oism

Photon energy (eV)

MCAFM

MCPM

MAPM

MCAFM

MCPM

MAPM

L2

L3

L2

L3

b

a

c MCAFMMAPM

MCAFM - MAPM





Figure S12: Control of piezoresponse vectors by the effective in-plane electric field (Eeff) of tip-based electric poling near the triple phase point. Sets of in-plane PFM images for double box poling regions of which the inner boxes were prepared by Eeff pointing to directions of 45° (a), 60° (b), 75° (c), and 90° (d). Each set consists of eight in-plane PFM images measured at different tip orientations as displayed by the cartoons of tip. The value of angle on the top-left corner in each image represents the in-plane piezoresponse direction which is perpendicular to tip orientation and gives positive red-colored contrast. All the images were obtained from a 13-nm-thick Bi0.9La0.1FeO3 sample at ambient conditions and their horizontal axes were parallel to the crystal axis of [100]. The directional angles are defined to be clockwise angles from the positive horizontal direction [100].

a 0˚ 41˚ 88˚ 141˚

176˚ 231˚ 270˚ 315˚

Eeff direction // 45 deg. (for inner box)

0˚ 41˚ 88˚ 141˚

176˚ 231˚ 270˚ 315˚

60 deg.b

0˚ 41˚ 88˚ 141˚

176˚ 231˚ 270˚ 315˚

75 deg.c

0˚ 41˚ 88˚ 141˚

176˚ 231˚ 270˚ 315˚

90 deg.d

+

_

0

Eeff

[100]

[010]

0

0

0

0

0

IP-P

FM si

gnal

0

0

0

0

0

IP-P

FM si

gnal

0

0

0

0

0

IP-P

FM si

gnal

0

0

0

0

0

Tip-sample angle (degrees)

IP-P

FM si

gnal

Eeff // 45°(inner box)

Eeff








Figure S12: Control of piezoresponse vectors by the effective in-plane electric field (Eeff) of tip-based electric poling near the triple phase point. Sets of in-plane PFM images for double box poling regions of which the inner boxes were prepared by Eeff pointing to directions of 45° (a), 60° (b), 75° (c), and 90° (d). Each set consists of eight in-plane PFM images measured at different tip orientations as displayed by the cartoons of tip. The value of angle on the top-left corner in each image represents the in-plane piezoresponse direction which is perpendicular to tip orientation and gives positive red-colored contrast. All the images were obtained from a 13-nm-thick Bi0.9La0.1FeO3 sample at ambient conditions and their horizontal axes were parallel to the crystal axis of [100]. The directional angles are defined to be clockwise angles from the positive horizontal direction [100].

a 0˚ 41˚ 88˚ 141˚

176˚ 231˚ 270˚ 315˚

Eeff direction // 45 deg. (for inner box)

0˚ 41˚ 88˚ 141˚

176˚ 231˚ 270˚ 315˚

60 deg.b

0˚ 41˚ 88˚ 141˚

176˚ 231˚ 270˚ 315˚

75 deg.c

0˚ 41˚ 88˚ 141˚

176˚ 231˚ 270˚ 315˚

90 deg.d

+

_

0

Eeff

[100]

[010]

0

0

0

0

0IP

-PFM

sign

al

0

0

0

0

0

IP-P

FM si

gnal

0

0

0

0

0

IP-P

FM si

gnal

0

0

0

0

0

Tip-sample angle (degrees)

IP-P

FM si

gnal


Eeff








Figure S13: Angle-resolved PFM analyses for the four different poling regions. Every position has multiple in-plane PFM signals acquired at eight different tip orientations. Fitting the data to a sinusoidal curve enables defining the amplitude and direction of the in-plane piezoresponse vector as a function of position. (Left) Tip-orientation-dependent in-plane PFM signal (black symbol) and the corresponding sinusoidal fitting curve (red line) are displayed for the representative 5 by 5 positions marked by the black rectangles on the corresponding vector PFM map. (Right) Maps of in-plane piezoresponse vectors. Color code for the directional representation follows the color circle in the Fig. S14c.

Figure S14: Correlation of the applied Eeff and the resultant in-plane piezoresponse vector at x = 0.1. a, Topographic images, out-of-plane PFM images, and color maps of the angle representing the direction of the local in-plane piezoresponse vector. Four different poling regions were written by applying Eeff along the directional angle 45°, 60°, 75°, and 90° to create inner boxes, respectively. b, The relation between the Eeff and the average of the piezoresponse vectors inside the black line boxes. c, Color code for the directional representation.

a

Pnet

[100]

[010]

b

c

•

0

2

nm

30 45 60 75 90 105Eeff direction (degrees)

0

15

30

45

60

75

90

P ne t

dire

ctio

n(d

egre

es)

Topography Out-of-plane PFMIn-plane PFM angle





1.017 24.3y x

×

Figure S15: Frequency dependence of dielectric properties for pure BFO (left) and La20%-doped BFO (right). a,b, Capacitance versus temperature at selected frequencies. c,d, Loss tangent versus temperature. e,f, Temperature derivative of the capacitance. A solid star is marked on the anomaly originating from water freezing. An empty diamond is marked on the anomaly arising from temperature control instability near the starting point of 400 K for the cooling run. The temperature-independent offset in capacitance became severe as the frequency approached 1 MHz, due to significant BNC cable impedances at high frequencies.

0 100 200 300 40026

28

30

32

34

Cap

acita

nce

(pF)

Temperature (K)0 100 200 300 400

26

28

30

32

34

Cap

acita

nce

(pF)

Temperature (K)

0 100 200 300 400-60

-30

0

30

60 1 MHz

dC/d

T (f

F/K

)

Temperature (K)

1 kHz 10 kHz 100 kHz

0 100 200 300 400

0

20

40

60 1 MHz

dC/d

T (f

F/K

)

Temperature (K)


0 100 200 300 4000.00

0.02

0.04

0.06

0.08

Tan δ

Temperature (K)0 100 200 300 400

0.00

0.02

0.04

0.06

0.08

Tan δ

Temperature (K)

★

◇

★ ◇

a bBiFeO3 Bi0.8La0.2FeO3

c

e

d

f





Figure S13: Angle-resolved PFM analyses for the four different poling regions. Every position has multiple in-plane PFM signals acquired at eight different tip orientations. Fitting the data to a sinusoidal curve enables defining the amplitude and direction of the in-plane piezoresponse vector as a function of position. (Left) Tip-orientation-dependent in-plane PFM signal (black symbol) and the corresponding sinusoidal fitting curve (red line) are displayed for the representative 5 by 5 positions marked by the black rectangles on the corresponding vector PFM map. (Right) Maps of in-plane piezoresponse vectors. Color code for the directional representation follows the color circle in the Fig. S14c.

Figure S14: Correlation of the applied Eeff and the resultant in-plane piezoresponse vector at x = 0.1. a, Topographic images, out-of-plane PFM images, and color maps of the angle representing the direction of the local in-plane piezoresponse vector. Four different poling regions were written by applying Eeff along the directional angle 45°, 60°, 75°, and 90° to create inner boxes, respectively. b, The relation between the Eeff and the average of the piezoresponse vectors inside the black line boxes. c, Color code for the directional representation.

a

Pnet

[100]

[010]

b

c

•

0

2

nm

30 45 60 75 90 105Eeff direction (degrees)

0

15

30

45

60

75

90

P ne t

dire

ctio

n(d

egre

es)

Topography Out-of-plane PFMIn-plane PFM angle





1.017 24.3y x

×

Figure S15: Frequency dependence of dielectric properties for pure BFO (left) and La20%-doped BFO (right). a,b, Capacitance versus temperature at selected frequencies. c,d, Loss tangent versus temperature. e,f, Temperature derivative of the capacitance. A solid star is marked on the anomaly originating from water freezing. An empty diamond is marked on the anomaly arising from temperature control instability near the starting point of 400 K for the cooling run. The temperature-independent offset in capacitance became severe as the frequency approached 1 MHz, due to significant BNC cable impedances at high frequencies.

0 100 200 300 40026

28

30

32

34

Cap

acita

nce

(pF)

Temperature (K)0 100 200 300 400

26

28

30

32

34

Cap

acita

nce

(pF)

Temperature (K)

0 100 200 300 400-60

-30

0

30

60 1 MHz

dC/d

T (f

F/K

)

Temperature (K)


0 100 200 300 400

0

20

40

60 1 MHz

dC/d

T (f

F/K

)

Temperature (K)


0 100 200 300 4000.00

0.02

0.04

0.06

0.08

Tan δ

Temperature (K)0 100 200 300 400

0.00

0.02

0.04

0.06

0.08

Tan δ

Temperature (K)

★

◇

★ ◇

a bBiFeO3 Bi0.8La0.2FeO3

c

e

d

f





Figure S16: Detection of dense stripe nanodomains. Low magnification images of a, topography, b, Out-of-plane(OOP) PFM, and c, In-plane(IP) PFM. The outer and inner boxes were made using a conductive diamond-coated silicon cantilever at a tip voltage of -4.1 V and +3.5 V, respectively. d, Another IP PFM scan on the inner box for a better resolution. e, Fast Fourier transformation of the IP PFM image in d indicating the existence of dense stripes tilted by 12° from [010] with a width of ~18 nm. f, High resolution IP PFM scan on the red dashed box was performed to magnify the nanodomains containing dense stripes. All the images were obtained from a 13-nm-thick Bi0.9La0.1FeO3 sample at ambient conditions

[100]

[010

]

Amplitude Amplitude

Topography OOP PFM IP PFM

500 nm

55 -1

FFT

200 nmAmplitude

a b c

d

e

Amplitude100 nm

f

Figure S17: PFM images for the poled regions in the main manuscript. Topography, OOP PFM, and IP PFM images for the main Fig. 3 (a) and the main Fig. 4 (b). These were taken right after the poling when at least 10 days earlier than the STXM measurements. The left and right PFM images for the huge poling areas (a few 10 m wide) were quickly measured with fast scan speeds (16 or 25.2 um/s) and relatively few scanning lines (39 or 82 nm/line) at excitation voltages (1.8 or 1.5 V) to minimize any PFM measurement effect, respectively. All the scale bars represent 5 m. The magnified poling damage area reveals a significant height change with a rough surface suggesting an irreversible damage. The root-mean-square roughness of the burned area is 0.37 nm three times larger than the other areas.

5 μm

Topography

OOP PFM

IP PFM

Topography

OOP PFM

IP PFM

0

10

nm

-1

1

mV

0

10

nm

-0.5

0.5

mV

-1

1

mV

-0.1

0.1

mV

a b

0

5nm





Figure S16: Detection of dense stripe nanodomains. Low magnification ima

electric-field-induced spin disorder-to-order transition ... · ko, kwang-eun kim, yong-jin kim,...

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