advances.scencemag.org/cg/content/full/2/7/e1600304/dc1 Supplementary Materals for Interface-drven topologcal Hall effect n SrRuO3-SrIrO3 blayer Jobu Matsuno, Naok Ogawa, Kenj Yasuda, Fumtaka Kagawa, Wataru Koshbae, Naoto Nagaosa, Yoshnor Tokura, Masash Kawasak Publshed 8 July 2016, Sc. Adv. 2, e1600304 (2016) DOI: 10.1126/scadv.1600304 Ths PDF fle ncludes: secton SI. Comparson between the Kerr rotaton angle and the magnetzaton secton SII. Detals of the calculaton of the skyrmon stablty n the multlayer secton SIII. Prelmnary magnetc force mcroscopy mages fg. S1. Kerr rotaton, magnetzaton, and topologcal Hall resstvty. fg. S2. Multlayer model and calculated skyrmon radus. fg. S3. Magnetc force mcroscopy mages for m = 5 measured at 50 K and varous magnetc feld.
secton SI. Comparson between the Kerr rotaton angle and the magnetzaton We performed the magneto-optc Kerr effect (MOKE) measurements wth a laser at 690 nm (= 1.80 ev), correspondng to hgher energy nterband transtons; an essental energy scale of spnorbt nteracton responsble for MOKE can be regarded as perturbaton and therefore the Kerr rotaton angle θk s theoretcally justfed to be proportonal to macroscopc magnetzaton M [P. N. Argyres, Phys. Rev. 97, 334 (1955).] Snce AHE s usually defned to be proportonal to magnetzaton M as a functon of magnetc feld at a fxed temperature, the AHE should be always proportonal to θk, rrespectve of the detaled magnetc structure. We also expermentally confrmed ths; n fg. S1A, we plot the Kerr rotaton angle θk measured by MOKE and the magnetzaton M measured by SQUID magnetometry for the m = 5 sample at 80 K, whch s exactly the same condton wth those n Fg. 2C. We can clearly see the good concdence between θk and M. Subsequently, the extracted THE from MOKE agrees well wth that from SQUID as ndcated n fg. S1B. We can hereby justfy usng θk nstead of M also n our case. The reason why we used the Kerr rotaton angle s that magnetzaton measured for thn flms nevtably ncludes a farly large contrbuton of the substrates, makng t dffcult to measure flm magnetzaton n ultrathn lmt of m = 4. Please note that the M-H measurement s much more dffcult than M-T one shown n Fg. 1A. Whle the latter s measured at 0.05 T, we need 3 T or more for the former; ths ndcates 60 tmes larger background sgnals derved from damagnetc substrates, whch made the flm contrbuton hardly dscernble. Therefore, we used Kerr rotaton angle as M for the sake of consstency through the data wth dfferent thckness (m). fg. S1. Kerr rotaton, magnetzaton, and topologcal Hall resstvty. (A) Magnetc feld dependence of Kerr rotaton angle (θk, orange) and magnetzaton (M, green) of m = 5 at 80 K. (B) Topologcal Hall resstvty deduced from MOKE (orange) and SQUID (green), respectvely.
secton SII. Detals of the calculaton of the skyrmon stablty n the multlayer We numercally examned a sngle skyrmon stablty n the model defned by the followng Hamltonan H J m n l l Dˆ ˆ ˆ, n ', ' y n1, n1, x x n1, n1, yˆ h ll' ' l1 n z l, The model s schematcally shown n fg. S2A, demonstratng that the DM nteracton D exsts only on the frst layer (l = 1). The drecton s fxed by the broken nverson symmetry whle we do not know f D s postve or negatve. The normalzed magnetc moment at the ste on the layer l s denoted as nl,. The unt vectors xˆ and ŷ defne the two-dmensonal square lattce on a layer. The calculaton was performed for a fnte sze system where the layer has 300 300 lattces wth perodc boundary condton. The last term represents the Zeeman energy wth external magnetc feld h perpendcular to the multlayer. Fgure S2B shows the layer-dependent radus of the stablzed sngle skyrmon wth a parameter set of J = 1.0, D = 0.3, h = 0.01. The skyrmon radus at layer l among the total m layers, rm(l), s defned by 2 r ( ) m l 1 z n l, 0 One can see that the layer l dependence of rm(l) s not strong; the skyrmon s nearly cylndrcal. Wthn the whole parameter range nvestgated, the skyrmon s always runnng through all the layers once t s stablzed and hence the monopole structure never appears. In such a stuaton, nl, s are almost parallel to each other between the layers,.e., nl, = n1,. Therefore, the orgnal Hamltonan turns nto the effectve one H eff mj n1, n1, ' ( D / m) yˆ n n xˆ xˆ 1, 1, n1, n1, yˆ ' h n z 1, We can mmedately derve Deff = D / m from ths two-dmensonal Hamltonan when one consders the Hamltonan per one layer,.e., Heff / m.
fg. S2. Multlayer model and calculated skyrmon radus. (A) Schematcs of the multlayer model used n the calculaton of the skyrmon stablty. (B) Estmated skyrmon radus at layer l among the total m (m = 6) layers, rm=6(l).
secton SIII. Prelmnary magnetc force mcroscopy (MFM) mages We measured the MFM mages for the m = 5 sample at 50 K at whch sgnfcant THE sgnal s observed. The result s shown wth the correspondng topologcal Hall resstvty and the Kerr rotaton angle n fg. S3. We observed the magnetzaton reversal process between 0.08 T and 0.50 T, correspondng to the θk( M )-H curve. The strkng feature s the sngle bubble doman of ~30 nm, for example observed at 0.20 T. Consderng the small nature of the doman, we speculate that t does not contan any Bloch lnes and hence can be a skyrmon; the observed mages are compatble wth the skyrmon pcture. Nevertheless, we would lke to pont out that ths cannot be a defnte proof of the skyrmon for the followng reasons: 1) Snce the MFM mages are lackng n the nformaton on n-plane magnetzaton, we cannot expermentally tell whether the bubble has a topologcal charge or not. 2) The spatal resoluton (= 20 nm) s nsuffcent to detect sngle skyrmon features wth the expected sze of ~10 nm. Measurements MFM was conducted under hgh vacuum wth a commercally avalable scannng probe mcroscope (attocube AFM/MFM I). We used SSS-QMFMR cantlever from Nanosensors. The measurement was performed at 50 K wth the cantlever exctaton ampltude of 20 nm and the lft heght of ~30 nm. The resonance frequency was 69.795 khz and the Q-factor was 3.3 10 4 under the measurement condton. We observed magnetzaton reversal of the cantlever at ~0.05 T and therefore the magnetzaton drecton was pontng upward durng the measurement ( 0.08 T). In order to extract ntrnsc magnetc contrast, we have subtracted the background data. For the mages measured at 0.14. 0.20, and 0.26 T, the background s the 0.08-T mage, whle we used the 0.50-T mage for 0.26, 0.32, 0.38, and 0.44 T. The reason why we used the two background data s that there s a magnetc-feld dependent offset n the MFM mages. At 0.26 T, we show the two mages from whch we subtracted each background data. We observed that they are n prncple consstent, despte the small dfference n the appearance.
fg. S3. Magnetc force mcroscopy mages for m = 5 measured at 50 K and varous magnetc feld. Magnetc feld was controlled from 2 T to 0.5 T. Raw mage measured at 0.08 T has been subtracted as a background from upper three mages, whle the background s the 0.50-T mage for lower four mages. At 0.20 T, the sze of the bubble doman enclosed by the dashed square s roughly 30 nm. The magnetc-feld dependence of the correspondng topologcal Hall resstvty and the Kerr rotaton angle s also shown.