Supplementary Fig. 1: Schematic of the typical phase shift of the resonance. Supplementary Fig. 2: Rule out the scanning effect at the sample edges.

Transcription

1 Supplemenary Fig. 1: Schemaic of he ypical phase shif of he resonance. The blue solid curve shows he phase signal as a funcion of frequency f a free resonance sae (when f = f 0, = 0). When he sysem is under aracive forces, he resonance will shif o he lower f and, giving a f and, as shown by he blue dashed curve. Supplemenary Fig. 2: Rule ou he scanning effec a he sample edges. The MFM images wih he fas scan direcion perpendicular o he srip (as shown by he black arrow) under 9 T a (a) 300 K and (b) 10 K. The black scale bars in (a) and (b) are 10 μm. The magneic srucures wih he fas scan direcion along he srip (as shown by he black arrow) a (c) 300K and (d) 10K. The black scale bars in (c) and (d) are 4 μm. MFM images acquired a 300 K and 10 K have a scale of 20 and 100 respecively. Scanned areas are 35 μm35 μm in (a) and (b), 20

2 μm4 μm in (c) and (d). The idenical resuls in differen scan direcions could rule ou he scanning effec. Supplemenary Fig. 3: Comparing he EFM and MFM daa under 9T. (a) Zero bias volage EFM daa a differen emperaures followed by he opography acquired by commercial nonmagneic conducive ip (BudgeSensors-ElecriMuli75, Cr/P coaing). (b) MFM daa a differen emperaures followed by he opography acquired by commercial magneic ip (Co/Cr coaing). All he EFM and MFM images have he same scale of 20 and 100 nm lif heigh. Scanned areas are 15 μm 15 μm. The black scale bar is 3 μm. Topography and elecrosaic signals give very small and limied conribuion a he sep edge and hey didn vary a lo when decreasing he emperaure.

3 Supplemenary Fig. 4: Magneic conras inversion. MFM images were aken under (a) 1 T, (b) 1.2 T and (c) 2 T by high coerciviy (around 1.5 T~ 2 T) Co/P probe a 10 K afer he sample and he ip were iniialized under -9 T field. All he images have he same scale of 60 and 100 nm lif heigh. Scanned areas are 21 μm21 μm. The black scale bar is 5 μm. The signals show a clear inversion when he field going hrough he coerciviy of he ip, which prove he edge signals are magneic in naure. Supplemenary Fig. 5: Ferromagneism of he edge phases. (a) The field dependen MFM images of he 3 μm LPCMO srip a 10 K. All he MFM images have a span of 80. Scanned areas are 20 μm20 μm. The black scale bar is 10 μm. The black arrows show he experimen sequence. (b) The normalized MFM signal of he srip as a funcion of magneic field calculaed from (a), which clearly shows hyseresis behavior and non-zero remanence.

4 Supplemenary Fig. 6: Srong size effec of he edge phases. MFM images under 9 T a (a) 300 K, (b) 200 K, (c) 120 K and (d) 10 K. MFM images acquired a 300 K and 200 K are given a scale of 20 while he images a 120 K and 10 K have a scale of 100. Scanned areas are 35 μm35 μm. The black scale bar is 10 μm. (e) The normalized MFM signals (using subsrae signals as baseline) of he circled posiions, labeled as srip edge (red), srip cener (green), film edge (blue) and film cener (black), are ploed as a funcion of emperaure. The sronger MFM signals indicae he enhancemen of ferromagneic edge phases in he narrow srip.

5 Supplemenary Fig. 7: Rule ou arificial edge damaging effec by Time of Fligh Secondary Ion Mass Specromery (TOF-SIMS). (a) The disribuion of La in he LPCMO srip. (b) The disribuion of Pr in he LPCMO srip. Scanned areas are 10 μm10 μm wih resoluion beer han 100 nm. The whie scale bar is 5 μm. The uniform disribuion of La and Pr demonsraes ha here is no chemical change a he srip edges afer he lihography. Supplemenary Fig. 8: Edge phases in LPCMO/SrLaGaO 4 (100). The MFM images of a 80 nm hick LPCMO on SrLaGaO 4 (100) under 9 T a differen emperaures followed by is opography. All he MFM images have he same scale of 20. Scanned areas are 22 μm22 μm. The black scale bar is 10 μm. The sysem also shows clear edge phases.

6 Supplemenary Fig. 9: Edge phases in LPCMO/LaAlO 3 (100). The MFM images of a 60 nm hick LPCMO on LaAlO 3 (100) under 9 T a differen emperaures followed by is opography. All he MFM images have he same scale of 20. Scanned areas are 25 μm25 μm. The black scale bar is 10 μm. The sysem also shows clear edge phases.

7 Supplemenary noe 1:The imaging process and inerpreaions of he MFM images In order o subrac he morphology conribuion from MFM signals, we perform he MFM imaging in he dual pass mode. In he firs pass, he ip scans he opography of samples in apping mode driven a is resonan frequency (f 0 ). Then i lifs up 100 nm and scans again following he same line profile in he second pass o record he phase signal () of is oscillaion, which reflecs he magneic srucures of he sample (Fig. 1e). From Fig. 1a and 1b, we can see clearly ha he morphology can be removed from he MFM image wih a 100 nm lif heigh and proper uning of he feedback loop. Wih 9 T magneic field applied o he sample, very small bu visible magneic conras (less han 1 ) appears in Fig. 2 a 300K due o he paramagneism of he LPCMO and he diamagneism of he SrTiO 3 (100) subsrae. The diamagneic signals of he subsrae are several orders weaker han he ferromagneic signals of he sample and remain nearly unchanged in he whole emperaure range. Therefore, we could use he subsrae signals as he zero base line o analyze he ferromagneic signals of he sample. MFM signals (phase shif Δ or frequency shifδf ) are proporional o he force gradien (F) aced on he ip caused by he ip-sample ineracions, approximaely wrien as : Δ = QF/k, where Q is he qualiy facor of he resonance and k is he spring consan of he canilever 1,2. In his work, he 9 T field was performed perpendicular o he sample surface so ha he momens of he ip and he FM domains will be driven ou of he plane and only he normal componen of he ferromagneic domain signals can be deeced. The aracive force wih negaive force gradien caused by heir ineracions makes he canilever effecively sofer, hereby reducing he resonan frequency of he canilever and generae a negaive phase shif a he resonance frequency f 0 (see Supplemenary Fig. 1). Therefore, we could

8 qualiaively inerpre he MFM images in Fig. 1c as following: The areas wih negaive phase signals are he FM saes; Since ferromagneic domains (micron meer scale) will generae nonuniform sray fields a he lif heigh, he phase signal or force gradien around hem will be non-zero. So he large areas wih zero phase signal are he CE saes or he subsraes; The posiive phase signals come from he opposie magneic flux around he FM domains which gives posiive force gradien. Areas wih posiive phase signals are also he CE saes or he subsraes. Alhough we can obain he absolue magneizaion of he FM domains since MFM signals don reflec he magneizaion direcly, semiquaniive comparisons are sill possible in our case. The ip-sample disance is keeping consan in he second pass so ha any variaion of he phase signals in he sample mus come from he relaive change of magneizaion. Sronger FM domains wih high magneizaion will inroduce higher magneic force gradien along normal direcion, hus producing a bigger phase change. In oher words, he edges indeed have significan sronger ferromagneic signals han he cener as is shown in Fig. 1 and Fig. 2. Supplemenary noe 2: Magneic origin of he edge sae signals MFM signals may include some opography and elecrosaic signals in some cases. However, in magnes, he magneic signals are usually much higher han opography and elecrosaic signals when he emperaure is well below he Curie emperaure. In order o clarify his issue, we also did he EFM measuremen using a nonmagneic conducive ip (BudgeSensors-ElecriMuli75, Cr/P coaing) under he same condiion on he same sample, as shown in Supplemenary Fig. 3. From he EFM images, we can conclude ha opography and elecrosaic signals give very small and limied conribuion wih zero bias volage a he sep edge and hey didn vary a lo when decreasing he emperaure.

9 we also conduced MFM measuremen by using he high coerciviy (1.5 T~2 T) Co/P MFM ips 3 o pick up he magneic conras inversion, hus showing heir magneic origin. The sample and he ip were iniialized under -9 T a 10 K. Then MFM images were acquired a 1 T, 1.2 T and 2 T o pick up he signal inversion. Since he coerciviy of he sample is around 300 Oe, we can ge posiive edge sae signals (repulsive force beween ip and sample) a 1 T and 1.2 T, and hen negaive ones afer going hrough he ip coerciviy a 2 T, as shown in Supplemenary Fig. 4. Therefore, we can confirm ha he opography and elecrosaic signals during he MFM process are neglecable and hese conrass are ruly magneic origin. Supplemenary noe 3: Deails of he double-exchange model According o Refs. 4, 5, he Hamilonian of he wo-orbial double-exchange model reads as: r (.. ) S S i j i, j, AF i j (1) ij H c c hc J The firs erm denoes he sandard double-exchange hopping process for he e g elecrons beween neares-neighbor sies i and j. The operaors c ( c i, ) annihilae (creae) an e g i, elecron a he orbial of he laice sie i. Wihin he sandard infinie Hund coupling approximaion, he spin of he e g elecrons is always parallel o he spin of he localized 2g degrees of freedom S, generaing he Berry phase: ij =cos( i /2)cos( j /2)+sin( i /2)sin( j /2)exp[-i( i - j )], (2) where and are he polar and azimuhal angles of he 2g spins, respecively. The hree neares-neighbor (NN) hopping direcions are denoed by r. Two e g orbials (a: x 2 -y 2 and b: 3z 2 -r 2 ) are involved in he double-exchange process for manganies, wih he hopping ampliudes given by:

10 x y x aa x ba y aa y ba x ab, x bb y 3 3 ab 0, (3) y bb z z aa z ba z ab z 0. bb The hopping 0 will be considered as he uni of energy. This hopping can be roughly esimaed o be ev 4, 5. The second erm of he Hamilonian is he aniferromagneic superexchange ineracion beween he NN 2g spins. The ypical value of he superexchange coupling is in he order of based on a variey of previous invesigaions for bulk manganies. This model Hamilonian does no include he elecron-laice coupling, e.g. he Jahn-Teller disroions. Even hough, his simplified model sill capures he main physics of manganies, and hus can give he phase diagram very close o he experiemnal one. Paricularly, he CE phase can be sablized (only) around half-doping (n~0.5) wih proper J 6 AF, in agreemen wih he experimenal phase diagram, even he Jahn-Teller disorion is no aken in accoun. Thus, i is accepable o sudy many properies of manganies using his simplified model. Magneic ground sae phase diagram of above double-exchange model on hree-dimensional laices is shown in Ref. 7.

11 Supplemenary noe 4: Mone Carlo simulaions and zero-emperaure opimizaion To solve above model Hamilonian, he Mone Carlo simulaion is applied o he classical spin variables S, while exac diagonalizaion is used for he fermionic secor (i.e. for he e g elecrons). The firs Mone Carlo seps are adoped for hermal equilibrium and he following Mone Carlo seps are used for measuremens. Readers can find more deails of he Mone Carlo mehod employed in he presen work in Refs. 4, 5. To simulae he ground sae properies, he Mone Carlo simulaion is performed a a low emperaure, e.g Then a zero-emperaure opimizaion of he classical spin variables S is performed o furher reduce he hermal flucuaion. Supplemenary References: 1. Nishi, R., Houda, I., Aramaa, T., Sugawara, Y. & Moria, S. Phase change deecion of aracive force gradien by using a quarz resonaor in nonconac aomic force microscopy. Appl. Surf. Sci. 157, (2000). 2. Said, R. A. Perurbaion deecion of elecric force gradiens using he phase shif mehod. J. Phys. D: Appl. Phys. 34, L7 L10 (2001). 3. Liou, S. H. & Yao, Y. D. Developmen of high coerciviy magneic force microscopy ips. J. Magn. Magn. Maer. 190, (1998). 4. Dagoo, E. Nanoscale Phase Separaion and Colossal Magneoresisance (Berlin: Springer, 2002). 5. Dagoo, E., Hoa, T. & Moreo, A. Colossal magneoresisan maerials: The key role of phase separaion. Phys. Repors 344, (2001). 6. Hoa, T. Orbial ordering phenomena in d- and f-elecron sysems. Rep. Prog. Phys. 69, 2061 (2006). 7. Dong, S., Zhang, X.T., Yu, R., Liu, J.-M. & Dagoo, E. Microscopic model for he ferroelecric field effec in oxide heerosrucures. Phys. Rev. B 84, (2011).